Manisha Thakurathi
ContactDepartment of PhysicsUniversity of Basel Klingelbergstrasse 82 CH4056 Basel, Switzerland

Education
20072009: Master of Science in Physics, G.B. Pant University, Pantnagar, India20102016: PhD, Indian Institute of Science, Bangalore, India with Prof. Diptiman Sen
April 2016present: Postdoc, University of Basel, Switzerland with Prof. Jelena Klinovaja  Prof. Daniel Loss
Publications
Show all abstracts.1.  Transport across a system with three pwave superconducting wires: effects of Majorana modes and interactions 
Oindrila Deb, Manisha Thakurathi, Diptiman Sen Eur. Phys. J. B (2016) 89: 19 ; arxiv 1508.00819.
We study the effects of Majorana modes and interactions between electrons on transport in a onedimensional system with a junction of three pwave superconductors (SCs) which are connected to normal metal leads. For sufficiently long SCs, there are zero energy Majorana modes at the junctions between the SCs and the leads,and, depending on the signs of the pwave pairings in the three SCs, there can also be one or three Majorana modes at the junction of the three SCs. We show that the various subgap conductances have peaks occurring at the energies of all these modes; we therefore get a rich pattern of conductance peaks. Next, we use a renormalization group approach to study the scattering matrix of the system at energies far from the SC gap. The fixed points of the renormalization group flows and their stabilities are studied; we find that the scattering matrix at the stable fixed point is highly symmetric even when the microscopic scattering matrix and the interaction strengths are not symmetric. We discuss the implications of this for the conductances. Finally we propose an experimental realization of this system which can produce different signs of the pwave pairings in the different SCs.
 
2.  Majorana modes and transport across junctions of superconductors and normal metals 
Manisha Thakurathi, Oindrila Deb, Diptiman Sen J. Phys. Condens. Matter 27 275702 (2015); arxiv 1412.0072.
We study Majorana modes and transport in onedimensional systems with a pwave superconductor (SC) and normal metal leads. For a system with a SC lying between two leads, it is known that there is a Majorana mode at the junction between the SC and each lead. If the pwave pairing Δ changes sign or if a strong impurity is present at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the subgap conductance between the leads and the SC. We derive an analytical expression as a function of Δ and the length L of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the shifts oscillate and decay exponentially as L is increased. The energy shifts exactly match the location of the peaks in the conductance. Using bosonization and the renormalization group method, we study the effect of interactions between the electrons on Δ and the strengths of an impurity inside the SC or the barriers between the SC and the leads; this in turn affects the Majorana modes and the conductance. Finally we propose a novel experimental realization of these systems, in particular of a system where Δ changes sign at one point inside the SC.
 
3.  Majorana edge modes in the Kitaev model 
Manisha Thakurathi, K. Sengupta, Diptiman Sen Phys. Rev. B 89, 235434; arxiv:1310.4701.
We study the Majorana modes, both equilibrium and Floquet, which can appear at the edges of the Kitaev model on the honeycomb lattice. We first present the analytical solutions known for the equilibrium Majorana edge modes for both zigzag and armchair edges of a semiinfinite Kitaev model and chart the parameter regimes of the model in which they appear. We then examine how edge modes can be generated if the Kitaev coupling on the bonds perpendicular to the edge is varied periodically in time as periodic δfunction kicks. We derive a general condition for the appearance and disappearance of the Floquet edge modes as a function of the drive frequency for a generic ddimensional integrable system. We confirm this general condition for the Kitaev model with a finite width by mapping it to a onedimensional model. Our numerical and analytical study of this problem shows that Floquet Majorana modes can appear on some edges in the kicked system even when the corresponding equilibrium Hamiltonian has no Majorana mode solutions on those edges. We support our analytical studies by numerics for finite sized system which show that periodic kicks can generate modes at the edges and the corners of the lattice.
 
4.  Majorana Fermions in superconducting wires: effects of longrange hopping, broken timereversal symmetry and potential landscapes 
Wade DeGottardi, Manisha Thakurathi, Smitha Vishveshwara, Diptiman Sen Phys. Rev. B 88, 165111 ; arxiv:1303.3304.
We present a comprehensive study of two of the most experimentally relevant extensions of Kitaev's spinless model of a 1D pwave superconductor: those involving (i) longer range hopping and superconductivity and (ii) inhomogeneous potentials. We commence with a pedagogical review of the spinless model and, as a means of characterizing topological phases exhibited by the systems studied here, we introduce bulk topological invariants as well as those derived from an explicit consideration of boundary modes. In timereversal invariant systems, we find that the longer range hopping leads to topological phases characterized by multiple Majorana modes. In particular, we investigate a spin model, which respects a duality and maps to a fermionic model with multiple Majorana modes; we highlight the connection between these topological phases and the broken symmetry phases in the original spin model. In the presence of timereversal symmetry breaking terms, we show that the topological phase diagram is characterized by an extended gapless regime. For the case of inhomogeneous potentials, we explore phase diagrams of periodic, quasiperiodic, and disordered systems. We present a detailed mapping between normal state localization properties of such systems and the topological phases of the corresponding superconducting systems. This powerful tool allows us to leverage the analyses of Hofstadter's butterfly and the vast literature on Anderson localization to the question of Majorana modes in superconducting quasiperiodic and disordered systems, respectively. We briefly touch upon the synergistic effects that can be expected in cases where longrange hopping and disorder are both present.
 
5.  Floquet generation of Majorana end modes and topological invariants 
Manisha Thakurathi, Aavishkar A. Patel, Diptiman Sen, Amit Dutta Phys. Rev. B 88, 155133; arxiv:1303.2300.
We show how Majorana end modes can be generated in a onedimensional system by varying some of the parameters in the Hamiltonian periodically in time. The specific model we consider is a chain containing spinless electrons with a nearestneighbor hopping amplitude, a pwave superconducting term and a chemical potential; this is equivalent to a spin1/2 chain with anisotropic XY couplings between nearest neighbors and a magnetic field applied in the zdirection. We show that varying the chemical potential (or magnetic field) periodically in time can produce Majorana modes at the ends of a long chain. We discuss two kinds of periodic driving, periodic deltafunction kicks and a simple harmonic variation with time. We discuss some distinctive features of the end modes such as the inverse participation ratio of their wave functions and their Floquet eigenvalues which are always equal to +/ 1 for timereversal symmetric systems. For the case of periodic deltafunction kicks, we use the effective Hamiltonian of a system with periodic boundary conditions to define two topological invariants. The first invariant is a wellknown winding number while the second invariant has not appeared in the literature before. The second invariant is more powerful in that it always correctly predicts the numbers of end modes with Floquet eigenvalues equal to +1 and 1, while the first invariant does not. We find that the number of end modes can become very large as the driving frequency decreases. We show that periodic deltafunction kicks in the hopping and superconducting terms can also produce end modes. Finally, we study the effect of electronphonon interactions (which are relevant at finite temperatures) and a random noise in the chemical potential on the Majorana modes.
 
6.  Fidelity susceptibility of onedimensional models with twisted boundary conditions 
Manisha Thakurathi, Diptiman Sen, Amit Dutta Phys. Rev. B 86, 245424; arxiv:1210.1382.
Recently it has been shown that the fidelity of the ground state of a quantum manybody system can be used to detect its quantum critical points (QCPs). If g denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for onedimensional models with large but finite size, the fidelity susceptibility χF can detect a QCP provided that the correlation length exponent satisfies ν < 2. We then show that χF can be used to locate a QCP even if ν ≥ 2
if we introduce boundary conditions labeled by a twist angle Nθ, where N is the system size. If the QCP lies at g = 0, we find that if N is kept constant, χF has a scaling form given by χF ∼ θ ^(−2/ν) f(g/θ^(1/ν) ) if θ ≪ 2π/N. We illustrate this both in a tightbinding model of fermions with a spatially varying chemical potential with amplitude h and period 2q in which ν = q, and in a XY spin1/2 chain in which ν = 2. Finally we show that when q is very large, the model has two additional QCPs at h = ±2 which cannot be detected by studying the energy spectrum but are clearly detected by χF . The peak value and width of χF seem to scale as nontrivial powers of q at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.
 
7.  Quenching across quantum critical points in periodic systems: dependence of scaling laws on periodicity 
Manisha Thakurathi, Wade DeGottardi, Diptiman Sen, Smitha Vishveshwara Phys. Rev. B 85, 165425; arxiv:1112.6092.
We study the quenching dynamics of a manybody system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin1/2 chain with equal xx and yy couplings and subject to a periodically varying magnetic field in the z direction or, equivalently, a tightbinding model of spinless fermions with a periodic local chemical potential, having period 2q, where q is a natural number. For a linear quench of the magnetic field strength (or potential strength) at rate 1/τ across a quantum critical point, we find that the density of
defects thereby produced scales as 1/τ q/(q+1), deviating from the 1/√τ scaling that is ubiquitous to a range of systems. We analyze this behavior by mapping the lowenergy physics of the system to a set of fermionic twolevel systems labeled by the lattice momentum k undergoing a nonlinear quench as well as by performing numerical simulations. We also find that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of τ although it may exhibit a crossover at intermediate values of τ . Finally, for the case where a zz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of q and interaction strength.
