Prof. Dr. Jelena Klinovaja
ContactDepartment of PhysicsUniversity of Basel Klingelbergstrasse 82 4056 Basel, Switzerland

Short Biography
Jelena Klinovaja received her Bachelor and Master degree in Applied Mathematics and Physics from the Moscow Institute of Physics and Technology (State University), Department of General and Applied Physics, in 2007 and 2009, resp, both with summa cum laude (5.0/5.0). Subsequently, she joined the group of Prof. Daniel Loss at the University of Basel, where she received her PhD in Theoretical Physics in 2012 with summa cum laude. In 2013, she was awarded a threeyear Harvard Fellowship to perform independent research in the area of the theoretical quantum condensed matter physics. Klinovaja was appointed as a tenure track assistant professor at the Department of Physics at the University of Basel in 2014. In February 2019 she was tenured and promoted to associate professor. Since 2020, she is a Deputy CoDirector of the NCCR SPIN (National Centre of Competence in Research: Spin Qubits in Silicon). In her career, she was offered several prestigious fellowships (in 2013, Pappalardo Fellowship from MIT and Yale Prize Postdoctoral Fellowship) and received research prizes such as the Swiss Physical Society Prize 2013 in Condensed Matter Physics (sponsored by IBM), Prize of the Faculty of Natural Sciences, University of Basel, for best PhD work, and Camille und Henry Dreyfus scholarship. In 2017, she was awarded the prestigious Starting Grant of the European Research Council (ERC). Later, in 2022, she was awarded the Consolidator Grant of the European Research Council (ERC).Research Summary
My group is interested in many aspects of the quantum theory of condensed matter systems with a special focus on topological effects and spin phenomena. We explore the physics of topological insulators, carbonbased systems (graphene, bilayer graphene, and carbon nanotubes), atomic chains, semiconducting 2DEGs, and nanowires. In our work, we not only study the properties of existing structures but also combine wellknown ingredients such as nonuniform magnetic fields, superconductivity, and spinorbit interaction to 'engineer' systems with exotic quantum properties, in particular in the presence of strong electronelectron interactions treated by quantum field theoretic methods. Part of our work is related to the physics of exotic bound states such as fractional fermions, Majorana fermions, and parafermions, particles that possess nonAbelian braid statistics and have attracted considerable attention in recent years, also due to their potential use for topological quantum computing.Open Positions
We are looking for outstanding, highly motivated, and enthusiastic graduate students and/or postdoctoral fellows. PhD candidates need to hold a Master's (or equivalent) degree in theoretical condensed matter physics. Postdoc candidates should have a PhD in theoretical condensed matter physics. To apply please submit the following documents per email to Prof. Jelena Klinovaja: a curriculum vitae
 a list of publications
 your academic records (Bachelor's, Master's or PhD diploma)
 a short statement of your research interests and how they relate to the work of our group
 three recommendation letters sent directly to Prof. Jelena Klinovaja
Publications
Show all abstracts.
Enhanced orbital magnetic field effects in Ge hole nanowires
Christoph Adelsberger, Stefano Bosco, Jelena Klinovaja, and Daniel Loss.
arXiv:2207.12050
Hole semiconductor nanowires (NW) are promising platforms to host spin qubits and Majorana bound states for topological qubits because of their strong spinorbit interactions (SOI). The properties of these systems depend strongly on the design of the cross section and on strain, as well as on external electric and magnetic fields. In this work, we analyze in detail the dependence of the SOI and g factors on the orbital magnetic field. We focus on magnetic fields aligned along the axis of the NW, where orbital effects are enhanced and result in a renormalization of the effective g factor up to 400%, even at small values of magnetic field. We provide an exact analytical solution for holes in Ge NWs and we derive an effective lowenergy model that enables us to investigate the effect of electric fields applied perpendicular to the NW. We also discuss in detail the role of strain, growth direction, and high energy valence bands in different architectures, including Ge/Si core/shell NWs, gatedefined onedimensional channels in planar Ge, and curved Ge quantum wells. In core/shell NWs grown along the [110] direction the g factor can be twice larger than for other growth directions which makes this growth direction advantageous for Majorana bound states. Also curved Ge quantum wells feature large effective g factors and SOI, again ideal for hosting Majorana bound states. Strikingly, because these quantities are independent of the electric field, hole spin qubits encoded in curved quantum wells are to good approximation not susceptible to charge noise, significantly boosting their coherence time.

Longdistance coupling of spin qubits via topological magnons
Bence Hetényi, Alexander Mook, Jelena Klinovaja, and Daniel Loss.
arXiv:2207.01264
We consider two distant spin qubits in quantum dots, both coupled to a twodimensional topological ferromagnet hosting chiral magnon edge states at the boundary. The chiral magnon is used to mediate entanglement between the spin qubits, realizing a fundamental building block of scalable quantum computing architectures: a longdistance twoqubit gate. Previous proposals for longdistance coupling with magnons involved offresonant coupling, where the detuning of the spinqubit frequency from the magnonic band edge provides protection against spontaneous relaxation. The topological magnon mode, on the other hand, lies inbetween two magnonic bands far away from any bulk magnon resonances, facilitating strong and highly tuneable coupling between the two spin qubits. Even though the coupling between the qubit and the chiral magnon is resonant for a wide range of qubit splittings, we find that the magnoninduced qubit relaxation is vastly suppressed if the coupling between the qubit and the ferromagnet is antiferromagnetic. A fast and highfidelity longdistance coupling protocol is presented capable of achieving spinqubit entanglement over micrometer distances with 1 MHz gate speed and up to 99.9% fidelities. The resulting spinqubit entanglement may be used as a probe for the longsought detection of topological edge magnons.

Fractional secondorder topological insulator from a threedimensional coupledwires construction
Katharina Laubscher, Pim Keizer, and Jelena Klinovaja
arXiv:2205.12939
We construct a threedimensional secondorder topological insulator with gapless helical hinge states from an array of weakly tunnelcoupled Rashba nanowires. For suitably chosen interwire tunnelings, we demonstrate that the system has a fully gapped bulk as well as fully gapped surfaces, but hosts a Kramers pair of gapless helical hinge states propagating along a path of hinges that is determined by the hierarchy of interwire tunnelings and the boundary termination of the system. Furthermore, the coupledwires approach allows us to incorporate electronelectron interactions into our description. At suitable filling factors of the individual wires, we show that sufficiently strong electronelectron interactions can drive the system into a fractional secondorder topological insulator phase with hinge states carrying only a fraction e/p of the electronic charge e for an odd integer p.

Superconducting diode effect due to magnetochiral anisotropy in topological insulator and Rashba nanowires
Henry F. Legg, Daniel Loss, and Jelena Klinovaja.
arXiv:2205.12939
The critical current of a superconductor can depend on the direction of current flow due to magnetochiral anisotropy when both inversion and timereversal symmetry are broken, an effect known as the superconducting (SC) diode effect. Here, we consider onedimensional (1D) systems in which superconductivity is induced via the proximity effect. In both topological insulator and Rashba nanowires, the SC diode effect due to a magnetic field applied along the spinpolarization axis and perpendicular to the nanowire provides a measure of inversion symmetry breaking in the presence of a superconductor. Furthermore, a strong dependence of the SC diode effect on an additional component of magnetic field applied parallel to the nanowire as well as on the position of the chemical potential can be used to detect that a device is in the region of parameter space where the phase transition to topological superconductivity is expected to arise.

Coupled superconducting spin qubits with spinorbit interaction
Maria Spethmann, XianPeng Zhang, Jelena Klinovaja, and Daniel Loss.
arXiv:2205.03843
Superconducting spin qubits, also known as Andreev spin qubits, promise to combine the benefits of superconducting qubits and spin qubits defined in quantum dots. While most approaches to control these qubits rely on controlling the spin degreeoffreedom via the supercurrent, superconducting spin qubits can also be coupled to each other via the superconductor to implement twoqubit quantum gates. We theoretically investigate the interaction between superconducting spin qubits in the weak tunneling regime and concentrate on the effect of spinorbit interaction (SOI), which can be large in semiconductorbased quantum dots and thereby offers an additional tuning parameter for quantum gates. We find analytically that the effective interaction between two superconducting spin qubits consists of Ising, Heisenberg, and DzyaloshinskiiMoriya interactions and can be tuned by the superconducting phase difference, the tunnel barrier strength, or the SOI parameters. The Josephson current becomes dependent on SOI and spin orientations. We demonstrate that this interaction can be used for fast controlled phaseflip gates with a fidelity >99.99%. We propose a scalable network of superconducting spin qubits which is suitable for implementing the surface code.

Prevalence of trivial zeroenergy subgap states in nonuniform helical spin chains on the surface of superconductors
Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja.
arXiv:2204.02324
Helical spin chains, consisting of magnetic (ad)atoms, on the surface of bulk superconductors are predicted to host Majorana bound states (MBSs) at the ends of the chain. Here, we investigate the prevalence of trivial zeroenergy bound states in these helical spin chain systems. First, we show that the Hamiltonian of a helical spin chain on a superconductor can be mapped to an effective Hamiltonian reminiscent of a semiconductor nanowire with strong Rashba spinorbit coupling. In particular, we show that a varying rotation rate between neighbouring magnetic moments maps to smooth nonuniform potentials in the effective nanowire Hamiltonian. Previously it has been found that trivial zeroenergy states are abundant in nanowire systems with smooth potentials. Therefore, we perform an extensive search for zeroenergy bound states in helical spin chain systems with varying rotation rates. Although bound states with near zeroenergy do exist for certain dimensionalities and rotation profiles, we find that zeroenergy bound states are far less prevalent than in semiconductor nanowire systems with equivalent nonuniformities. In particular, utilising varying rotation rates, we do not find zeroenergy bound states in the most experimentally relevant setup consisting of a onedimensional helical spin chain on the surface of a threedimensional superconductor, even for profiles that produce near zeroenergy states in equivalent one and two dimensional systems. Although our findings do not rule them out, the much reduced prevalence of zeroenergy bound states in long nonuniform helical spin chains compared with equivalent semiconductor nanowires, as well as the ability to measure states locally via STM, should reduce the experimental barrier to identifying MBSs in such systems.

Fully tunable longitudinal spinphoton interactions in Si and Ge quantum dots
Stefano Bosco, Pasquale Scarlino, Jelena Klinovaja, and Daniel Loss.
arXiv:2203.17163
Spin qubits in silicon and germanium quantum dots are promising platforms for quantum computing, but entangling spin qubits over micrometer distances remains a critical challenge. Current prototypical architectures maximize transversal interactions between qubits and microwave resonators, where the spin state is flipped by nearly resonant photons. However, these interactions cause backaction on the qubit, that yield unavoidable residual qubitqubit couplings and significantly affect the gate fidelity. Strikingly, residual couplings vanish when spinphoton interactions are longitudinal and photons couple to the phase of the qubit. We show that large longitudinal interactions emerge naturally in stateoftheart hole spin qubits. These interactions are fully tunable and can be parametrically modulated by external oscillating electric fields. We propose realistic protocols to measure these interactions and to implement fast and highfidelity twoqubit entangling gates. These protocols work also at high temperatures, paving the way towards the implementation of largescale quantum processors.

Topological Hybrids of Magnons and Magnon Bound Pairs
Alexander Mook, Rhea Hoyer, Jelena Klinovaja, and Daniel Loss.
arXiv:2203.12374
We consider quantum condensed matter systems without particlenumber conservation. Since the particle number is not a good quantum number, states belonging to different particlenumber sectors can hybridize, which causes topological anticrossings in the spectrum. The resulting spectral gaps support chiral edge excitations whose wavefunction is a superposition of states in the two hybridized sectors. This situation is realized in fully saturated spinanisotropic quantum magnets without spin conservation, in which single magnons hybridize with magnon bound pairs, i.e., twomagnon bound states. The resulting chiral edge excitations are exotic composites that carry mixed spinmultipolar character, inheriting spindipolar and spinquadrupolar character from their singleparticleness and twoparticleness, respectively. In contrast to established topological magnons, the topological effects discussed here are of genuine quantum mechanical origin and vanish in the classical limit. We discuss implications for both intrinsic anomalous Halltype transport and beyondspintronics computation paradigms. We conclude that fully polarized quantum magnets are a promising platform for topology caused by hybridizations between particlenumber sectors, complementing the field of ultracold atoms working with a conserved number of particles.

RKKY interaction at helical edges of topological superconductors
Katharina Laubscher, Dmitry Miserev, Vardan Kaladzhyan, Daniel Loss, and Jelena Klinovaja.
arXiv:2203.08137
We study spin configurations of magnetic impurities placed close to the edge of a twodimensional topological superconductor both analytically and numerically. First, we demonstrate that the spin of a single magnetic impurity close to the edge of a topological superconductor tends to align along the edge. The strong easyaxis spin anisotropy behind this effect originates from the interaction between the impurity and the gapless helical Majorana edge states. We then compute the RudermanKittelKasuyaYosida (RKKY) interaction between two magnetic impurities placed close to the edge. We show that, in the limit of large interimpurity distances, the RKKY interaction between the two impurities is mainly mediated by the Majorana edge states and leads to a ferromagnetic alignment of both spins along the edge. This effect can be used to detect helical Majorana edge states.

Crossed Andreev reflection in spinpolarized chiral edge states due to Meissner effect
Tamás Haidekker Galambos, Flavio Ronetti, Daniel Loss, and Jelena Klinovaja.
arXiv:2203.05894
We consider a hybrid quantum Hallsuperconductor system, where a superconducting finger with oblique profile is wedged into a twodimensional electron gas in the presence of a perpendicular magnetic field, as considered by Lee et al., Nat. Phys. 13, 693 (2017). The electron gas is in the quantum Hall regime at filling factor ν=1. Due to the Meissner effect, the perpendicular magnetic field close to the quantum Hallsuperconductor boundary is distorted and gives rise to an inplane component of the magnetic field. This component enables nonlocal crossed Andreev reflection between the spinpolarized chiral edge states running on opposite sides of the superconducting finger, thus opening a gap in the spectrum of the edge states without the need of spinorbit interaction or nontrivial magnetic textures. We compute numerically the transport properties of this setup and show that a negative resistance exists as consequence of nonlocal Andreev processes. We also obtain numerically the zeroenergy local density of states, which systematically shows peaks stable to disorder. The latter result is compatible with the emergence of Majorana bound states.

Magnetoelectric Cavity Magnonics in Skyrmion Crystals
Alexander Mook, Tomoki Hirosawa, Jelena Klinovaja, and Daniel Loss
arXiv:2203.03241
We present a theory of magnetoelectric magnonphoton coupling in cavities hosting noncentrosymmetric magnets. Analogously to nonreciprocal phenomena in multiferroics, the magnetoelectric coupling is timereversal and inversion asymmetric. This asymmetry establishes a means for exceptional tunability of magnonphoton coupling, which can be switched on and off by reversing the magnetization direction. Taking the multiferroic skyrmionhost Cu2OSeO3 as an example, we reveal the electrical activity of skyrmion eigenmodes and propose it for magnonphoton splitting of "magnetically dark" elliptic modes. Furthermore, we predict a cavityinduced magnonmagnon coupling between magnetoelectrically active skyrmion excitations. Our study highlights magnetoelectric cavity magnonics as a novel platform for realizing quantumhybrid systems and the quantum control of topological magnetic textures.

Instability of the ferromagnetic quantum critical point in strongly interacting 2D and 3D electron gases with arbitrary spinorbit splitting
Dmitry Miserev, Daniel Loss, and Jelena Klinovaja.
arXiv:2201.10995
In this work we revisit itinerant ferromagnetism in 2D and 3D electron gases with arbitrary spinorbit splitting and strong electronelectron interaction. We identify the resonant scattering processes close to the Fermi surface that are responsible for the instability of the ferromagnetic quantum critical point at low temperatures. In contrast to previous theoretical studies, we show that such processes cannot be fully suppressed even in presence of arbitrary spinorbit splitting. A fully selfconsistent nonperturbative treatment of the electronelectron interaction close to the phase transition shows that these resonant processes always destabilize the ferromagnetic quantum critical point and lead to a firstorder phase transition. Characteristic signatures of these processes can be measured via the nonanalytic dependence of the spin susceptibility on magnetic field both far away or close to the phase transition.

Metallization and proximity superconductivity in topological insulator nanowires
Henry F. Legg, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 105, 155413 (2022)
A heterostructure consisting of a topological insulator (TI) nanowire brought into proximity with a superconducting layer provides a promising route to achieve topological superconductivity and associated Majorana bound states (MBSs). Here, we study effects caused by such a coupling between a thin layer of an swave superconductor and a TI nanowire. We show that there is a distinct phenomenology arising from the metallization of states in the TI nanowire by the superconductor. In the strong coupling limit, required to induce a large superconducting pairing potential, we find that metallization results in a shift of the TI nanowire subbands (∼20 meV) as well as it leads to a small reduction in the size of the subband gap opened by a magnetic field applied parallel to the nanowire axis. Surprisingly, we find that metallization effects in TI nanowires can also be beneficial. Most notably, coupling to the superconductor induces a potential in the portion of the TI nanowire close to the interface with the superconductor, this breaks inversion symmetry and at finite momentum lifts the spin degeneracy of states within a subband. As such coupling to a superconductor can create or enhance the subband splitting that is key to achieving topological superconductivity. This is in stark contrast to semiconductors, where it has been shown that metallization effects always reduce the equivalent subbandsplitting caused by spinorbit coupling. We also find that in certain geometries metallization effects can reduce the critical magnetic required to enter the topological phase. We conclude that, unlike in semiconductors, the metallization effects that occur in TI nanowires can be relatively easily mitigated, for instance by modifying the geometry of the attached superconductor or by compensation of the TI material.

Quasiparticle poisoning in trivial and topological Josephson junctions
Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja.
arXiv:2112.05650
We study theoretically a short singlechannel Josephson junction between superconductors in the trivial and topological phases. The junction is assumed to be biased by a small current and subjected to quasiparticle poisoning. We find that the presence of quasiparticles leads to a voltage signal from the Josephson junction that can be observed both in the trivial and in the topological phase. Quite remarkably, these voltage signatures are sufficiently different in the two phases such that they can serve as means to clearly distinguish between trivial Andreev and topological Majorana bound states in the system. Moreover, these voltage signatures, in the trivial and topological phase, would allow one to measure directly the quasiparticle poisoning rates and to test various approaches for protection against quasiparticle poisoning.

NonMajorana zero energy modes in diluted spin chains proximitized to a superconductor
Felix Küster, Sascha Brinker, Richard Hess, Daniel Loss, Stuart Parkin, Jelena Klinovaja, Samir Lounis, and Paolo Sessi.
arXiv:2112.05708
Spin chains proximitized with superconducting condensates have emerged as one of the most promising platforms for the realization of Majorana modes. The recent use of atomic manipulation techniques raised great expectations for successfully creating and controlling such chains. Here, we craft diluted spin chains atombyatom following seminal theoretical proposal suggesting indirect coupling mechanisms as a viable route to trigger topological superconductivity. We demonstrate that, starting from deep Shiba states, it is possible to cross the quantum phase transition, a necessary condition for the emergence of topological superconductivity, for very short chains. This transition is associated with the emergence of highly localized zero energy end modes. The use of a substrate with highly anisotropic Fermi surface enables to create spin chains characterized by distinct magnetic configurations along various crystallographic directions. By scrutinizing a large set of parameters we reveal the ubiquitous existence of zero energy boundary modes. Although mimicking signatures generally assigned to Majorana modes, the end modes are identified as topologically trivial Shiba states. These results highlight the important role of the local environment, showing that it cannot be completely eliminated also in diluted systems where the effect is expected to be minimized. Our work demonstrates that zero energy modes in spin chains proximitized to supercondcutors are not necessarily a link to Majorana modes while simultaneously identifying new experimental platforms, driving mechanisms, and test protocols for the determination of topologically non trivial superconducting phases.

Hole Spin Qubits in Ge Nanowire Quantum Dots: Interplay of Orbital Magnetic Field, Strain, and Growth Direction
Christoph Adelsberger, Mónica Benito, Stefano Bosco, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 105, 075308 (2022)
Hole spin qubits in quasi onedimensional structures are a promising platform for quantum information processing because of the strong spinorbit interaction (SOI). We present analytical results and discuss device designs that optimize the SOI in Ge semiconductors. We show that at the magnetic field values at which qubits are operated, orbital effects of magnetic fields can strongly affect the response of the spin qubit. We study onedimensional hole systems in Ge under the influence of electric and magnetic fields applied perpendicularly to the device. In our theoretical description, we include these effects exactly. The orbital effects lead to a strong renormalization of the gfactor. We find a sweetspot of the nanowire (NW) gfactor where charge noise is strongly suppressed and present an effective lowenergy model that captures the dependence of the SOI on the electromagnetic fields. Moreover, we compare properties of NWs with square and circular crosssections with ones of gatedefined onedimensional channels in twodimensional Ge heterostructures. Interestingly, the effective model predicts a flat band ground state for finetuned electric and magnetic fields. By considering a quantum dot (QD) harmonically confined by gates, we demonstrate that the NW gfactor sweet spot is retained in the QD. Our calculations reveal that this sweet spot can be designed to coincide with the maximum of the SOI, yielding highly coherent qubits with large Rabi frequencies. We also study the effective gfactor of NWs grown along different high symmetry axes and find that our model derived for isotropic semiconductors is valid for the most relevant growth directions of nonisotropic Ge NWs. Moreover, a NW grown along one of the three main crystallographic axes shows the largest SOI. Our results show that the isotropic approximation is not justified in Ge in all cases.

Fractional spin excitations and conductance in the spiral staircase Heisenberg ladder
Flavio Ronetti, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 105, 134413 (2022)
We investigate theoretically the spiral staircase Heisenberg spin1/2 ladder in the presence of antiferromagnetic longrange spin interactions and a uniform magnetic field. As a special case we also consider the Kondo necklace model. If the magnetizations of the two chains forming the ladder satisfy a certain resonance condition, involving interchain couplings as perturbations, the system is in a partially gapped magnetic phase hosting excitations characterized by fractional spins, whose values can be changed by the magnetic field. We show that these fractional spin excitations can be probed by spin currents in a transport setup with a spin conductance that reveals the fractionalized spin. In some special cases, the spin conductance reaches universal values in units of (gμB)2/h, where g is the gfactor, μB the Bohr magneton, and h the Planck constant. We obtain our results with the help of bosonization and numerical density matrix renormalization group methods.

Zeroenergy Andreev bound states in ironbased superconductor Fe(Te,Se)
Zhe Hou and Jelena Klinovaja.
arXiv:2109.08200
Majorana bound states have been predicted to exist in vortices of topological superconductors (SC). A realization of the FuKane model, based on a threedimensional topological insulator brought into proximity to an swave SC, in ironbased SC Fe(Te,Se) has attracted strong interest after pronounced zeroenergy bias peaks were observed in several experiments. Here, we show that, by taking into account inhomogeneities of the chemical potential or the presence of potential impurities on the surface of Fe(Te,Se), the emergence of these zeroenergy bias peaks can be explained by trivial Andreev bound states (ABSs) whose energies are close to zero. Our numerical simulations reveal that the ABSs behave similarly to Majorana bound states. ABSs are localized only on the, say, top surface and cannot be distinguished from their topological counterparts in transport experiments performed with STM tips. Thus, such ABSs deserve a careful investigation of their own.

Giant magnetochiral anisotropy from quantum confined surface states of topological insulator nanowires
Henry F. Legg, Matthias Rößler, Felix Münning, Dingxun Fan, Oliver Breunig, Andrea Bliesener, Gertjan Lippertz, Anjana Uday, A. A. Taskin, Daniel Loss, Jelena Klinovaja, and Yoichi Ando.
Nature Nanotechnology (2022)
Wireless technology relies on the conversion of alternating electromagnetic fields to direct currents, a process known as rectification. While rectifiers are normally based on semiconductor diodes, quantum mechanical nonreciprocal transport effects that enable highly controllable rectification have recently been discovered. One such effect is magnetochiral anisotropy (MCA), where the resistance of a material or a device depends on both the direction of current flow and an applied magnetic field. However, the size of rectification possible due to MCA is usually extremely small, because MCA relies on electronic inversion symmetry breaking which typically stems from intrinsic spinorbit coupling  a relativistic effect  in a noncentrosymmetric environment. Here, to overcome this limitation, we artificially break inversion symmetry via an applied gate voltage in thin topological insulator (TI) nanowire heterostructures and theoretically predict that such a symmetry breaking can lead to a giant MCA effect. Our prediction is confirmed via experiments on thin bulkinsulating (Bi1−xSbx)2Te3 TI nanowires, in which we observe the largest ever reported size of MCA rectification effect in a normal conductor  over 10000 times greater than in a typical material with a large MCA  and its behaviour is consistent with theory. Our findings present new opportunities for future technological applications of topological devices.

Lasercontrolled real and reciprocalspace topology in multiferroic insulators
Tomoki Hirosawa, Jelena Klinovaja, Daniel Loss, and Sebastian A.Diaz.
Phys. Rev. Lett. 128, 037201 (2022)
Magnetic materials in which it is possible to control the topology of their magnetic order in real space or the topology of their magnetic excitations in reciprocal space are highly soughtafter as platforms for alternative data storage and computing architectures. Here we show that multiferroic insulators, owing to their magnetoelectric coupling, offer a natural and advantageous way to address these two different topologies using laser fields. We demonstrate that via a delicate balance between the energy injection from a highfrequency laser and dissipation, single skyrmionsarchetypical topological magnetic texturescan be set into motion with a velocity and propagation direction that can be tuned by the laser field amplitude and polarization, respectively. Moreover, we uncover an ultrafast Floquet magnonic topological phase transition in a laserdriven skyrmion crystal and we propose a new diagnostic tool to reveal it using the magnonic thermal Hall conductivity.

Helical Liquids in Semiconductors
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Semicond. Sci. Technol. 36, 123003 (2021)
Onedimensional helical liquids can appear at boundaries of certain condensed matter systems. Two prime examples are the edge of a quantum spin Hall insulator, also known as a twodimensional topological insulator, and the hinge of a threedimensional secondorder topological insulator. For these materials, the presence of a helical state at the boundary serves as a signature of their nontrivial bulk topology. Additionally, these boundary states are of interest themselves, as a novel class of strongly correlated lowdimensional systems with interesting potential applications. Here, we review existing results on such helical liquids in semiconductors. Our focus is on the theory, though we confront it with existing experiments. We discuss various aspects of the helical states, such as their realization, topological protection and stability, or possible experimental characterization. We lay emphasis on the hallmark of these states, being the prediction of a quantized electrical conductance. Since so far reaching a wellquantized conductance remained challenging experimentally, a large part of the review is a discussion of various backscattering mechanisms which have been invoked to explain this discrepancy. Finally, we include topics related to proximityinduced topological superconductivity in helical states, as an exciting application towards topological quantum computation with the resulting Majorana bound states.

Universality of Abelian and nonAbelian Wannier functions in one dimension
Kiryl Piasotski, Mikhail Pletyukhov, Clara S. Weber, Jelena Klinovaja, Dante M. Kennes, and Herbert Schoeller.
Phys. Rev. Research 3, 033167 (2021)
Within a Dirac model in 1+1 dimensions, a prototypical model to describe lowenergy physics for a wide class of lattice models, we propose a fieldtheoretical version for the representation of Wannier functions, the ZakBerry connection, and the geometric tensor. In two natural Abelian gauges we present universal scaling of the Dirac Wannier functions in terms of four fundamental scaling functions that depend only on the phase γ of the gap parameter and the charge correlation length ξ in an insulator. The two gauges allow for a universal lowenergy formulation of the surface charge and surface fluctuation theorem, relating the boundary charge and its fluctuations to bulk properties. Our analysis describes the universal aspects of Wannier functions for the wide class of onedimensional generalized AubryAndréHarper lattice models. In the lowenergy regime of small gaps we demonstrate universal scaling of all lattice Wannier functions and their moments in the corresponding Abelian gauges. We also discuss nonAbelian lattice gauges and find that lattice Wannier functions of maximal localization show universal scaling and are uniquely related to the Dirac Wannier function of the lower band. Our results present evidence that universal aspects of Wannier functions and of the boundary charge are uniquely related and can be elegantly described within universal lowenergy theories.

Local and nonlocal quantum transport due to Andreev bound states in finite Rashba nanowires with superconducting and normal sections
Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 104, 075405 (2021)
We analyze Andreev bound states (ABSs) that form in normal sections of a Rashba nanowire that is only partially covered by a superconducting layer. These ABSs are localized close to the ends of the superconducting section and can be pinned to zero energy over a wide range of magnetic field strengths even if the nanowire is in the nontopological regime. For finitesize nanowires (typically ≲1 μm in current experiments), the ABS localization length is comparable to the length of the nanowire. The probability density of an ABS is therefore nonzero throughout the nanowire and differentialconductance calculations reveal a correlated zerobias peak (ZBP) at both ends of the nanowire. When a second normal section hosts an additional ABS at the opposite end of the superconducting section, the combination of the two ABSs can mimic the closing and reopening of the bulk gap in local and nonlocal conductances accompanied by the appearance of the ZBP. These signatures are reminiscent of those expected for Majorana bound states (MBSs) but occur here in the nontopological regime. Our results demonstrate that conductance measurements of correlated ZBPs at the ends of a typical superconducting nanowire or an apparent closing and reopening of the bulk gap in the local and nonlocal conductance are not conclusive indicators for the presence of MBSs.

Majorana bound states in semiconducting nanostructures
Katharina Laubscher and Jelena Klinovaja.
Journal of Applied Physics 130, 081101 (2021)
In this Tutorial, we give a pedagogical introduction to Majorana bound states (MBSs) arising in semiconducting nanostructures. We start by briefly reviewing the wellknown Kitaev chain toy model in order to introduce some of the basic properties of MBSs before proceeding to describe more experimentally relevant platforms. Here, our focus lies on simple `minimal' models where the Majorana wave functions can be obtained explicitly by standard methods. In a first part, we review the paradigmatic model of a Rashba nanowire with strong spinorbit interaction (SOI) placed in a magnetic field and proximitized by a conventional swave superconductor. We identify the topological phase transition separating the trivial phase from the topological phase and demonstrate how the explicit Majorana wave functions can be obtained in the limit of strong SOI. In a second part, we discuss MBSs engineered from proximitized edge states of twodimensional (2D) topological insulators. We introduce the JackiwRebbi mechanism leading to the emergence of bound states at mass domain walls and show how this mechanism can be exploited to construct MBSs. Due to their recent interest, we also include a discussion of Majorana corner states in 2D secondorder topological superconductors. This Tutorial is mainly aimed at graduate students  both theorists and experimentalists  seeking to familiarize themselves with some of the basic concepts in the field.

Tuning interactions between spins in a superconductor
Hao Ding, Yuwen Hu, Mallika T. Randeria, Silas Hoffman, Oindrila Deb, Jelena Klinovaja Daniel Loss, and Ali Yazdani.
Proc. Natl. Acad. Sci. USA 118, 14 (2021)
We consider a threedimensional topological insulator (TI) wire with a nonuniform chemical potential induced by gating across the crosssection. This inhomogeneity in chemical potential lifts the degeneracy between two onedimensional surface state subbands. A magnetic field applied along the wire, due to orbital effects, breaks timereversal symmetry and lifts the Kramers degeneracy at zeromomentum. If placed in proximity to an swave superconductor, the system can be brought into a topological phase at relatively weak magnetic fields. Majorana bound states (MBSs), localized at the ends of the TI wire, emerge and are present for an exceptionally large region of parameter space in realistic systems. Unlike in previous proposals, these MBSs occur without the requirement of a vortex in the superconducting pairing potential, which represents a significant simplification for experiments. Our results open a pathway to the realisation of MBSs in present day TI wire devices.

Majorana bound states in topological insulators without a vortex
Henry F. Legg, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 104, 165405 (2021)
We consider a threedimensional topological insulator (TI) wire with a nonuniform chemical potential induced by gating across the crosssection. This inhomogeneity in chemical potential lifts the degeneracy between two onedimensional surface state subbands. A magnetic field applied along the wire, due to orbital effects, breaks timereversal symmetry and lifts the Kramers degeneracy at zeromomentum. If placed in proximity to an swave superconductor, the system can be brought into a topological phase at relatively weak magnetic fields. Majorana bound states (MBSs), localized at the ends of the TI wire, emerge and are present for an exceptionally large region of parameter space in realistic systems. Unlike in previous proposals, these MBSs occur without the requirement of a vortex in the superconducting pairing potential, which represents a significant simplification for experiments. Our results open a pathway to the realisation of MBSs in present day TI wire devices.

Majorana Bound States Induced by Antiferromagnetic Skyrmion Textures
Sebastian A. Diaz, Daniel Loss, Jelena Klinovaja, and Silas Hoffman.
Phys. Rev. B 104, 214501 (2021)
Majorana bound states are zeroenergy states predicted to emerge in topological superconductors and intense efforts seeking a definitive proof of their observation are still ongoing. A standard route to realize them involves antagonistic orders: a superconductor in proximity to a ferromagnet. Here we show this issue can be resolved using antiferromagnetic rather than ferromagnetic order. We propose to use a chain of antiferromagnetic skyrmions, in an otherwise collinear antiferromagnet, coupled to a bulk conventional superconductor as a novel platform capable of supporting Majorana bound states that are robust against disorder. Crucially, the collinear antiferromagnetic region neither suppresses superconductivity nor induces topological superconductivity, thus allowing for Majorana bound states localized at the ends of the chain. Our model introduces a new class of systems where topological superconductivity can be induced by editing antiferromagnetic textures rather than locally tuning material parameters, opening avenues for the conclusive observation of Majorana bound states.

Fractional boundary charges with quantized slopes in interacting one and twodimensional systems
Katharina Laubscher, Clara S. Weber, Dante M. Kennes, Mikhail Pletyukhov, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 104, 035432 (2021)
We study fractional boundary charges (FBCs) for two classes of strongly interacting systems. First, we study strongly interacting nanowires subjected to a periodic potential with a period that is a rational fraction of the Fermi wavelength. For sufficiently strong interactions, the periodic potential leads to the opening of a charge density wave gap at the Fermi level. The FBC then depends linearly on the phase offset of the potential with a quantized slope determined by the period. Furthermore, different possible values for the FBC at a fixed phase offset label different degenerate ground states of the system that cannot be connected adiabatically. Next, we turn to the fractional quantum Hall effect (FQHE) at odd filling factors ν=1/(2l+1), where l is an integer. For a Corbino disk threaded by an external flux, we find that the FBC depends linearly on the flux with a quantized slope that is determined by the filling factor. Again, the FBC has 2l+1 different branches that cannot be connected adiabatically, reflecting the (2l+1)fold degeneracy of the ground state. These results allow for several promising and strikingly simple ways to probe strongly interacting phases via boundary charge measurements.

YuShibaRusinov States and Ordering of Magnetic Impurities Near the Boundary of a Superconducting Nanowire
Oindrila Deb, Silas Hoffman, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 103, 165403 (2021)
We theoretically study the spectrum induced by one and two magnetic impurities near the boundary of a onedimensional nanowire in proximity to a conventional swave superconductor and extract the ground state magnetic configuration. We show that the energies of the subgap states, supported by the magnetic impurities, are strongly affected by the boundary for distances less than the superconducting coherence length. In particular, when the impurity is moved towards the boundary, multiple quantum phase transitions periodically occur in which the parity of the superconducting condensate oscillates between even and odd. We find that the magnetic ground state configuration of two magnetic impurities depends not only on the distance between them but also explicitly on their distance away from the boundary of the nanowire. As a consequence, the magnetic ground state can switch from ferromagnetic to antiferromagnetic while keeping the interimpurity distance unaltered by simultaneously moving both impurities away from the boundary. The ground state magnetic configuration of two impurities is found analytically in the weak coupling regime and exactly for an arbitrary impurity coupling strength using numerical tightbinding simulations.

Insulating regime of an underdamped currentbiased Josephson junction supporting Z3 and Z4 parafermions
Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 103, 180505 (2021)
We study analytically a currentbiased topological Josephson junction supporting Zn parafermions. First, we show that in an infinitesize system a pair of parafermions on the junction can be in n different states; the 2\pi n periodicity of the phase potential of the junction results in a significant suppression of the maximal current Im for an insulating regime of the underdamped junction. Second, we study the behaviour of a realistic finitesize system with avoided level crossings characterized by splitting \delta. We consider two limiting cases: when the phase evolution may be considered adiabatic, which results in decreased periodicity of the effective potential, and the opposite case, when LandauZener transitions restore the 2 \pi n periodicity of the phase potential. The resulting current Im is exponentially different in the opposite limits, which allows us to propose a new detection method to establish the appearance of parafermions in the system experimentally, based on measuring Im at different values of the splitting \delta.

Clock model and parafermions in Rashba nanowires
Flavio Ronetti, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 103, 235410 (2021)
We consider a semiconducting nanowire with Rashba spinorbit interaction subjected to a magnetic field and in the presence of strong electronelectron interactions. When the ratio between Fermi and Rashba momenta is tuned to 1/2, two competing resonant multiparticle scattering processes are present simultaneously and the interplay between them brings the system into a gapless critical parafermion phase. This critical phase is described by a selfdual sineGordon model, which we are able to map explicitly onto the lowenergy sector of the ℤ4 parafermion clock chain model. Finally, we show that by alternating regions in which only one of these two processes is present one can generate localized zeroenergy parafermion bound states.

Interactionstabilized topological magnon insulator in ferromagnets
Alexander Mook, Kirill Plekhanov, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. X 11, 021061 (2021)
Condensed matter systems admit topological collective excitations above a trivial ground state, an example being Chern insulators formed by Dirac bosons with a gap at finite energies. However, in contrast to electrons, there is no particlenumber conservation law for collective excitations. This gives rise to particle numbernonconserving manybody interactions whose influence on singleparticle topology is an open issue of fundamental interest in the field of topological quantum materials. Taking magnons in ferromagnets as an example, we uncover topological magnon insulators that are stabilized by interactions through opening Cherninsulating gaps in the magnon spectrum. This can be traced back to the fact that the particlenumber nonconserving interactions break the effective timereversal symmetry of the harmonic theory. Hence, magnonmagnon interactions are a source of topology that can introduce chiral edge states, whose chirality depends on the magnetization direction. Importantly, interactions do not necessarily cause detrimental damping but can give rise to topological magnons with exceptionally long lifetimes. We identify two mechanisms of interactioninduced topological phase transitionsone driven by an external field, the other by temperatureand show that they cause unconventional sign reversals of transverse transport signals, in particular of the thermal Hall conductivity. We identify candidate materials where this manybody mechanism is expected to occur, such as the metalorganic kagomelattice magnet Cu(1,3benzenedicarboxylate), the van der Waals honeycomblattice magnet CrI3, and the multiferroic kamiokite (Fe2Mo3O8). Our results demonstrate that interactions can play an important role in generating nontrivial topology.

Chiral Hinge Magnons in SecondOrder Topological Magnon Insulators
Alexander Mook, Sebastian A.Diaz, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 104, 024406 (2021)
When interacting spins in condensed matter order ferromagnetically, their ground state wave function is topologically trivial. Nonetheless, in two dimensions, the ferromagnetic state can support spin excitations with nontrivial topology, an exotic state known as topological magnon insulator (TMI). Here, we theoretically unveil and numerically confirm a novel ferromagnetic state in three dimensions dubbed secondorder TMI, whose hallmarks are excitations at its hinges, where facets intersect. Since ferromagnetism naturally comes with broken timereversal symmetry, the hinge magnons are chiral, rendering backscattering impossible. Hence, they trace out a threedimensional path about the sample unimpeded by defects and are topologically protected by the spectral gap. They are remarkably robust against disorder and simultaneously highly tunable by atomiclevel engineering of the sample termination. Our findings empower magnonics with the tools of higher order topology, a promising route to combine lowenergy information transfer free of Joule heating with threedimensional vertical integration.

Universality of Boundary Charge Fluctuations
Clara S. Weber, Kiryl Piasotski, Mikhail Pletyukhov, Jelena Klinovaja, Daniel Loss, Herbert Schoeller, and Dante M. Kennes.
Phys. Rev. Lett. 126, 016803 (2021)
We establish the quantum fluctuations $\Delta Q_B^2$ of the charge $Q_B$ accumulated at the boundary of an insulator as an integral tool to characterize phase transitions where a direct gap closes (and reopens), typically occurring for insulators with topological properties. The power of this characterization lies in its capability to treat different kinds of insulators on equal footing; being applicable to transitions between topological and nontopological band, Anderson, and Mott insulators alike. In the vicinity of the phase transition we find a universal scaling $\Delta Q_B^2 (E_g)$ as function of the gap size E_g and determine its generic form in various dimensions. For prototypical phase transitions with a massive Diraclike bulk spectrum we demonstrate a scaling with the inverse gap in one dimension and a logarithmic one in two dimensions.

Magnetic phase transitions in twodimensional twovalley semiconductors with inplane magnetic field
Dmitry Miserev, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 103, 024401 (2021)
A twodimensional electron gas (2DEG) in twovalley semiconductors has two discrete degrees of freedom given by the spin and valley quantum numbers. We analyze the zerotemperature magnetic instabilities of twovalley semiconductors with SOI, inplane magnetic field, and electronelectron interaction. The interplay of an applied inplane magnetic field and the SOI results in noncollinear spin quantization in different valleys. Together with the exchange intervalley interaction this results in a rich phase diagram containing four nontrivial magnetic phases. The negative nonanalytic cubic correction to the free energy, which is always present in an interacting 2DEG, is responsible for first order phase transitions. Here, we show that nonzero ground state values of the order parameters can cut this cubic nonanalyticity and drive certain magnetic phase transitions second order. We also find two tricritical points at zero temperature which together with the line of second order phase transitions constitute the quantum critical sector of the phase diagram. The phase transitions can be tuned externally by electrostatic gates or by the inplane magnetic field.

Quadrupole spin polarization as signature of secondorder topological superconductors
Kirill Plekhanov, Niclas Müller, Yanick Volpez, Dante M. Kennes, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 103, L041401 (2021)
We study theoretically secondorder topological superconductors characterized by the presence of pairs of zeroenergy Majorana corner states. We uncover a quadrupole spin polarization at the system edges that provides a striking signature to identify topological phases, thereby complementing standard approaches based on zerobias conductance peaks due to Majorana corner states. We consider two different classes of secondorder topological superconductors with broken timereversal symmetry and show that both classes are characterized by a quadrupolar structure of the spin polarization that disappears as the system passes through the topological phase transition. This feature can be accessed experimentally using spinpolarized scanning tunneling microscopes. We study different models hosting secondorder topological phases, both analytically and numerically, and using Keldysh techniques we provide numerical simulations of the spinpolarized currents probed by scanning tips.

Pinning of Andreev bound states to zero energy in twodimensional superconductorsemiconductor Rashba heterostructures
Olesia Dmytruk, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 102, 245431 (2020)
We consider a twodimensional electron gas with Rashba spinorbit interaction (SOI) partially covered by an swave superconductor, where the uncovered region remains normal but is exposed to an effective Zeeman field applied perpendicular to the plane. We find analytically and numerically Andreev bound states (ABSs) formed in the normal region and show that, due to SOI and by tuning the parameters of the system deeply into the topologically trivial phase, one can reach a regime where the energy of the lowest ABS becomes pinned close to zero as a function of Zeeman field. The energy of such an ABS is shown to decay as an inverse powerlaw in Zeeman field. We also consider a superconductorsemiconductor heterostructure with a superconducting vortex at the center and in the presence of strong SOI, and find again ABSs that can get pinned close to zero energy in the nontopological phase.

Kramers pairs of Majorana corner states in a topological insulator bilayer
Katharina Laubscher, Danial Chughtai, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 102, 195401 (2020)
We consider a system consisting of two tunnelcoupled twodimensional topological insulators proximitized by a top and bottom superconductor with a phase difference of π between them. We show that this system exhibits a timereversal invariant secondorder topological superconducting phase characterized by the presence of a Kramers pair of Majorana corner states at all four corners of a rectangular sample. We furthermore investigate the effect of a weak timereversal symmetry breaking perturbation and show that an inplane Zeeman field leads to an even richer phase diagram exhibiting two nonequivalent phases with two Majorana corner states per corner as well as an intermediate phase with only one Majorana corner state per corner. We derive our results analytically from continuum models describing our system. In addition, we also provide independent numerical confirmation of the resulting phases using discretized lattice representations of the models, which allows us to demonstrate the robustness of the topological phases and the Majorana corner states against parameter variations and potential disorder.

Fermi Surface Resonance and Quantum Criticality in Strongly Interacting Fermi Gases
Dmitry Miserev, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 103, 075104 (2021)
Fermions in the Fermi gas obey the Pauli exclusion principle restricting any two fermions from filling the same quantum state. Strong interaction between fermions can completely change the properties of the Fermi gas. In our theoretical study we find a new exotic quantum phase in strongly interacting Fermi gases constrained to a certain condition imposed on the Fermi surfaces which we call the Fermi surface resonance. The new phase is quantum critical which can be identified by the powerlaw frequency tail of the spectral density and divergent static susceptibilities. An especially striking feature of the new phase is the anomalous powerlaw temperature dependence of the dc resistivity that is similar to strange metals. The new quantum critical phase can be experimentally found in ordinary semiconductor heterostructures.

Critical current for an insulating regime of an underdamped currentbiased topological Josephson junction
Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 033448 (2020)
We study analytically an underdamped currentbiased topological Josephson junction. First, we consider a simplified model at zero temperature, where the parity of the nonlocal fermionic state formed by Majorana bound states (MBSs) localized on the junction is fixed, and show that a transition from insulating to conducting state in this case is governed by singlequasiparticle tunneling rather than by Cooper pair tunneling in contrast to a nontopological Josephson junction. This results in a significantly lower critical current for the transition from insulating to conducting state. We propose that, if the length of the system is finite, the transition from insulating to conducting state occurs at exponentially higher bias current due to hybridization of the states with different parities as a result of the overlap of MBSs localized on the junction and at the edges of the topological nanowire forming the junction. Finally, we discuss how the appearance of MBSs can be established experimentally by measuring the critical current for an insulating regime at different values of the applied magnetic field.

Majorana zero modes and their bosonization
Victor Chua, Katharina Laubscher, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 102, 155416 (2020)
The simplest continuum model of a onedimensional noninteracting superconducting fermionic symmetryprotected topological (SPT) phase is analyzed in great detail using analytic methods. A full exact diagonalization of the meanfield Bogoliubovde Gennes Hamiltonian is carried out with open boundaries and finite lengths. Majorana zero modes are derived and studied in great detail. Thereafter exact operator bosonization in both open and closed geometries is carried out. The complementary viewpoints provided by fermionic and bosonic formulations of the superconducting SPT phase are then reconciled. In particular, we provide a complete and exact account of how the topological Majorana zero modes manifest in a bosonized description of an SPT phase.

Magnonic Quadrupole Topological Insulator in Antiskyrmion Crystals
Tomoki Hirosawa, Sebastian A. Diaz, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Lett. 125, 207204 (2020)
When the crystalline symmetries that protect a higherorder topological phase are not preserved at the boundaries of the sample, gapless hinge modes or ingap corner states cannot be stabilized. Therefore, careful engineering of the sample termination is required. Similarly, magnetic textures, whose quantum fluctuations determine the supported magnonic excitations, tend to relax to new configurations that may also break crystalline symmetries when boundaries are introduced. Here we uncover that antiskyrmion crystals provide an experimentally accessible platform to realize a magnonic topological quadrupole insulator, whose hallmark signature are robust magnonic corner states. Furthermore, we show that tuning an applied magnetic field can trigger the selfassembly of antiskyrmions carrying a fractional topological charge along the sample edges. Crucially, these fractional antiskyrmions restore the symmetries needed to enforce the emergence of the magnonic corner states. Using the machinery of nested Wilson loops, adapted to magnonic systems supported by noncollinear magnetic textures, we demonstrate the quantization of the bulk quadrupole moment, edge dipole moments, and corner charges.

Majorana bound states in topological insulators with hidden Dirac points
Ferdinand Schulz, Kirill Plekhanov, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 033215 (2020)
We address the issue whether it is possible to generate Majorana bound states at the magneticsuperconducting interface in twodimensional topological insulators with hidden Dirac points in the spectrum. In this case, the Dirac point of edge states is located at the energies of the bulk states such that two types of states are strongly hybridized. Here, we show that welldefined Majorana bound states can be obtained even in materials with hidden Dirac point provided that the width of the magnetic strip is chosen to be comparable with the localization length of the edge states. The obtained topological phase diagram allows one to extract precisely the position of the Dirac point in the spectrum. In addition to standard zerobias peak features caused by Majorana bound states in transport experiments, we propose to supplement future experiments with measurements of charge and spin polarization. In particular, we demonstrate that both observables flip their signs at the topological phase transition, thus, providing an independent signature of the presence of topological superconductivity. All features remain stable against substantially strong disorder.

Superconducting Quantum Interference in Edge State Josephson Junctions
Tamás Haidekker Galambos, Silas Hoffman, Patrik Recher, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Lett. 125, 157701 (2020)
We study superconducting quantum interference in a Josephson junction linked via edge states in twodimensional (2D) insulators. We consider two scenarios in which the 2D insulator is either a topological or a trivial insulator supporting onedimensional (1D) helical or nonhelical edge states, respectively. In equilibrium, we find that the qualitative dependence of critical supercurrent on the flux through the junction is insensitive to the helical nature of the mediating states and can, therefore, not be used to verify the topological features of the underlying insulator. However, upon applying a finite voltage bias smaller than the superconducting gap to a relatively long junction, the finitefrequency interference pattern in the nonequilibrium transport current is qualitatively different for helical edge states as compared to nonhelical ones.

Rational boundary charge in onedimensional systems with interaction and disorder
Mikhail Pletyukhov, Dante M. Kennes, Kiryl Piasotski, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller.
Phys. Rev. Research 2, 033345 (2020)
We study the boundary charge $Q_B$ of generic semiinfinite onedimensional insulators with translational invariance and show that nonlocal symmetries (i.e., including translations) lead to rational quantizations $p/q$ of $Q_B$. In particular, we find that (up to an unknown integer) the quantization of $Q_B$ is given in integer units of $ρ/2$ and $(ρ−1)/2$, where $ρ$ is the average charge per site (which is a rational number for an insulator). This is a direct generalization of the known halfinteger quantization of $Q_B$ for systems with local inversion or local chiral symmetries to any rational value. Quite remarkably, this rational quantization remains valid even in the presence of shortranged electronelectron interactions as well as static random disorder (breaking translational invariance). This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions. We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models. Furthermore, for the special case of halffilling $ρ=1/2$, we present explicit results in singlechannel and nearestneighbor hopping models and identify Weyl semimetal physics at gap closing points. Our general framework also allows us to shed new light on the wellknown rational quantization of soliton charges at domain walls

Quantum Damping of Skyrmion Crystal Eigenmodes due to Spontaneous Quasiparticle Decay
Alexander Mook, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Research 2, 033491 (2020)
The elementary excitations of skyrmion crystals experience both emergent magnetic fields and anharmonic interactions brought about by the topologically nontrivial noncollinear texture. The resulting flat bands cause strong spontaneous quasiparticle decay, dressing the eigenmodes of skyrmion crystals with a finite zerotemperature quantum lifetime. Sweeping the flat bands through the spectrum by changing the magnetic field leads to an externally controllable energyselective magnon breakdown. In particular, we uncover that the three fundamental modes, i.e., the anticlockwise, breathing, and clockwise mode, exhibit distinct decay behavior, with the clockwise (anticlockwise) mode being the least (most) stable mode out of the three.

Transport signatures of topological phases in double nanowires probed by spinpolarized STM
Manisha Thakurathi, Denis Chevallier, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 023197 (2020)
We study a doublenanowire setup proximity coupled to an swave superconductor and search for the bulk signatures of the topological phase transition that can be observed experimentally, for example, with an STM tip. Three bulk quantities, namely, the charge, the spin polarization, and the pairing amplitude of intrawire superconductivity are studied in this work. The spin polarization and the pairing amplitude flip sign as the system undergoes a phase transition from the trivial to the topological phase. In order to identify promising ways to observe bulk signatures of the phase transition in transport experiments, we compute the spin current flowing between a local spinpolarized probe, such as an STM tip, and the doublenanowire system in the Keldysh formalism. We find that the spin current contains information about the sign flip of the bulk spin polarization and can be used to determine the topological phase transition point.

Magnetic field independent subgap states in hybrid Rashba nanowires
C. Jünger, R. Delagrange, D. Chevallier, S. Lehmann, K.A. Dick, C. Thelander, J. Klinovaja, D. Loss. A. Baumgartner, and C. Schönenberger.
Phys. Rev. Lett. 125, 017701 (2020)
Subgap states in semiconductingsuperconducting nanowire hybrid devices are controversially discussed as potential topologically nontrivial quantum states. One source of ambiguity is the lack of an energetically and spatially well defined tunnel spectrometer. Here, we use quantum dots directly integrated into the nanowire during the growth process to perform tunnel spectroscopy of discrete subgap states in a long nanowire segment. In addition to subgap states with a standard magnetic field dependence, we find topologically trivial subgap states that are independent of the external magnetic field, i.e. that are pinned to a constant energy as a function of field. We explain this effect qualitatively and quantitatively by taking into account the strong spinorbit interaction in the nanowire, which can lead to a decoupling of Andreev bound states from the field due to a spatial spin texture of the confined eigenstates.

Universal conductance dips and fractional excitations in a twosubband quantum wire
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Research 2, 043208 (2020)
We theoretically investigate a quantum wire based on a quasionedimensional Kondo lattice formed by localized spins and itinerant electrons, where the lowest two subbands of the quantum wire are populated. We uncover a backscattering mechanism involving helically ordered spins and Coulomb interaction between the electrons. The combination of these ingredients results in scattering resonances and partial gaps which give rise to nonstandard plateaus and conductance dips at certain electron densities. The positions and values of these dips are independent of material parameters, serving as direct transport signatures of this mechanism. While our theory describes a generic Kondo lattice, an experimentally relevant realization is provided by quantum wires made out of IIIV semiconductors hosting nuclear spins such as InAs. Observation of the universal conductance dips would not only confirm the presence of a nuclear spin helix but also identify a strongly correlated fermion system hosting fractional excitations, resembling the fractional quantum Hall states even without external magnetic fields.

Majorana and parafermion corner states from two coupled sheets of bilayer graphene
Katharina Laubscher, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 013330 (2020)
We consider a setup consisting of two coupled sheets of bilayer graphene in the regime of strong spinorbit interaction, where electrostatic confinement is used to create an array of effective quantum wires. We show that for suitable interwire couplings the system supports a topological insulator phase exhibiting Kramers partners of gapless helical edge states, while the additional presence of a small inplane magnetic field and weak proximityinduced superconductivity leads to the emergence of zeroenergy Majorana corner states at all four corners of a rectangular sample, indicating the transition to a secondorder topological superconducting phase. The presence of strong electronelectron interactions is shown to promote the above phases to their exotic fractional counterparts. In particular, we find that the system supports a fractional topological insulator phase exhibiting fractionally charged gapless edge states and a fractional secondorder topological superconducting phase exhibiting zeroenergy Z_{2m} parafermion corner states, where m is an odd integer determined by the position of the chemical potential.

Firstorder magnetic phasetransition of mobile electrons in monolayer MoS2
Jonas Gaël Roch, Dmitry Miserev, Guillaume Froehlicher, Nadine Leisgang, Lukas Sponfeldner, Kenji Watanabe, Takashi Taniguchi, Jelena Klinovaja, Daniel Loss, and Richard John Warburton.
Phys. Rev. Lett. 124, 187602 (2020)
Evidence is presented for a firstorder magnetic phase transition in a gated twodimensional semiconductor, monolayerMoS2. The phase boundary separates a spinpolarised (ferromagnetic) phase at low electron density and a paramagnetic phase at high electron density. Abrupt changes in the optical response signal an abrupt change in the magnetism. The magnetic order is thereby controlled via the voltage applied to the gate electrode of the device. Accompanying the change in magnetism is a large change in the electron effective mass.

Surface charge theorem and topological constraints for edge states: Analytical study of onedimensional nearestneighbor tightbinding models
Mikhail Pletyukhov, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller.
Phys. Rev. B 101, 165304 (2020)
For a wide class of noninteracting tightbinding models in one dimension with nondegenerate bands we propose an analytic continuation of Bloch states for complex quasimomentum useful for an analytical understanding of boundary physics in halfinfinite systems. By finding the solution for all bulk and edge states, we prove the localization of the boundary charge in the insulating regime and show that all edge states leave a corresponding fingerprint in the density from the bulk states. We determine the explicit form of the density given by an exponential decay with localization length proportional to the inverse gap and a preexponential function following a powerlaw with generic exponent −1/2 at large distances. Introducing a phase variable that shifts the lattice continuously towards the boundary, we determine the topological constraints for the phasedependence of the edge states connecting adjacent bands. The constraints are shown to be equivalent to the possible quantization values for a topological index proposed in an accompanying letter (see arXiv) defined in terms of the change of the boundary charge when the boundary is shifted by one site. By analysing the phasedependence of poles in the complex plane during the continuous shift of the boundary by one site, we show that the phasedependence of the model parameters can always be chosen such that no edge state crosses the chemical potential in a certain gap. This clarifies the result found in the accompanying letter that the underlying reason for the topological constraints is charge conservation and particlehole duality alone but does not require any edge state physics. The topological index characterizing universal properties of the boundary charge is compared to the Zak phase and the Chern number and is shown to contain more information useful for a generic discussion of topological properties of onedimensional systems.

Topological invariants to characterize universality of boundary charge in onedimensional insulators beyond symmetry constraints
Mikhail Pletyukhov, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller.
Phys. Rev. B 101, 161106(R) (2020)
FIn the absence of any symmetry constraints we address universal properties of the boundary charge QB for a wide class of tightbinding models with nondegenerate bands in one dimension. We provide a precise formulation of the bulkboundary correspondence by splitting QB via a gauge invariant decomposition in a Friedel, polarisation, and edge part. We reveal the topological nature of QB by proving the quantization of a topological index I=ΔQB−ρ¯, where ΔQB is the change of QB when shifting the lattice by one site towards a boundary and ρ¯ is the average charge per site. For a single band we find this index to be given by the winding number of the fundamental phase difference of the Bloch wave function between two adjacent sites. For a given chemical potential we establish a central topological constraint I∈{−1,0} related to charge conservation and particlehole duality. Our results are shown to be stable against disorder and we propose generalizations to multichannel and interacting systems.

From Andreev to Majorana bound states in hybrid superconductorsemiconductor nanowires
Elsa Prada, Pablo SanJose, Michiel W. A. de Moor, Attila Geresdi, Eduardo J. H. Lee, Jelena Klinovaja, Daniel Loss, Jesper Nygård, Ramón Aguado, and Leo P. Kouwenhoven.
Nature Reviews Physics 2, 575 (2020)
Electronic excitations above the ground state must overcome an energy gap in superconductors with spatiallyhomogeneous pairing. In contrast, inhomogeneous superconductors such as those with magnetic impurities, weak links or heterojunctions containing normal metals can host subgap electronic excitations that are generically known as Andreev bound states (ABSs). With the advent of topological superconductivity, a new kind of ABS with exotic qualities, known as Majorana bound state (MBS), has been discovered. We review the main properties of all such subgap states and the stateoftheart techniques for their detection. We focus on hybrid superconductorsemiconductor nanowires, possibly coupled to quantum dots, as one of the most flexible and promising experimental platforms. We discuss how the combined effect of spinorbit coupling and Zeeman energy in these wires triggers the transition from ABSs into MBSs and show theoretical progress beyond minimal models in understanding experiments, including the possibility of a new type of robust Majorana zero mode without the need of a band topological transition. We examine the role of spatial nonlocality, a special property of MBS wavefunctions that, together with nonAbelian braiding, is the key ingredient for realizing topological quantum computing.

MagneticallyConfined Bound States in Rashba Systems
Flavio Ronetti, Kirill Plekhanov, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 022052(R) (2020)
In the absence of any symmetry constraints we address universal properties of the boundary charge QB for a wide class of tightbinding models with nondegenerate bands in one dimension. We provide a precise formulation of the bulkboundary correspondence by splitting QB via a gauge invariant decomposition in a Friedel, polarisation, and edge part. We reveal the topological nature of QB by proving the quantization of a topological index I=ΔQB−ρ¯, where ΔQB is the change of QB when shifting the lattice by one site towards a boundary and ρ¯ is the average charge per site. For a single band we find this index to be given by the winding number of the fundamental phase difference of the Bloch wave function between two adjacent sites. For a given chemical potential we establish a central topological constraint I∈{−1,0} related to charge conservation and particlehole duality. Our results are shown to be stable against disorder and we propose generalizations to multichannel and interacting systems.

Electronic transport in onedimensional Floquet topological insulators via topological and nontopological edge states
Niclas Muller, Dante M. Kennes, Jelena Klinovaja. Daniel Loss, and Herbert Schoeller.
Phys. Rev. B 101, 155417 (2020)
Based on probing electronic transport properties, we propose an experimental test for the recently discovered rich topological phase diagram of onedimensional Floquet topological insulators with Rashba spinorbit interaction [Kennes et al., Phys. Rev. B 100, 041104(R) (2019)]. Using the KeldyshFloquet formalism, we compute electronic transport properties of these nanowires, where we propose to couple the leads in such a way, as to primarily address electronic states with a large relative weight at one edge of the system. By tuning the Fermi energy of the leads to the center of the topological gap, we are able to directly address the topological edge states, granting experimental access to the topological phase diagram. Surprisingly, we find conductance values similar or even larger in magnitude to those corresponding to topological edge states, when tuning the lead Fermi energy to special values in the bulk, which coincide with bifurcation points of the dispersion relation in complex quasimomentum space. These peaks reveal the presence of narrow bands of states whose wave functions are linear combinations of delocalized bulk states and exponentially localized edge states, where the amplitude of the edgestate component is sharply peaked at the aforementioned bifurcation point, resulting in an unusually large relative edgeweight. We discuss the transport properties of these nontopological edge states and explain their emergence in terms of an intuitive yet quantitative physical picture. The mechanism giving rise to these states is not specific to the model we consider here, suggesting that they may be present in a wide class of systems.

TimeReversal Invariant Topological Superconductivity in Planar Josephson Bijunction
Yanick Volpez, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 023415 (2020)
We consider a Josephson bijunction consisting of a thin SIS piJosephson junction sandwiched between twodimensional semiconducting layers with strong Rashba spinorbit interaction. Each of these layers forms an SNS junction due to proximityinduced superconductivity. The SIS junction is assumed to be thin enough such that the two Rashba layers are tunnelcoupled. We show that, by tuning external gates, this system can be controllably brought into a timereversal invariant topological superconducting phase with a Kramers pair of Majorana bound states being localized at the end of the normal region for a large parameter phase space. In particular, in the strong spinorbit interaction limit, the topological phase can be accessed already in the regime of small tunneling amplitudes.

Hinge Modes and Surface States in SecondOrder Topological ThreeDimensional Quantum Hall Systems induced by Charge Density Modulation
Pawel Szumniak, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 102, 125126 (2020)
We consider a system of weakly coupled onedimensional wires forming a threedimensional stack in the presence of a spatially periodic modulation of the chemical potential along the wires, equivalent to a charge density wave (CDW). An external static magnetic field is applied parallel to the wire axes. We show that, for a certain parameter regime, due to interplay between the CDW and magnetic field, the system can support a secondorder topological phase characterized by the presence of chiral quasi1D Quantum Hall Effect (QHE) hinge modes. Interestingly, we demonstrate that direction of propagation of the hinge modes depends on the phase of the CDW and can be reversed only by electrical means without the need of changing the orientation of the magnetic field. Furthermore, we show that the system can also support 2D chiral surface QHE states, which can coexist with onedimensional hinge modes, realizing a scenario of a hybrid highorder topology. We show that the hinge modes are robust against static disorder.

Chiral Magnonic Edge States in Ferromagnetic Skyrmion Crystals Controlled by Magnetic Fields
Sebastian A. Diaz, Tomoki Hirosawa, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Research 2, 013231 (2020)
Achieving control over magnon spin currents in insulating magnets  where dissipation due to Joule heating is highly suppressed  is an active area of research that could lead to energyefficient spintronics applications. However, magnon spin currents supported by conventional systems with uniform magnetic order have proven hard to control. An alternative approach that relies on topologically protected magnonic edge states of spatially periodic magnetic textures has recently emerged. A prime example of such textures is the ferromagnetic skyrmion crystal which hosts chiral edge states providing a platform for magnon spin currents. Here, we show, for the first time, an external magnetic field can drive a topological phase transition in the spin wave spectrum of a ferromagnetic skyrmion crystal. The topological phase transition is signaled by the closing of a lowenergy bulk magnon gap at a critical field. In the topological phase, below the critical field, two topologically protected chiral magnonic edge states lie within this gap, but they unravel in the trivial phase, above the critical field. Remarkably, the topological phase transition involves an inversion of two magnon bands that at the Γ point correspond to the breathing and anticlockwise modes of the skyrmions in the crystal. Our findings suggest that an external magnetic field could be used as a knob to switch on and off magnon spin currents carried by topologically protected chiral magnonic edge states.

Interaction Driven Floquet Engineering of Topological Superconductivity in Rashba Nanowires
Manisha Thakurathi, Pavel P. Aseev, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 013292 (2020)
We analyze, analytically and numerically, a periodically driven Rashba nanowire proximity coupled to an swave superconductor using bosonization and renormalization group analysis in the regime of strong electronelectron interactions. Due to the repulsive interactions, the superconducting gap is suppressed, whereas the Floquet Zeeman gap is enhanced, resulting in a higher effective value of gfactor compared to the noninteracting case. The flow equations for different coupling constants, velocities, and Luttingerliquid parameters explicitly establish that even for small initial values of the Floquet Zeeman gap compared to the superconducting proximity gap, the interactions drive the system into the topological phase and the interband interaction term helps to achieve larger regions of the topological phase in parameter space.

Hinge states in a system of coupled Rashba layers
Kirill Plekhanov, Flavio Ronetti, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 2, 013083 (2020)
We consider a system of stacked tunnelcoupled twodimensional electron and holegas layers with Rashba spinorbit interactions subjected to a staggered Zeeman field. The interplay of different intralayer tunnel couplings results in a phase transition to a topological insulator phase in three dimensions hosting gapless surface states. The staggered Zeeman field further enriches the topological phase diagram by generating a secondorder topological insulator phase hosting gapless hinge states. The emergence of the topological phases is proven analytically in the regime of small Zeeman field and confirmed by numerical simulations in the nonperturbative region of the phase diagram. The topological phases are stable against external perturbations and disorder.

Coherent backaction between spins and an electronic bath: NonMarkovian dynamics and low temperature quantum thermodynamic electron cooling
Stephanie Matern, Daniel Loss, Jelena Klinovaja, and Bernd Braunecker.
Phys. Rev. B 100, 134308 (2019)
We provide a general analytical framework for calculating the dynamics of a spin system in contact with a bath beyond the Markov approximation. The approach is based on a systematic expansion of the NakashimaZwanzig master equation in the weakcoupling limit but makes no assumption on the time dynamics and includes all quantum coherent memory effects leading to nonMarkovian dynamics. Our results describe, for the free induction decay, the full time range from the nonMarkovian dynamics at short times, to the wellknown exponential thermal decay at long times. We provide full analytic results for the entire time range using a bath of itinerant electrons as an archetype for universal quantum fluctuations. Furthermore, we propose a quantum thermodynamic scheme to employ the temperature insensitivity of the nonMarkovian decay to transport heat out of the electron system and thus, by repeated reinitialisation of a cluster of spins, to efficiently cool the electrons at very low temperatures.

Floquet SecondOrder Topological Superconductor Driven via Ferromagnetic Resonance
Kirill Plekhanov, Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 1, 032013(R) (2019)
We consider a Floquet triplelayer setup composed of a twodimensional electron gas with spinorbit interactions, proximity coupled to an swave superconductor and to a ferromagnet driven at resonance. The ferromagnetic layer generates a timeoscillating Zeeman field which competes with the induced superconducting gap and leads to a topological phase transition. The resulting Floquet states support a secondorder topological superconducting phase with a pair of localized zeroenergy Floquet Majorana corner states. Moreover, the phase diagram comprises a Floquet helical topological superconductor, hosting a Kramers pair of Majorana edge modes protected by an effective timereversal symmetry, as well as a gapless Floquet Weyl phase. The topological phases are stable against disorder and parameter variations and are within experimental reach.

Fractional Topological Superconductivity and Parafermion Corner States
Katharina Laubscher, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Research 1, 032017(R) (2019)
We consider a system of weakly coupled Rashba nanowires in the strong spinorbit interaction (SOI) regime. The nanowires are arranged into two tunnelcoupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is \pi. We show that in such a system strong electronelectron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of Z_{2m} parafermion edge modes, where m is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak inplane magnetic field, the system is driven into a secondorder topological superconducting phase hosting zeroenergy Z_{2m} parafermion bound states localized at two opposite corners of a rectangular sample. As a special case, zeroenergy Majorana corner states emerge in the noninteracting limit m=1, where the chemical potential is tuned to the SOI energy of the single nanowires.

Majorana fermions in magnetic chains
Remy Pawlak, Silas Hoffman, Jelena Klinovaja, Daniel Loss, and Ernst Meyer.
Progress in Particle and Nuclear Physics 107, 1 (2019)
Majorana fermions have recently garnered a great attention outside the field of particle physics, in condensed matter physics. In contrast to their particle physics counterparts, Majorana fermions are zero energy, chargeless, spinless, composite quasiparticles, residing at the boundaries of socalled topological superconductors. Furthermore, in opposition to any particles in the standard model, Majorana fermions in solidstate systems obey nonAbelian exchange statistics that make them attractive candidates for decoherencefree implementations of quantum computers. In this review, we report on the recent advances to realize synthetic topological superconductors supporting Majorana fermions with an emphasis on chains of magnetic impurities on the surface of superconductors. After outlining the theoretical underpinning responsible for the formation of Majorana fermions, we report on the subsequent experimental efforts to build topological superconductors and the resulting evidence in favor of Majorana fermions, focusing on scanning tunneling microscopy and the hunt for zerobias peaks in the measured current. We conclude by summarizing the open questions in the field and propose possible experimental measurements to answer them.

Degeneracy lifting of Majorana bound states due to electronphonon interactions
Pavel P. Aseev, Pasquale Marra, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 99, 205435 (2019)
We study theoretically how electronphonon interaction affects the energies and level broadening (inverse lifetime) of Majorana bound states (MBSs) in a clean topological nanowire at low temperatures. At zero temperature, the energy splitting between the right and left MBSs remains exponentially small with increasing nanowire length L. At finite temperatures, however, the absorption of thermal phonons leads to the broadening of energy levels of the MBSs that does not decay with system length, and the coherent absorption/emission of phonons at opposite ends of the nanowire results in MBSs energy splitting that decays only as an inverse powerlaw in L. Both effects remain exponential in temperature. In the case of quantized transverse motion of phonons, the presence of Van Hove singularities in the phonon density of states causes additional resonant enhancement of both the energy splitting and the level broadening of the MBSs. This is the most favorable case to observe the phononinduced energy splitting of MBSs as it becomes much larger than the broadening even if the topological nanowire is much longer than the coherence length. We also calculate the charge and spin associated with the energy splitting of the MBSs induced by phonons. We consider both a spinless lowenergy continuum model, which we evaluate analytically, as well as a spinful lattice model for a Rashba nanowire, which we evaluate numerically.

Majorana Bound States in Double Nanowires with Reduced Zeeman Thresholds due to Supercurrents
Olesia Dmytruk, Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 99, 245416 (2019)
We study the topological phase diagram of a setup composed of two nanowires with strong Rashba spinorbit interaction subjected to an external magnetic field and brought into the proximity to a bulk swave superconductor in the presence of a supercurrent flowing through it. The supercurrent reduces the critical values of the Zeeman energy and crossed Andreev superconducting pairing required to reach the topological phase characterized by the presence of one Majorana bound state localized at each system end. We demonstrate that, even in the regime of the crossed Andreev pairing being smaller than the direct proximity pairing, a relatively weak magnetic field drives the system into the topological phase due to the presence of the supercurrent.

Entangling Spins in Double Quantum Dots and Majorana Bound States
Marko J. Rancic, Silas Hoffman, Constantin Schrade, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 99, 165306 (2019)
We study the coupling between a singlettriplet qubit realized in a double quantum dot to a topological qubit realized by spatially wellseparated Majorana bound states. We demonstrate that the singlettriplet qubit can be leveraged for readout of the topological qubit and for supplementing the gate operations that cannot be performed by braiding of Majorana bound states. Furthermore, we extend our setup to a network of singlettriplet and topological hybrid qubits that paves the way to scalable faulttolerant quantum computing.

Spontaneous Symmetry Breaking in Monolayers of Transition Metal Dichalcogenides
Dmitry Miserev, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 100, 014428 (2019)
We analyze magnetic phases of monolayers of transition metal dichalcogenides that are twovalley materials with electronelectron interactions. The exchange intervalley scattering makes twovalley systems less stable to the spin fluctuations but more stable to the valley fluctuations. We predict a first order ferromagnetic phase transition governed by the nonanalytic and negative cubic term in the thermodynamic potential that results in a large spontaneous spin magnetization. Finite spin orbit interaction leads to the outofplane Ising order of the ferromagnetic phase. Our theoretical prediction is consistent with the recent experiment on electrondoped monolayers of MoS2 reported by Roch et al. [1].

Topological Magnons and Edge States in Antiferromagnetic Skyrmion Crystals
Sebastian Diaz, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Lett. 122, 187203 (2019)
Antiferromagnetic skyrmion crystals are magnetic phases predicted to exist in antiferromag nets with DzyaloshinskiiMoriya interactions. Their spatially periodic noncollinear magnetic tex ture gives rise to topological bulk magnon bands characterized by nonzero Chern numbers. We find topologicallyprotected chiral magnonic edge states over a wide range of magnetic fields and DzyaloshinskiiMoriya interaction values. Moreover, and of particular importance for experimen tal realizations, edge states appear at the lowest possible energies, namely, within the first bulk magnon gap. Thus, antiferromagnetic skyrmion crystals show great promise as novel platforms for topological magnonics.

Chiral 1D Floquet topological insulators beyond rotating wave approximation
Dante M. Kennes, Niclas Muller, Mikhail Pletyukhov, Clara Weber, Christoph Bruder, Fabian Hassler, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller.
Phys. Rev. B 100, 041103(R) (2019)
We study onedimensional (1D) Floquet topological insulators with chiral symmetry going beyond the standard rotating wave approximation. The occurrence of many anticrossings between Floquet replicas leads to a dramatic extension of phase diagram regions with stable topological edge states (TESs). We present an explicit construction of all TESs in terms of a truncated Floquet Hamiltonian in frequency space, prove the bulkboundary correspondence, and analyze the stability of the TESs in terms of their localization lengths. We propose experimental tests of our predictions in curved bilayer graphene.

Second Order Topological Superconductivity in $\pi$Junction Rashba Layers
Yanick Volpez, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. Lett. 122, 126402 (2019)
We consider a Josephson junction bilayer consisting of two tunnelcoupled twodimensional electron gas layers with Rashba spinorbit interaction, proximitized by a top and bottom swave superconductor with phase difference $\phi$ close to $\pi$. We show that, in the presence of a finite weak inplane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If $\phi=\pi$, in a rectangular geometry, these zeroenergy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference $\phi$ deviates from $\pi$ by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference $\phi$ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling $\phi$ by magnetic flux, the other involves an additional layer of randomlyoriented magnetic impurities responsible for the phase shift of $\pi$ in the proximityinduced superconducting pairing.

Zeroenergy Andreev bound states from quantum dots in proximitized Rashba nanowires
Christopher Reeg, Olesia Dmytruk, Denis Chevallier, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 98, 245407 (2018)
We study an analytical model of a Rashba nanowire that is partially covered by and coupled to a thin superconducting layer, where the uncovered region of the nanowire forms a quantum dot. We find that, even if there is no topological superconducting phase possible, there is a trivial Andreev bound state that becomes pinned exponentially close to zero energy as a function of magnetic field strength when the length of the quantum dot is tuned with respect to its spinorbit length such that a resonance condition of FabryPerot type is satisfied. In this case, we find that the Andreev bound state remains pinned near zero energy for Zeeman energies that exceed the characteristic spacing between Andreev bound state levels but that are smaller than the spinorbit energy of the quantum dot. Importantly, as the pinning of the Andreev bound state depends only on properties of the quantum dot, we conclude that this behavior is unrelated to topological superconductivity. To support our analytical model, we also perform a numerical simulation of a hybrid system while explicitly incorporating a thin superconducting layer, showing that all qualitative features of our analytical model are also present in the numerical results.

From fractional boundary charges to quantized Hall conductance
Manisha Thakurathi, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 98, 245404 (2018)
We study the fractional boundary charges (FBCs) occurring in nanowires in the presence of periodically modulated chemical potentials and connect them to the FBCs occurring in a twodimensional electron gas in the presence of a perpendicular magnetic field in the integer quantum Hall effect (QHE) regime. First, we show that in nanowires the FBCs take fractional values and change linearly as a function of phase offset of the modulated chemical potential. This linear slope takes quantized values determined by the period of the modulation and depends only on the number of the filled bands. Next, we establish a mapping from the onedimensional system to the QHE setup, where we again focus on the properties of the FBCs. By considering a cylinder topology with an external flux similar to the Laughlin construction, we find that the slope of the FBCs as function of flux is linear and assumes universal quantized values, also in the presence of arbitrary disorder. We establish that the quantized slopes give rise to the quantization of the Hall conductance. Importantly, the approach via FBCs is valid for arbitrary flux values and disorder. The slope of the FBCs plays the role of a topological invariant for clean and disordered QHE systems. Our predictions for the FBCs can be tested experimentally in nanowires and in Corbino disk geometries in the integer QHE regime.

Lifetime of Majorana qubits in Rashba nanowires with nonuniform chemical potential
Pavel Aseev, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 98, 155414 (2018)
We study the lifetime of topological qubits based on Majorana bound states hosted in a one dimensional Rashba nanowire (NW) with proximityinduced superconductivity and nonuniform chemical potential needed for manipulation and readout. If nearby gates tune the chemical potential locally so that part of the NW is in the trivial phase, Andreev bound states (ABSs) can emerge which are localized at the interface between topological and trivial phases with energies significantly less than the gap. The emergence of such subgap states strongly decreases the Majorana qubit lifetime at finite temperatures due to local perturbations that can excite the system into these ABSs. Using Keldysh formalism, we study such excitations caused by fluctuating charges in capacitively coupled gates and calculate the corresponding Majorana lifetimes due to thermal noise, which are shown to be much shorter than those in NWs with uniform chemical potential.

Renormalization of quantum dot gfactor in superconducting Rashba nanowires
Olesia Dmytruk, Denis Chevallier, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 98, 165403 (2018)
We study analytically and numerically the renormalization of the gfactor in semiconducting Rashba nanowires (NWs), consisting of a normal and superconducting section. If the potential barrier between the sections is high, a quantum dot (QD) is formed in the normal section. For harmonic (hardwall) confinement, the effective gfactor of all QD levels is suppressed exponentially (powerlaw) in the product of the spinorbit interaction (SOI) wavevector and the QD length. If the barrier between the two sections is removed, the gfactor of the emerging Andreev bound states is suppressed less strongly. In the strong SOI regime and if the chemical potential is tuned to the SOI energy in both sections, the gfactor saturates to a universal constant. Remarkably, the effective gfactor shows a pronounced peak at the SOI energy as function of the chemical potentials. In addition, if the SOI is uniform, the gfactor renormalization as a function of the chemical potential is given by a universal dependence which is independent of the QD size. This prediction provides a powerful tool to determine experimentally whether the SOI in the whole NW is uniform and, moreover, gives direct access to the SOI strengths of the NW via gfactor measurements. In addition, it allows one to find the optimum position of the chemical potential for bringing the NW into the topological phase at large magnetic fields.

Majorana Kramers pairs in higherorder topological insulators
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Lett. 121, 196801 (2018)
We propose a tunefree scheme to realize Kramers pairs of Majorana bound states in recently discovered higherorder topological insulators (HOTIs). We show that, by bringing two hinges of a HOTI into the proximity of an swave superconductor, the competition between local and crossedAndreev pairing leads to formation of Majorana Kramers pairs, when the latter pairing dominates over the former. We demonstrate that such a topological superconductivity is stabilized by moderate electronelectron interactions. The proposed setup avoids the application of a magnetic field or local voltage gates, and requires weaker interactions comparing to nonhelical nanowires.

Proximity effect in a twodimensional electron gas coupled to a thin superconducting layer
Christopher Reeg, Daniel Loss, and Jelena Klinovaja.
Beilstein Journal of Nanotechnology 9, 1263 (2018)
There have recently been several experiments studying induced superconductivity in semiconducting twodimensional electron gases that are strongly coupled to thin superconducting layers, as well as probing possible topological phases supporting Majorana bound states in such setups. We show that a large band shift is induced in the semiconductor by the superconductor in this geometry, thus making it challenging to realize a topological phase. Additionally, we show that while increasing the thickness of the superconducting layer reduces the magnitude of the band shift, it also leads to a more significant renormalization of the semiconducting material parameters and does not reduce the challenge of tuning into a topological phase.

Conductance of fractional Luttinger liquids at finite temperatures
Pavel Aseev, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 98, 045416 (2018)
We study the electrical conductance in singlemode quantum wires with Rashba spinorbit inter action subjected to externally applied magnetic fields in the regime in which the ratio of spinorbit momentum to the Fermi momentum is close to an odd integer, so that a combined effect of multi electron interaction and applied magnetic field leads to a partial gap in the spectrum. We study how this partial gap manifests itself in the temperature dependence of the fractional conductance of the quantum wire. We use two complementing techniques based on bosonization: refermionization of the model at a particular value of the interaction parameter and a semiclassical approach within a dilute soliton gas approximation of the functional integral. We show how the lowtemperature fractional conductance can be affected by the finite length of the wire, by the properties of the contacts, and by a shift of the chemical potential, which takes the system away from the resonance condition. We also predict an internal resistivity caused by a dissipative coupling between gapped and gapless modes.

Rashba Sandwiches with Topological Superconducting Phases
Yanick Volpez, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 97, 195421 (2018)
We introduce a versatile heterostructure harboring various topological superconducting phases characterized by the presence of helical, chiral, or unidirectional edge states. Changing parameters, such as an effective Zeeman field or chemical potential, one can tune between these three topolog ical phases in the same setup. Our model relies only on conventional nontopological ingredients. The bilayer setup consists of an swave superconductor sandwiched between two twodimensional electron gas layers with strong Rashba spinorbit interaction. The interplay between two different pairing mechanisms, proximity induced direct and crossed Andreev superconducting pairings, gives rise to multiple topological phases. In particular, helical edge states occur if crossed Andreev su perconducting pairing is dominant. In addition, an inplane Zeeman field leads to a 2D gapless topological phase with unidirectional edge states, which were previously predicted to exist only in noncentrosymmetric superconductors. If the Zeeman field is tilted out of the plane, the system is in a topological phase hosting chiral edge states.

Boundary spin polarization as robust signature of topological phase transition in Majorana nanowires
Marcel Serina, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 98, 035419 (2018)
We show that the boundary charge and spin can be used as alternative signatures of the topological phase transition in topological models such as semiconducting nanowires with strong Rashba spinorbit interaction in the presence of a magnetic field and in proximity to an swave superconductor. We identify signatures of the topological phase transition that do not rely on the presence of Majorana zeroenergy modes and, thus, can serve as independent probes of topological properties. The boundary spin component along the magnetic field, obtained by summing contributions from all states below the Fermi level, has a pronounced peak at the topological phase transition point. Generally, such signatures can be observed at boundaries between topological and trivial sections in nanowires and are stable against disorder.

Metallization of Rashba wire by superconducting layer in the strongproximity regime
Christopher Reeg, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 97, 165425 (2018)
Semiconducting quantum wires defined within twodimensional electron gases and strongly coupled to thin superconducting layers have been extensively explored in recent experiments as promising platforms to host Majorana bound states. We study numerically such a geometry, consisting of a quasionedimensional wire coupled to a disordered threedimensional superconducting layer. We find that, in the strongcoupling limit of a sizable proximityinduced superconducting gap, all transverse subbands of the wire are significantly shifted in energy relative to the chemical potential of the wire. For the lowest subband, this band shift is comparable in magnitude to the spacing between quantized levels that arise due to the finite thickness of the superconductor (which typically is ~500 meV for a 10nmthick layer of Aluminum); in higher subbands, the band shift is much larger. Additionally, we show that the width of the system, which is usually much larger than the thickness, and moderate disorder within the superconductor have almost no impact on the induced gap or band shift. We provide a detailed discussion of the ramifications of our results, arguing that a huge band shift and significant renormalization of semiconducting material parameters in the strongcoupling limit make it challenging to realize a topological phase in such a setup, as the strong coupling to the superconductor essentially metallizes the semiconductor. This metallization of the semiconductor can be tested experimentally through the measurement of the band shift.

Effects of nuclear spins on the transport properties of the edge of twodimensional topological insulators
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 97, 125432 (2018)
The electrons in the edge channels of twodimensional topological insulators can be described as a helical TomonagaLuttinger liquid. They couple to nuclear spins embedded in the host materials through the hyperfine interaction, and are therefore subject to elastic spinflip backscattering on the nuclear spins. We investigate the nuclearspininduced edge resistance due to such backscattering by performing a renormalizationgroup analysis. Remarkably, the effect of this backscattering mechanism is stronger in a helical edge than in nonhelical channels, which are believed to be present in the trivial regime of InAs/GaSb quantum wells. In a system with sufficiently long edges, the disordered nuclear spins lead to an edge resistance which grows exponentially upon lowering the temperature. On the other hand, electrons from the edge states mediate an anisotropic RudermanKittelKasuyaYosida nuclear spinspin interaction, which induces a spiral nuclear spin order below the transition temperature. We discuss the features of the spiral order, as well as its experimental signatures. In the ordered phase, we identify two backscattering mechanisms, due to charge impurities and magnons. The backscattering on charge impurities is allowed by the internally generated magnetic field, and leads to an Andersontype localization of the edge states. The magnonmediated backscattering results in a powerlaw resistance, which is suppressed at zero temperature. Overall, we find that in a sufficiently long edge the nuclear spins, whether ordered or not, suppress the edge conductance to zero as the temperature approaches zero.

Majorana Kramers pairs in Rashba double nanowires with interactions and disorder
Manisha Thakurathi, Pascal Simon, Ipsita Mandal, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 97, 045415 (2018)
We analyze the effects of electronelectron interactions and disorder on a Rashba doublenanowire setup coupled to an swave superconductor, which has been recently proposed as a versatile platform to generate Kramers pairs of Majorana bound states in the absence of magnetic fields. We identify the regime of parameters for which these Kramers pairs are stable against interaction and disorder effects. We use bosonization, perturbative renormalization group, and replica techniques to derive the flow equations for various parameters of the model and evaluate the corresponding phase diagram with topological and disorderdominated phases. We confirm aforementioned results by considering a more microscopic approach which starts from the tunneling Hamiltonian between the threedimensional swave superconductor and the nanowires. We find again that the interaction drives the system into the topological phase and, as the strength of the source term coming from the tunneling Hamiltonian increases, strong electronelectron interactions are required to reach the topological phase.

Topological Phase Detection in Rashba Nanowires with a Quantum Dot
Denis Chevallier, Pawel Szumniak, Silas Hoffman, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 97, 045404 (2018)
We study theoretically the detection of the topological phase transition occurring in Rashba nanowires with proximityinduced superconductivity using a quantum dot. The bulk states lowest in energy of such a nanowire have a spin polarization parallel or antiparallel to the applied magnetic field in the topological or trivial phase, respectively. We show that this property can be probed by the quantum dot created at the end of the nanowire by external gates. By tuning one of the two spinsplit levels of the quantum dot to be in resonance with nanowire bulk states, one can detect the spin polarization of the lowest band via transport measurement. This allows one to determine the topological phase of the Rashba nanowire independently of the presence of Majorana bound states.

Suppression of the overlap between Majorana fermions by orbital magnetic effects in semiconductingsuperconducting nanowires
Olesia Dmytruk and Jelena Klinovaja.
Phys. Rev. B 97, 155409 (2018)
We study both analytically and numerically the role of orbital effects caused by a magnetic field applied along the axis of a semiconducting Rashba nanowire in the topological regime hosting Majorana fermions. We demonstrate that the orbital effects can be effectively taken into account in a onedimensional model by shifting the chemical potential, and, thus modifying the topological criterion. We focus on the energy splitting between two Majorana fermions in a finite nanowire and find a striking interplay between orbital and Zeeman effects on this splitting. In the limit of strong spinorbit interaction, we find regimes where the amplitude of the oscillating splitting stays constant or even decays with increasing magnetic field, in stark contrast to the commonly studied case where orbital effects of the magnetic field are neglected. The period of these oscillations is found to be almost constant in many parameter regimes.

DIII Topological Superconductivity with Emergent TimeReversal Symmetry
Christopher Reeg, Constantin Schrade, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 96, 161407 (2017)
We find a new class of topological superconductors which possess an emergent timereversal symmetry that is present only after projecting to an effective lowdimensional model. We show that a topological phase in symmetry class DIII can be realized in a noninteracting system coupled to an swave superconductor only if the physical timereversal symmetry of the system is broken, and we provide three general criteria that must be satisfied in order to have such a phase. We also provide an explicit model which realizes the class DIII topological superconductor in 1D. We show that, just as in timereversal invariant topological superconductors, the topological phase is characterized by a Kramers pair of Majorana fermions that are protected by the emergent timereversal symmetry.

Finitesize effects in a nanowire strongly coupled to a thin superconducting shell
Christopher Reeg, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 96, 125426 (2017)
We study the proximity effect in a onedimensional nanowire strongly coupled to a finite superconductor with a characteristic size which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al shells which form a very strong contact with either InAs or InSb nanowires. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finitesize effects become very detrimental when the level spacing of the superconductor greatly exceeds its energy gap. Without any finetuning of the size of the superconductor (on the scale of the Fermi wavelength), the tunneling energy scale must be larger than the level spacing in order to reach the hard gap regime which is seen ubiquitously in the experiments. However, in this regime, the large tunneling energy scale induces a large shift in the effective chemical potential of the nanowire and pushes the topological phase transition to magnetic field strengths which exceed the critical field of Al.

Magnonic topological insulators in antiferromagnets
Kouki Nakata, Se Kwon Kim, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 96, 224414 (2017)
Extending the notion of symmetry protected topological phases to insulating antiferromagnets (AFs) described in terms of opposite magnetic dipole moments associated with the magnetic Neel order, we establish a bosonic counterpart of topological insulators in semiconductors. Making use of the AharonovCasher effect, induced by electric field gradients, we propose a magnonic analog of the quantum spin Hall effect (magnonic QSHE) for edge states that carry helical magnons. We show that such up and down magnons form the same Landau levels and perform cyclotron motion with the same frequency but propagate in opposite direction. The insulating AF becomes characterized by a topological Z_2 number consisting of the Chern integer associated with each helical magnon edge state. Focusing on the topological Hall phase for magnons, we study bulk magnon effects such as magnonic spin, thermal, Nernst, and Ettinghausen effects, as well as the thermomagnetic properties of helical magnon transport both in topologically trivial and nontrivial bulk AFs and establish the magnonic WiedemannFranz law. We show that our predictions are within experimental reach with current device and measurement techniques.

ThreeDimensional Fractional Topological Insulators in Coupled Rashba Layers
Yanick Volpez, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 96, 085422 (2017)
We propose a model of threedimensional topological insulators consisting of weakly coupled electron and holegas layers with Rashba spinorbit interaction stacked along a given axis. We show that in the presence of strong electronelectron interactions the system realizes a fractional strong topological insulator, where the rotational symmetry and condensation energy arguments still allow us to treat the problem as quasionedimensional with bosonization techniques. We also show that if Rashba and Dresselhaus spinorbit interaction terms are equally strong, by doping the system with magnetic impurities, one can bring it into the Weyl semimetal phase.

Lowfield Topological Threshold in Majorana Double Nanowires
Constantin Schrade, Manisha Thakurathi, Christopher Reeg, Silas Hoffman, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 96, 035306 (2017)
A hard proximityinduced superconducting gap has recently been observed in semiconductor nanowire systems at low magnetic fields. However, in the topological regime at high magnetic fields a soft gap reemerges and represents a fundamental obstacle to topologically protected quantum information processing with Majorana bound states. Here we show that this obstacle can be overcome in a setup of double Rashba nanowires which are coupled to an swave superconductor and subjected to an external magnetic field along the wires. Specifically, we demonstrate that the required field strength for the topological threshold can be significantly reduced by the destructive interference of direct and crossedAndreev pairing in this setup; precisely down to the regime in which current experimental technology allows for a hard superconducting gap. We also show that the resulting Majorana bound states exhibit sufficiently short localization lengths which makes them ideal candidates for future braiding experiments.

Spindependent coupling between quantum dots and topological quantum wires
Silas Hoffman, Denis Chevallier, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 96, 045440 (2017)
Considering Rashba quantum wires with a proximityinduced superconducting gap as physical realizations of Majorana fermions and quantum dots, we calculate the overlap of the Majorana wave functions with the local wave functions on the dot. We determine the spindependent tunneling amplitudes between these two localized states and show that we can tune into a fully spin polarized tunneling regime by changing the distance between dot and Majorana fermion. Upon directly applying this to the tunneling model Hamiltonian, we calculate the effective magnetic field on the quantum dot flanked by two Majorana fermions. The direction of the induced magnetic field on the dot depends on the occupation of the nonlocal fermion formed from the two Majorana end states which can be used as a readout for such a Majorana qubit.

Nuclear spininduced localization of the edge states in twodimensional topological insulators
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 96, 081405 (2017)
We investigate the influence of nuclear spins on the resistance of helical edge states of twodimensional topological insulators (2DTIs). Via the hyperfine interaction, nuclear spins allow electron backscattering, otherwise forbidden by time reversal symmetry. We identify two backscattering mechanisms, depending on whether the nuclear spins are ordered or not. Their temperature dependence is distinct but both give resistance, which increases with the edge length, decreasing temperature, and increasing strength of the electronelectron interaction. Overall, we find that the nuclear spins will typically shut down the conductance of the 2DTI edges at zero temperature.

Spin and Charge Signatures of Topological Superconductivity in Rashba Nanowires
Pawel Szumniak, Denis Chevallier, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 96, 041401(R) (2017)
We consider a Rashba nanowire with proximity gap which can be brought into the topological phase by tuning external magnetic field or chemical potential. We study spin and charge of the bulk quasiparticle states when passing through the topological transition for open and closed systems. We show, analytically and numerically, that the spin of bulk states around the topological gap reverses its sign when crossing the transition due to band inversion, independent of the presence of Majorana fermions in the system. This spin reversal can be considered as a bulk signature of topological superconductivity that can be accessed experimentally. We find a similar behaviour for the charge of the bulk quasiparticle states, also exhibiting a sign reversal at the transition. We show that these signatures are robust against random static disorder.

Destructive interference of direct and crossed Andreev pairing in a system of two nanowires coupled via an swave superconductor
Christopher R. Reeg, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 96, 081301(R) (2017)
We consider a system of two onedimensional nanowires coupled via an swave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only direct pairing, where one wire is decoupled from the superconductor, or only crossed Andreev pairing, where each nanowire is considered as a spinpolarized edge of a quantum Hall state, we find that the gap is always reduced, by orders of magnitude in certain cases, when both types of pairing are present. Our analytical results are further supported by numerical calculations on a tightbinding lattice. Finally, we show that treating the proximity effect by integrating out the superconductor cannot reproduce the results of our exact diagonalization.

Quantum dynamics of skyrmions in chiral magnets
Christina Psaroudaki, Silas Hoffman, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. X 7, 041045 (2017)
We study the quantum propagation of a skyrmion in chiral magnetic insulators by generalizing the micromagnetic equations of motion to a finite temperature path integral formalism, using field theoretic tools. Promoting the center of the skyrmion to a dynamic quantity, the fluctuations around the skyrmionic configuration give rise to a timedependent damping of the skyrmion motion. From the frequency dependence of the damping kernel, we are able to identify the skyrmion mass, thus providing a microscopic description of the kinematic properties of skyrmions. When the free energy is translationally invariant we find the skyrmion mass is finite only at finite temperature. However, if defects are present or a magnetic trap is applied, the skyrmion mass acquires a finite value, even at vanishingly small temperature. We demonstrate that a skyrmion in a confined geometry provided by a magnetic trap behaves as a massive particle owing to its quasione dimensional confinement.

Finitetemperature conductance of strongly interacting quantum wire with a nuclear spin order
Pavel Aseev, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 95, 125440 (2017)
We study the temperature dependence of the electrical conductance of a clean strongly interacting quantum wire in the presence of a helical nuclear spin order. The nuclear spin helix opens a temperaturedependent partial gap in the electron spectrum. Using a bosonization framework we describe the gapped electron modes by sineGordonlike kinks. We predict an internal resistivity caused by an Ohmiclike friction these kinks experience via interacting with gapless excitations. As a result, the conductance rises from G=e^2/h at temperatures below the critical temperature when nuclear spins are fully polarized to G=2e^2/h at higher temperatures when the order is destroyed, featuring a relatively wide plateau in the intermediate regime. The theoretical results are compared with the experimental data for GaAs quantum wires obtained recently by Scheller et al. [Phys. Rev. Lett. 112, 066801 (2014)].

Magnonic quantum Hall effect and WiedemannFranz law
Kouki Nakata, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 95, 125429 (2017)
Casher effect, a magnon moving in an electric field acquires a geometric phase and forms Landau levels in an electric field gradient of sawtooth form. At low temperatures, the lowest energy band being almost flat carries a Chern number associated with a Berry curvature. Appropriately defining the thermal conductance for bosons, we find that the magnon Hall conductances get quantized and show a universal thermomagnetic behavior, i.e., are independent of materials, and obey a WiedemannFranz law for magnon transport. We consider magnons with quadratic and linear (Diraclike) dispersions. Finally, we show that our predictions are within experimental reach for ferromagnets and skyrmion lattices with current device and measurement techniques.

Floquet Majorana and ParaFermions in Driven Rashba Nanowires
Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 95, 155407 (2017)
We study a periodically driven nanowire with Rashbalike conduction and valence bands in the presence of a magnetic field. We identify topological regimes in which the system hosts zeroenergy Majorana fermions. We further investigate the effect of strong electronelectron interactions that give rise to parafermion zero energy modes hosted at the nanowire ends. The first setup we consider allows for topological phases by applying only static magnetic fields without the need of superconductivity. The second setup involves both superconductivity and timedependent magnetic fields and allows one to generate topological phases without finetuning of the chemical potential. Promising candidate materials are graphene nanoribbons due to their intrinsic particlehole symmetry.

Tomography of Majorana Fermions with STM Tips
Denis Chevallier and Jelena Klinovaja.
Phys. Rev. B 94, 035417 (2016)
We investigate numerically the possibility to detect the spatial profile of Majorana fermions (MFs) modeling STM tips that are made of either normal or superconducting material. In both cases, we are able to resolve the localization length and the oscillation period of the MF wavefunction. We show that the tunneling between the substrate and the tip, necessary to get the information on the wave function oscillations, has to be smaller in the case of a superconducting STM. In the strong tunneling regime, the differential conductance saturates making it more difficult to observe the exponential decay of MFs. The temperature broadening of the profile is strongly suppressed in case of the superconducting lead resulting, generally, in better resolution.

Fractional boundary charges in quantum dot arrays with density modulation
JinHong Park, Guang Yang, Jelena Klinovaja, Peter Stano, and Daniel Loss.
Phys. Rev. B 94, 075416 (2016)
We show that fractional charges can be realized at the boundaries of a linear array of tunnel coupled quantum dots in the presence of a periodically modulated onsite potential. While the charge fractionalization mechanism is similar to the one in polyacetylene, here the values of fractional charges can be tuned to arbitrary values by varying the phase of the onsite potential or the total number of dots in the array. We also find that the fractional boundary charges, unlike the ingap bound states, are stable against static random disorder. We discuss the minimum array size where fractional boundary charges can be observed.

Universal Quantum Computation with Hybrid SpinMajorana Qubits
Silas Hoffman, Constantin Schrade, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 94, 045316 (2016)
We theoretically propose a set of universal quantum gates acting on a hybrid qubit formed by coupling a quantum dot spin qubit and Majorana fermion qubit. First, we consider a quantum dot tunnelcoupled to two topological superconductors. The effective spinMajorana exchange facilitates a hybrid CNOT gate for which either qubit can be the control or target. The second setup is a modular scalable network of topological superconductors and quantum dots. As a result of the exchange interaction between adjacent spin qubits, a CNOT gate is implemented that acts on neighboring Majorana qubits, and eliminates the necessity of interqubit braiding. In both setups the spinMajorana exchange interaction allows for a phase gate, acting on either the spin or the Majorana qubit, and for a SWAP or hybrid SWAP gate which is sufficient for universal quantum computation without projective measurements.

Majorana bound states in magnetic skyrmions
Guang Yang, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 93, 224505 (2016)
Magnetic skyrmions are highly mobile nanoscale topological spin textures. We show, both analytically and numerically, that a magnetic skyrmion of an even azimuthal winding number placed in proximity to an swave superconductor hosts a zeroenergy Majorana bound state in its core, when the exchange coupling between the itinerant electrons and the skyrmion is strong. This Majorana bound state is stabilized by the presence of a spinorbit interaction. We propose the use of a superconducting trijunction to realize nonAbelian statistics of such Majorana bound states.

Topological Phases of Inhomogeneous Superconductivity
Silas Hoffman, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 93, 165418 (2016)
We theoretically consider the effect of a spatially periodic modulation of the superconducting order parameter on the formation of Majorana fermions induced by a onedimensional system with magnetic impurities brought into close proximity to an swave superconductor. When the magnetic exchange energy is larger than the interimpurity electron hopping we model the effective system as a chain of coupled Shiba states. While in the opposite regime, the effective system is accurately described by a quantum wire model. Upon including a spatially modulated superconducting pairing, we find, for sufficiently large magnetic exchange energy, the system is able to support a single pair of Majorana fermions with one Majorana fermion on the left end of the system and one on the right end. When the modulation of superconductivity is large compared to the magnetic exchange energy, the Shiba chain returns to a trivially gapped regime while the quantum wire enters a new topological phase capable of supporting two pairs of Majorana fermions.

Chiral and NonChiral Edge States in Quantum Hall Systems with Charge Density Modulation
Pawel Szumniak, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 93, 245308 (2016)
We consider a system of weakly coupled wires with quantum Hall effect (QHE) and in the presence of a spatially periodic modulation of the chemical potential along the wire, equivalent to a charge density wave (CDW). We investigate the competition between the two effects which both open a gap. We show that by changing the ratio between the amplitudes of the CDW modulation and the tunneling between wires, one can switch between nontopological CDWdominated phase to topological QHEdominated phase. Both phases host edge states of chiral and nonchiral nature robust to onsite disorder. However, only in the topological phase, the edge states are immune to disorder in the phase shifts of the CDWs. We provide analytical solutions for filling factor n=1 and study numerically effects of disorder as well as present numerical results for higher filling factors.

Theory of time reversal topological superconductivity in double Rashba wires  symmetries of Cooper pair and Andreev bound states
Hiromi Ebisu, Bo Lu, Jelena Klinovaja, and Yukio Tanaka.
Prog. Theor. Exp. Phys. 083I01 (2016)
We study the system of double Rashba wires brought into the proximity to an swave superconductor. The time reversal invariant topological superconductivity is realized if the interwire pairing corresponding to crossed Andreev reflection dominates over the standard intrawire pairing. We derive the topological criterion and show that the system hosts zero energy Andreev bound states such as a Kramers pair of Majorana fermions. We classify symmetry of the Cooper pairs focusing on the four degrees of freedom, i.e., frequency, spin, spatial parity inside wires, and spatial parity between wires. The magnitude of the oddfrequency pairing is strongly enhanced in the topological state. We also explore properties of junctions occurring in such double wire systems. If one section of the junction is in the topological state and the other is in the trivial state, the energy dispersion of Andreev bound states is proportional to \pm sin(\phi), where \phi denotes the macroscopic phase difference between two sections. This behavior can be intuitively explained by the couplings of a Kramers pair of Majorana fermions and spinsinglet swave Cooper pair and can also be understood by analyzing an effective continuum model of the s+p/swave superconductor hybrid system.

From Coupled Rashba Electron and Hole Gas Layers to 3D Topological Insulators
Luka Trifunovic, Daniel Loss, and Jelena Klinovaja.
Phys. Rev. B 93, 205406 (2016)
We introduce a system of stacked twodimensional electron and hole gas layers with Rashba spin orbit interaction and show that the tunnel coupling between the layers induces a strong threedimensional (3D) topological insulator phase. At each of the twodimensional bulk boundaries we find the spectrum consisting of a single anistropic Dirac cone, which we show by analytical and numerical calculations. Our setup has a unitcell consisting of four tunnel coupled Rashba layers and presents a synthetic strong 3D topological insulator and is distinguished by its rather high experimental feasibility.

Topological Floquet Phases in Driven Coupled Rashba Nanowires
Jelena Klinovaja, Peter Stano, and Daniel Loss.
Phys. Rev. Lett. 116, 176401 (2016)
We consider periodicallydriven arrays of weakly coupled wires with conduction and valence bands of Rashba type and study the resulting Floquet states. This nonequilibrium system can be tuned into nontrivial phases such as of topological insulators, Weyl semimetals, and dispersionless zeroenergy edge mode regimes. In the presence of strong electronelectron interactions, we generalize these regimes to the fractional case, where elementary excitations have fractional charges e/m with m being an odd integer.

LongDistance Entanglement of Spin Qubits via Quantum Hall Edge States
Guang Yang, ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 93, 075301 (2016)
The implementation of a functional quantum computer involves entangling and coherent manipulation of a large number of qubits. For qubits based on electron spins confined in quantum dots, which are among the most investigated solidstate qubits at present, architectural challenges are often encountered in the design of quantum circuits attempting to assemble the qubits within the very limited space available. Here, we provide a solution to such challenges based on an approach to realizing entanglement of spin qubits over long distances. We show that longrange RudermanKittelKasuyaYosida interaction of confined electron spins can be established by quantum Hall edge states, leading to an exchange coupling of spin qubits. The coupling is anisotropic and can be either Isingtype or XYtype, depending on the spin polarization of the edge state. Such a property, combined with the dependence of the electron spin susceptibility on the chirality of the edge state, can be utilized to gain valuable insights into the topological nature of various quantum Hall states.

Antiferromagnetic nuclear spin helix and topological superconductivity in $^{13}$C nanotubes
ChenHsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Phys. Rev B 92, 235435 (2015)
We investigate the RKKY interaction arising from the hyperfine coupling between localized nuclear spins and conduction electrons in interacting 13C carbon nanotubes. Using the Luttinger liquid formalism, we show that the RKKY interaction is sublattice dependent, consistent with the spin susceptibility calculation in noninteracting carbon nanotubes, and it leads to an antiferromagnetic nuclear spin helix in finitesize systems. The transition temperature reaches up to tens of millikelvins, due to a strong boost by a positive feedback through the Overhauser field from ordered nuclear spins. Similar to GaAs nanowires, the formation of the helical nuclear spin order gaps out half of the conduction electrons, and is therefore observable as a reduction of conductance by a factor of two in a transport experiment. The nuclear spin helix leads to a density wave combining spin and charge degrees of freedom in the electron subsystem, resulting in synthetic spinorbit interaction, which induces nontrivial topological phases. As a result, topological superconductivity with Majorana fermion bound states can be realized in the system in the presence of proximityinduced superconductivity without the need of fine tuning the chemical potential. We present the phase diagram as function of system parameters, including the pairing gaps, the gap due to the nuclear spin helix, and the Zeeman field perpendicular to the helical plane.

Supercurrent Reversal in TwoDimensional Topological Insulators
Alexander Zyuzin, Mohammad Alidoust, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 92, 174515 (2015)
We theoretically demonstrate that a supercurrent across a twodimensional topological insulator subjected to an external magnetic field unambiguously reveals the existence of edgemode superconductivity. When the edge states of a narrow twodimensional topological insulator are hybridized, an external magnetic field can close the hybridization gap, thus driving a quantum phase transition from insulator to semimetal states of the topological insulator. Importantly, we find a sign reversal of the supercurrent at the quantum phase transition which offers a simple and experimentally feasible way to observe intrinsic properties of topological insulators including edgemode superconductivity.

Proximityinduced Josephson $\pi$Junctions in Topological Insulators
Constantin Schrade, A. A. Zyuzin, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Lett. 115, 237001 (2015)
We study two microscopic models of topological insulators in contact with an swave superconductor. In the first model the superconductor and the topological insulator are tunnel coupled via a layer of scalar and of randomly oriented spin impurities. Here, we require that spinflip tunneling dominates over spinconserving one. In the second model the tunnel coupling is realized by an array of singlelevel quantum dots with randomly oriented spins. It is shown that the tunnel region forms a $\pi$junction where the effective order parameter changes sign. Interestingly, due to the random spin orientation the effective descriptions of both models exhibit timereversal symmetry. We then discuss how the proposed $\pi$junctions support topological superconductivity without magnetic fields and can be used to generate and manipulate Kramers pairs of Majorana fermions by gates.

Probing Atomic Structure and Majorana Wavefunctions in MonoAtomic Fechains on Superconducting PbSurface
Remy Pawlak, Marcin Kisiel, Jelena Klinovaja, Tobias Meier, Shigeki Kawai, Thilo Glatzel, Daniel Loss, and Ernst Meyer.
npj Quantum Information 2, 16035 (2016)
Motivated by the striking promise of quantum computation, Majorana bound states (MBSs) in solidstate systems have attracted wide attention in recent years. In particular, the wavefunction localization of MBSs is a key feature and crucial for their future implementation as qubits. Here, we investigate the spatial and electronic characteristics of topological superconducting chains of iron atoms on the surface of Pb(110) by combining scanning tunneling microscopy (STM) and atomic force microscopy (AFM). We demonstrate that the Fe chains are monoatomic, structured in a linear fashion, and exhibit zerobias conductance peaks at their ends which we interprete as signature for a Majorana bound state. Spatially resolved conductance maps of the atomic chains reveal that the MBSs are well localized at the chain ends (below 25 nm), with two localization lengths as predicted by theory. Our observation lends strong support to use MBSs in Fe chains as qubits for quantum computing devices.

Fractional Charge and Spin States in Topological Insulator Constrictions
Jelena Klinovaja and Daniel Loss.
Phys. Rev. B 92 121410(R) (2015)
We investigate theoretically properties of twodimensional topological insulator constrictions both in the integer and fractional regimes. In the presence of a perpedicular magnetic field, the constriction functions as a spin filter with nearperfect efficiency and can be switched by electric fields only. Domain walls between different topological phases can be created in the constriction as an interface between tunneling, magnetic fields, charge density wave, or electronelectron interactions dominated regions. These domain walls host nonAbelian bound states with fractional charge and spin and result in degenerate ground states with parafermions. If a proximity gap is induced bound states give rise to an exotic Josephson current with 8\piperidiodicity.

Impurity Induced Quantum Phase Transitions and Magnetic Order in Conventional Superconductors: Competition between Bound and Quasiparticle states
Silas Hoffman, Jelena Klinovaja, Tobias Meng, and Daniel Loss.
Phys. Rev. B 92, 125422 (2015)
We theoretically study bound states generated by magnetic impurities within conventional swave superconductors, both analytically and numerically. In determining the effect of the hybridization of two such bound states on the energy spectrum as a function of magnetic exchange coupling, relative angle of magnetization, and distance between impurities, we find that quantum phase transitions can be modulated by each of these parameters. Accompanying such transitions, there is a change in the preferred spin configuration of the impurities. Although the interaction between the impurity spins is overwhelmingly dominated by the quasiparticle contribution, the ground state of the system is determined by the bound state energies. Selfconsistently calculating the superconducting order parameter, we find a discontinuity when the system undergoes a quantum phase transition as indicated by the bound state energies.

Superconducting Gap Renomalization around two Magnetic Impurities: From Shiba to Andreev Bound States
Tobias Meng, Jelena Klinovaja, Silas Hoffman, Pascal Simon, and Daniel Loss.
Phys. Rev. B 92, 064503 (2015)
We study the renormalization of the gap of an swave superconductor in the presence of two magnetic impurities. For weakly bound Shiba states, we analytically calculate the part of the gap renormalization that is sensitive to the relative orientation of the two impurity spins. For strongly exchange coupled impurities, a quantum phase transition from a subgap Shiba state to a supragap Andreev state is identified and discussed by solving the gap equation selfconsistently by numerics.

Integer and Fractional Quantum Anomalous Hall Effect in a Strip of Stripes Model
Jelena Klinovaja, Yaroslav Tserkovnyak, and Daniel Loss.
Phys. Rev. B 91, 085426 (2015)
We study the quantum anomalous Hall effect in a strip of stripes model coupled to a magnetic texture with zero total magnetization and in the presence of strong electronelectron interactions. A helical magnetization along the stripes and a spinselective coupling between the stripes gives rise to a bulk gap and chiral edge modes. Depending on the ratio between the period of the magnetic structure and the Fermi wavelength, the system can exhibit the integer or fractional quantum anomalous Hall effect. In the fractional regime, the quasiparticles have fractional charges and nontrivial Abelian braid statistics.

Majorana Fermions in Ge/Si Hole Nanowires
Franziska Maier, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 90, 195421 (2014)
We investigate theoretically the longdistance coupling and spin exchange in an array of quantum dot spin qubits in the presence of microwaves. We find that photon assisted cotunneling is boosted at resonances between photon and energies of virtually occupied excited states and show how to make it spin selective. We identify configurations that enable fast switching and spin echo sequences for efficient and nonlocal manipulation of spin qubits. We devise configurations in which the nearresonantly boosted cotunneling provides nonlocal coupling which, up to certain limit, does not diminish with distance between the manipulated dots before it decays weakly with inverse distance.

Fast LongDistance Control of Spin Qubits by Photon Assisted Cotunneling
Peter Stano, Jelena Klinovaja, Floris R. Braakman, Lieven M. K. Vandersypen, and Daniel Loss.
Phys. Rev. B 92, 075302 (2015)
We investigate theoretically the longdistance coupling and spin exchange in an array of quantum dot spin qubits in the presence of microwaves. We find that photon assisted cotunneling is boosted at resonances between photon and energies of virtually occupied excited states and show how to make it spin selective. We identify configurations that enable fast switching and spin echo sequences for efficient and nonlocal manipulation of spin qubits. We devise configurations in which the nearresonantly boosted cotunneling provides nonlocal coupling which, up to certain limit, does not diminish with distance between the manipulated dots before it decays weakly with inverse distance.

Fermionic and Majorana Bound States in Hybrid Nanowires with NonUniform SpinOrbit Interaction
Jelena Klinovaja and Daniel Loss.
Eur. Phys. J. B 88, 62 (2015)
We study intragap bound states in the topological phase of a Rashba nanowire in the presence of a magnetic field and with nonuniform spin orbit interaction (SOI) and proximityinduced superconductivity gap. We show that fermionic bound states (FBS) can emerge inside the proximity gap. They are localized at the junction between two wire sections characterized by different directions of the SOI vectors, and they coexist with Majorana bound states (MBS) localized at the nanowire ends. The energy of the FBS is determined by the angle between the SOI vectors and the lengthscale over which the SOI changes compared to the Fermi wavelength and the localization length. We also consider doublejunctions and show that the two emerging FBSs can hybridize and form a double quantum dotlike structure inside the gap. We find explicit analytical solutions of the bound states and their energies for certain parameter regimes such as weak and strong SOI. The analytical results are confirmed and complemented by an independent numerical tightbinding model approach. Such FBS can act as quasiparticle traps and thus can have implications for topological quantum computing schemes based on braiding MBSs.

Helical nuclear spin order in a strip of stripes in the Quantum Hall regime
Tobias Meng, Peter Stano, Jelena Klinovaja, and Daniel Loss.
Eur. Phys. J. B 87, 203 (2014)
We investigate nuclear spin effects in a twodimensional electron gas in the quantum Hall regime modeled by a weakly coupled array of interacting quantum wires. We show that the presence of hyperfine interaction between electron and nuclear spins in such wires can induce a phase transition, ordering electrons and nuclear spins into a helix in each wire. Electronelectron interaction effects, pronounced within the onedimensional stripes, boost the transition temperature up to tens to hundreds of millikelvins in GaAs. We predict specific experimental signatures of the existence of nuclear spin order, for instance for the resistivity of the system at transitions between different quantum Hall plateaus.

Quantum Spin Hall Effect in Strip of Stripes Model
Jelena Klinovaja and Yaroslav Tserkovnyak.
Phys. Rev. B 90, 115426 (2014)
We consider quantum spin Hall effect in an anisotropic strip of stripes and address both integer and fractional filling factors. The first model is based on a gradient of spinorbit interaction in the direction perpendicular to the stripes. The second model is based on two weakly coupled strips with reversed dispersion relations. We demonstrate that these systems host helical modes, modes in which opposite spins propagate in opposite directions. In the integer regime, the modes carry an elementary electron charge whereas in the fractional regime they carry fractional charges, and their excitations possess anyonic braiding statistics. These simple quasionedimensional models can serve as a platform for understanding effects arising due to electronelectron correlations in topological insulators.

Kramers Pairs of Majorana Fermions and Parafermions in Fractional Topological Insulators
Jelena Klinovaja, Amir Yacoby, and Daniel Loss.
Phys. Rev. B 90, 155447 (2014)
We propose a scheme based on topological insulators to generate Kramers pairs of Majorana fermions or parafermions in the complete absence of magnetic fields. Our setup consists of two topological insulators whose edge states are brought close to an swave superconductor. The resulting proximity effect leads to an interplay between a nonlocal crossed Andreev pairing, which is dominant in the strong electronelectron interaction regime, and usual superconducting pairing, which is dominant at large separation between the two topological insulator edges. As a result, there are zeroenergy bound states localized at interfaces between spatial regions dominated by the two different types of pairing. Due to the preserved timereversal symmetry, the bound states come in Kramers pairs. If the topological insulators carry fractional edge states, the zeroenergy bound states are parafermions, otherwise, they are Majorana fermions.

Renormalization of anticrossings in interacting quantum wires with Rashba and Dresselhaus spinorbit couplings
Tobias Meng, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 89, 205133 (2014)
We discuss how electronelectron interactions renormalize the spinorbit induced anticrossings between different subbands in ballistic quantum wires. Depending on the ratio of spinorbit coupling and subband spacing, electronelectron interactions can either increase or decrease anticrossing gaps. When the anticrossings are closing due to a special combination of Rashba and Dresselhaus spinorbit couplings, their gap approaches zero as an interaction dependent power law of the spinorbit couplings, which is a consequence of Luttinger liquid physics. Monitoring the closing of the anticrossings allows to directly measure the related renormalization group scaling dimension in an experiment. If a magnetic field is applied parallel to the spinorbit coupling direction, the anticrossings experience different renormalizations. Since this difference is entirely rooted in electronelectron interactions, unequally large anticrossings also serve as a direct signature of Luttinger liquid physics. Electronelectron interactions furthermore increase the sensitivity of conductance measurements to the presence of anticrossing.

TimeReversal Invariant Parafermions in Interacting Rashba Nanowires
Jelena Klinovaja and Daniel Loss.
Phys. Rev. B 90, 045118 (2014)
We propose a scheme to generate pairs of timereversal invariant parafermions. Our setup consists of two quantum wires with opposite Rashba spin orbit interactions coupled to an swave superconductor, in the presence of electronelectron interactions. The zeroenergy bound states localized at the wire ends arise from the interplay between two types of proximity induced superconductivity: the usual intrawire superconductivity and the interwire superconductivity due to crossed Andreev reflections. If the latter dominates, which is the case for strong electronelectron interactions, the system supports Kramers pair of parafermions. Moreover, the scheme can be extended to a twodimensional sea of timereversal invariant parafermions.

Parafermions in Interacting Nanowire Bundle
Jelena Klinovaja and Daniel Loss.
Phys. Rev. Lett. 112, 246403 (2014)
We propose a scheme to induce Z_3 parafermion modes, exotic zeroenergy bound states that possess nonAbelian statistics. We consider a minimal setup consisting of a bundle of four tunnel coupled nanowires hosting spinless electrons that interact strongly with each other. The hallmark of our setup is that it relies only on simple onedimensional wires, uniform magnetic fields, and strong interactions, but does not require the presence of superconductivity or exotic quantum Hall phases.

Transport signature of fractional Fermions in Rashba nanowires
Diego Rainis, Arijit Saha, Jelena Klinovaja, Luka Trifunovic, and Daniel Loss.
Phys. Rev. Lett. 112, 196803 (2014)
We study theoretically transport through a semiconducting nanowire (NW) in the presence of Rashba spin orbit interaction, uniform magnetic field, and spatially modulated magnetic field. The system is fully gapped, and the interplay between the spin orbit interaction and the magnetic fields leads to fractionally charged fermion (FF) bound states of JackiwRebbi type at each end of the nanowire. We investigate the transport and noise behavior of a N/NW/N system, where the wire is contacted by two normal leads (N), and we look for possible signatures that could help in the experimental detection of such states. We find that the differential conductance and the shot noise exhibit a subgap structure which fully reveals the presence of the FF state. Our predictions can be tested in standard twoterminal measurements through InSb/InAs nanowires.

Topological Superconductivity and Majorana Fermions in RKKY Systems
Jelena Klinovaja, Peter Stano, Ali Yazdani, and Daniel Loss.
Phys. Rev. Lett. 111, 186805 (2013)
We consider quasi onedimensional RKKY systems in proximity to an swave superconductor. We show that a $2k_F$peak in the spin susceptibility of the superconductor in the onedimensional limit supports helical order of localized magnetic moments via RKKY interaction, where $k_F$ is the Fermi wavevector. The magnetic helix is equivalent to a uniform magnetic field and very strong spinorbit interaction (SOI) with an effective SOI length $1/2k_F$. We find the conditions to establish such a magnetic state in atomic chains and semiconducting nanowires with magnetic atoms or nuclear spins. Generically, these systems are in a topological phase with Majorana fermions. The inherent selftuning of the helix to $2k_F$ eliminates the need to tune the chemical potential.

Correlations between Majorana fermions through a superconductor
A.A. Zyuzin, Diego Rainis, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. Lett. 111, 056802 (2013)
We consider a model of ballistic quasione dimensional semiconducting wire with intrinsic spinorbit interaction placed on the surface of a bulk swave superconductor (SC), in the presence of an external magnetic field. This setup has been shown to give rise to a topological superconducting state in the wire, characterized by a pair of Majoranafermion (MF) bound states formed at the two ends of the wire. Here we demonstrate that, besides the wellknown direct overlapinduced energy splitting, the two MF bound states may hybridize via elastic correlated tunneling processes through virtual quasiparticles states in the SC, giving rise to an additional energy splitting between MF states from the same as well as from different wires.

Integer and Fractional Quantum Hall Effect in a Strip of Stripes
Jelena Klinovaja and Daniel Loss.
Eur. Phys. J. B 87, 171 (2014)
We study anisotropic stripe models of interacting electrons in the presence of magnetic fields in the quantum Hall regime with integer and fractional filling factors. The model consists of an infinite strip of finite width that contains periodically arranged stripes (forming supercells) to which the electrons are confined and between which they can hop with associated magnetic phases. The interacting electron system within the onedimensional stripes are described by Luttinger liquids and shown to give rise to charge and spin density waves that lead to periodic structures within the stripe with a reciprocal wavevector 8k_F. This wavevector gives rise to Umklapp scattering and resonant scattering that results in gaps and chiral edge states at all known integer and fractional filling factors \nu. The integer and odd denominator filling factors arise for a uniform distribution of stripes, whereas the even denominator filling factors arise for a nonuniform stripe distribution. We calculate the Hall conductance via the Streda formula and show that it is given by \sigma_H=\nu e^2/h for all filling factors. We show that the composite fermion picture follows directly from the condition of the resonant Umklapp scattering.

Spintronics in MoS_2 monolayer quantum wires
Jelena Klinovaja and Daniel Loss.
Phys. Rev. B 88, 075404 (2013)
We study analytically and numerically spin effects in MoS_2 monolayer armchair quantum wires and quantum dots. The interplay between intrinsic and Rashba spin orbit interactions induced by an electric field leads to helical modes, giving rise to spin filtering in timereversal invariant systems. The Rashba spin orbit interaction can also be generated by spatially varying magnetic fields. In this case, the system can be in a helical regime with nearly perfect spin polarization. If such a quantum wire is brought into proximity to an swave superconductor, the system can be tuned into a topological phase, resulting in midgap Majorana fermions localized at the wire ends.

Local Spin Susceptibilities of LowDimensional Electron Systems
Peter Stano, Jelena Klinovaja, Amir Yacoby, and Daniel Loss.
Phys. Rev. B 88, 045441 (2013)
We investigate, assess, and suggest possibilities for a measurement of the local spin susceptibility of a conducting lowdimensional electron system. The basic setup of the experiment we envisage is a sourceprobe one. Locally induced spin density (e.g. by a magnetized atomic force microscope tip) extends in the medium according to its spin susceptibility. The induced magnetization can be detected as a dipolar magnetic field, for instance, by an ultrasensitive nitrogenvacancy center based detector, from which the spatial structure of the spin susceptibility can be deduced. We find that onedimensional systems, such as semiconducting nanowires or carbon nanotubes, are expected to yield a measurable signal. The signal in a twodimensional electron gas is weaker, though materials with high enough $g$factor (such as InGaAs) seem promising for successful measurements.

Topological Edge States and Fractional Quantum Hall Effect from Umklapp Scattering
Jelena Klinovaja and Daniel Loss.
Phys. Rev. Lett. 111, 196401 (2013)
We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant Umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. These gaps result in plateaus for the Hall conductivity exactly at the known fillings n/m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices.

Fractional Fermions with NonAbelian Statistics
Jelena Klinovaja and Daniel Loss.
Phys. Rev. Lett. 110, 126402 (2013)
We introduce a novel class of lowdimensional topological tightbinding models that allow for bound states that are fractionally charged fermions and exhibit nonAbelian braiding statistics. The proposed model consists of a double (single) ladder of spinless (spinful) fermions in the presence of magnetic fields. We study the system analytically in the continuum limit as well as numerically in the tightbinding representation. We find a topological phase transition with a topological gap that closes and reopens as a function of system parameters and chemical potential. The topological phase is of the type BDI and carries two degenerate midgap bound states that are localized at opposite ends of the ladders. We show numerically that these bound states are robust against a wide class of perturbations.

RKKY interaction in carbon nanotubes and graphene nanoribbons
Jelena Klinovaja and Daniel Loss.
Phys. Rev. B 87, 045422 (2013)
We study RudermannKittelKasuyaYosida (RKKY) interaction in carbon nanotubes (CNTs) and graphene nanoribbons in the presence of spin orbit interactions and magnetic fields. For this we evaluate the static spin susceptibility tensor in real space in various regimes at zero temperature. In metallic CNTs the RKKY interaction depends strongly on the sublattice and, at the Dirac point, is purely ferromagnetic (antiferromagnetic) for the localized spins on the same (different) sublattice, whereas in semiconducting CNTs the spin susceptibility depends only weakly on the sublattice and is dominantly ferromagnetic. The spin orbit interactions break the SU(2) spin symmetry of the system, leading to an anisotropic RKKY interaction of Ising and MoryiaDzyaloshinsky form, besides the usual isotropic Heisenberg interaction. All these RKKY terms can be made of comparable magnitude by tuning the Fermi level close to the gap induced by the spin orbit interaction. We further calculate the spin susceptibility also at finite frequencies and thereby obtain the spin noise in real space via the fluctuationdissipation theorem.

Giant spin orbit interaction due to rotating magnetic fields in graphene nanoribbons
Jelena Klinovaja and Daniel Loss.
Phys. Rev. X 3, 011008 (2013)
We theoretically study graphene nanoribbons in the presence of spatially varying magnetic fields produced e.g. by nanomagnets. We show both analytically and numerically that an exceptionally large Rashba spin orbit interaction (SOI) of the order of 10 meV can be produced by the nonuniform magnetic field. As a consequence, helical modes exist in armchair nanoribbons that exhibit nearly perfect spin polarization and are robust against boundary defects. This paves the way to realizing spin filter devices in graphene nanoribbons in the temperature regime of a few Kelvins. If a nanoribbon in the helical regime is in proximity contact to an swave superconductor, the nanoribbon can be tuned into a topological phase sustaining Majorana fermions.

Helical States in Curved Bilayer Graphene
Jelena Klinovaja, Gerson J. Ferreira, and Daniel Loss.
Phys. Rev. B 86, 235416 (2012)
We study spin effects of quantum wires formed in bilayer graphene by electrostatic confinement. With a proper choice of the confinement direction, we show that in the presence of magnetic field, spinorbit interaction induced by curvature, and intervalley scattering, bound states emerge that are helical. The localization length of these helical states can be modulated by the gate voltage which enables the control of the tunnel coupling between two parallel wires. Allowing for proximity effect via an swave superconductor, we show that the helical modes give rise to Majorana fermions in bilayer graphene.

Transition from fractional to Majorana fermions in Rashba nanowires
Jelena Klinovaja, Peter Stano, and Daniel Loss.
Phys. Rev. Lett. 109, 236801 (2012)
We study hybrid superconductingsemiconducting nanowires in the presence of Rashba spinorbit interaction as well as helical magnetic fields. We show that the interplay between them leads to a competition of phases with two topological gaps closing and reopening, resulting in unexpected reentrance behavior. Besides the topological phase with localized Majorana fermions (MFs) we find new phases characterized by fractionally charged fermion (FF) bound states of JackiwRebbi type. The system can be fully gapped by the magnetic fields alone, giving rise to FFs that transmute into MFs upon turning on superconductivity. We find explicit analytical solutions for MF and FF bound states and determine the phase diagram numerically by determining the corresponding Wronskian null space. We show by renormalization group arguments that electronelectron interactions enhance the Zeeman gaps opened by the fields.

Towards a realistic transport modeling for a superconducting nanowire with Majorana fermions
Diego Rainis, Luka Trifunovic, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 87, 024515 (2013)
Motivated by recent experiments searching for Majorana fermions (MFs) in hybrid semiconductingsuperconducting nanostructures and by subsequent theoretical interpretations, we consider the so far most realistic model (including disorder) and analyze its transport behavior numerically. In particular, we include in the model superconducting contacts used in the experiments to extract the current. We show that important new features emerge that are absent in simpler models, such as the enhanced visibility of the topological gap for increased spinorbit interaction. We find oscillations of the zero bias peak as function of magnetic field and explain their origin. Even taking into account all the possible (known) ingredients of the experiments and exploring many parameter regimes for MFs, we are not able to reach a satisfactory agreement with the reported data. Thus, a different physical origin for the observed zerobias peak cannot be excluded.

Exchangebased CNOT gates for singlettriplet qubits with spin orbit interaction
Jelena Klinovaja, Dimitrije Stepanenko, Bertrand I. Halperin, and Daniel Loss.
Phys. Rev. B 86, 085423 (2012)
We propose a scheme for implementing the CNOT gate over qubits encoded in a pair of electron spins in a double quantum dot. The scheme is based on exchange and spin orbit interactions and on local gradients in Zeeman fields. We find that the optimal device geometry for this implementation involves effective magnetic fields that are parallel to the symmetry axis of the spin orbit interaction. We show that the switching times for the CNOT gate can be as fast as a few nanoseconds for realistic parameter values in GaAs semiconductors. Guided by recent advances in surface codes, we also consider the perpendicular geometry. In this case, leakage errors due to spin orbit interaction occur but can be suppressed in strong magnetic fields.

Composite Majorana Fermion Wavefunctions in Nanowires
Jelena Klinovaja and Daniel Loss.
Phys. Rev. B 86, 085408 (2012)
We consider Majorana fermions (MFs) in quasionedimensional nanowire systems containing normal and superconducting sections where the topological phase based on Rashba spin orbit interaction can be tuned by magnetic fields. We derive explicit analytic solutions of the MF wavefunction in the weak and strong spin orbit interaction regimes. We find that the wavefunction for one single MF is a composite object formed by superpositions of different MF wavefunctions which have nearly disjoint supports in momentum space. These contributions are coming from the extrema of the spectrum, one centered around zero momentum and the other around the two Fermi points. As a result, the various MF wavefunctions have different localization lengths in real space and interference among them leads to pronounced oscillations of the MF probability density. For a transparent normalsuperconducting junction we find that in the topological phase the MF leaks out from the superconducting into the normal section of the wire and is delocalized over the entire normal section, in agreement with recent numerical results by Chevallier et al. (arXiv:1203.2643).

Electricfieldinduced Majorana Fermions in Armchair Carbon Nanotubes
Jelena Klinovaja, Suhas Gangadharaiah, and Daniel Loss.
Phys. Rev. Lett. 108, 196804 (2012)
We consider theoretically an armchair carbon nanotube (CNT) in the presence of an electric field and in contact with an swave superconductor. We show that the proximity effect opens up superconducting gaps in the CNT of different strengths for the exterior and interior branches of the two Dirac points. For strong proximity induced superconductivity the interior gap can be of the pwave type, while the exterior gap can be tuned by the electric field to be of the swave type. Such a setup supports a single Majorana bound state at each end of the CNT. In the case of a weak proximity induced superconductivity, the gaps in both branches are of the pwave type. However, the temperature can be chosen in such a way that the smallest gap is effectively closed. Using renormalization group techniques we show that the Majorana bound states exist even after taking into account electronelectron interactions.

Carbon nanotubes in electric and magnetic fields
Jelena Klinovaja, Manuel J. Schmidt, Bernd Braunecker, and Daniel Loss.
Phys. Rev. B 84, 085452 (2011)
We derive an effective lowenergy theory for metallic (armchair and nonarmchair) singlewall nanotubes in the presence of an electric field perpendicular to the nanotube axis, and in the presence of magnetic fields, taking into account spinorbit interactions and screening effects on the basis of a microscopic tightbinding model. The interplay between electric field and spinorbit interaction allows us to tune armchair nanotubes into a helical conductor in both Dirac valleys. Metallic nonarmchair nanotubes are gapped by the surface curvature, yet helical conduction modes can be restored in one of the valleys by a magnetic field along the nanotube axis. Furthermore, we discuss electric dipole spin resonance in carbon nanotubes, and find that the Rabi frequency shows a pronounced dependence on the momentum along the nanotube.

Helical modes in carbon nanotubes generated by strong electric fields
Jelena Klinovaja, Manuel J. Schmidt, Bernd Braunecker, and Daniel Loss.
Phys. Rev. Lett. 106, 156809 (2011)
Helical modes, conducting opposite spins in opposite directions, are shown to exist in metallic armchair nanotubes in an allelectric setup. This is a consequence of the interplay between spinorbit interaction and strong electric fields. The helical regime can also be obtained in chiral metallic nanotubes by applying an additional magnetic field. In particular, it is possible to obtain helical modes at one of the two Dirac points only, while the other one remains gapped. Starting from a tightbinding model we derive the effective lowenergy Hamiltonian and the resulting spectrum.

Spinselective Peierls transition in interacting onedimensional conductors with spinorbit interaction
Bernd Braunecker, George I. Japaridze, Jelena Klinovaja, and Daniel Loss.
Phys. Rev. B 82, 045127 (2010)
Interacting onedimensional conductors with Rashba spinorbit coupling are shown to exhibit a spinselective Peierlstype transition into a mixed spinchargedensitywave state. The transition leads to a gap for onehalf of the conducting modes, which is strongly enhanced by electronelectron interactions. The other half of the modes remains in a strongly renormalized gapless state and conducts opposite spins in opposite directions, thus providing a perfect spin filter. The transition is driven by magnetic field and by spinorbit interactions. As an example we show for semiconducting quantum wires and carbon nanotubes that the gap induced by weak magnetic fields or intrinsic spinorbit interactions can get renormalized by 1 order of magnitude up to 10  30 K.

A mixed scenario for the reconstruction of a charged helium surface
Valerii Shikin and Elena Klinovaya.
Low Temp. Phys. 36, 142 (2010)
A mixed scenario for the periodic reconstruction of a charged surface of a liquid when the liquid is close to occupancy saturation by 2D charges is discussed. It is shown that the unit cell of the periodic structure arising is a modified multicharged dimple.