### Areas of Research

My research interests span a wide range of topics in theoretical condensed matter physics including topological superconductors (TS), topological insulators (TI), topological quantum computation (TQC), cuprate high temperature superconductivity, and the physics of ultra-cold atoms. Below I give a detailed discussion of some of these topics and briefly outline the rest. My work on Majorana fermions (MFs) in topological states proximity induced on spin-orbit coupled semiconductors, which began in the summer of 2009 on a visit to the University of Maryland, College Park, has recently led to experimental success claimed by at least six different groups worldwide (TU-Delft, Netherlands; Lund, Sweden; Purdue, USA; Weizmann, Israel; Harvard, USA; UIUC, USA). Earlier, my proposal to realize half quantum vortices (HQVs) in p-wave superconductors, on subsequent development by others, led to experimental claims of its realization in strontium ruthenate. My future research will include studies of topological states in a variety of condensed matter systems in detail which I discuss below at some length. Following this, I briefly outline my other areas of interest such as cuprate high transition temperature superconductivity and topological states in ultra-cold atom systems.

#### Topological Systems, Majorana Fermions, Quantum Computation

I am interested in the chiral and topological superconducting states to realize Majorana fermion quasiparticles in the order parameter defects of nanoscale systems with strong spin-orbit interactions. MFs, which were introduced in high energy physics by the Italian physicist E. Majorana more than seven decades ago, have recently made a decisive and convincing comeback to condensed matter physics in part due to my efforts in the last few years. The TS states I focus on support a certain kind of unusual zero energy quantum states in various order parameter defects which satisfy the Majorana condition . The Majorana condition is at the heart of non-Abelian quantum statistics and, as proposed by Kitaev, such non-Abelian particles can potentially serve as building blocks for a topological quantum computer. In this sense this research is intimately connected to research in quantum computation.

In my research, I study the formation of the appropriate chiral superconducting states using an explicit route to discrete symmetry breaking, namely, structural asymmetry resulting in spin-orbit couplings. In recent works, I have proposed schemes for realizing TS states and MFs in semiconductor heterostructures spurring leading experimental groups world-wide to experimentally look for MF excitations in the proposed systems. This body of work, briefly outlined below, has given rise to a new sub-field - TS states in spin-orbit coupled semiconductor heterostructures - and has recently seen a great flurry of activities.

Consider a 2D semiconductor quantum well situated in a heterostructure which explicitly breaks the symmetry. Because of broken space inversion (SI) symmetry, the spin-orbit coupling in the quantum well takes the Rashba form,, which yields an inherent chirality to the one-particle band structure. In the presence of proximity-induced *s*-wave superconductivity and a transverse Zeeman splitting, this chiral system enters into a chiral TS state with Majorana fermions as the lowest energy excitations at the order parameter defects.

The transformative nature of this idea is due to the fact that a topological quantum state with MFs, a new state of matter, arises out of completely ordinary physical effects – spin-orbit coupling and Zeeman splitting in an ordinary heterostructure of a superconductor () and a semiconductor (InAs, InSb). Thus, it requires neither special materials (*e.g.,* topological insulators (TI), chiral *p*-wave superconductors) nor exceptionally low temperatures and high magnetic fields (as in the fractional quantum Hall systems). By this proposal, non-Abelian particles such as MFs therefore seem tantalizingly within experimental reach and at least six different experimental groups have recently claimed success in observing MF excitations in semiconductor heterostructures following this lead. The experimental success has generated a great deal of interest in both condensed matter and TQC communities, and has been covered in technical journals (Science, PRL, Nature Physics) as well as in popular science press such as Physics Today, Nature News, and Scientific American etc.

Another example of chiral TS states which has been and will continue to be a focus of my research are the superconductors/superfluids in natural systems and cold fermions. In strontium ruthenate, a supposedly quasi-2D chiral p-wave superconductor, I proposed a way to stabilize half-quantum vortices (HQV’s), excitations known to carry MFs, by applying a moderate magnetic field ~ 250 G. The basic reason why such vortices have a higher energy than the ordinary (full-quantum) vortices in triplet superconductors is that the HQV's involve a rotation of the *d* vector in the 2D plane resulting in an extra domain-wall energy that makes them energetically costlier. Following ideas from the He3 literature I have shown that HQVs can be stabilized in strontium ruthenate by applying a magnetic field (much less than the upper critical field ()) that facilitates the rotation of the vector in the 2D plane.

Following my work, more work has been done on this problem applied to mesoscopic samples and finally HQV’s have recently been claimed to be realized inexperimental work at UIUC. In the context of cold atom superfluids, with collaborators Zhang and Dassarma I wrote the very first paper on using artificially generated spin-orbit coupling and Zeeman field to create topological superfluid states and MFs. Recent experimental breakthrough at NIST realizing spin-orbit couplings for ultra-cold atoms has opened exciting new possibilities for the observation of MFs in cold atomic systems following our original proposal.

Broadly, my future research will include investigations of the many important questions for the topological semiconductor heterostructures, i) suitable new materials, ii) simplification of the heterostructure, iii) robustness of the MFs to thermal effects and quenched disorder, iv) experimental detections (a “smoking gun” experiment?) of MFs, and finally, v) possible TQC architectures using 2D systems and/or 1D quantum wire network. Other systems of interest include hole doped quantum wires, topological insulators, carbon nanotubes, organic superconductors, non-centrosymmetric superconductors, and ferromagnetic superconductors, all of which have been subjects of my recent research.

Generally speaking, I shall focus on the superconducting states which are gapped in the bulk, break the time reversal (TR) invariance, are characterized by an integer (Z) Chern (or TKNN) invariant, and have edge states with particular handedness or chirality (because of which these states are called chiral). Within this class, I will be most concerned with those states which are further classified by a Z2 topological invariant, which is given by the sign of the Pfaffian of the relevant BCS Hamiltonian at . The superconducting states with the above Pfaffian invariant admit (do not admit) gapless, non-degenerate, anyonic Majorana modes at vortices and sample edges which make them non-Abelian (Abelian) quantum matter. In line with my continuing research on the class D TS systems (protected by the particle-hole or charge-conjugation symmetry), I shall take a broad perspective on the candidate quantum systems asking questions such as i) under what conditions such states are experimentally realizable, and ii) how the MFs, constituents of a true new state of matter, can be experimentally observed. The appealing prospect of fault-tolerant quantum computation using such systems is but one motivation for this research.

In the realm of time-reversal invariant (TRI) topological superconductivity (class DIII) my goals are, (1) to establish a relationship between TRI superconductivity and the associated invariant and chiral superconductivity (class BDI) and the corresponding invariant, as is apparent in my recent work, (2) to study model and realistic TRI topological superconducting systems, specifically in the quasi-one-dimensional limit, (3) explore the edge states of the exciting new material transition metal dichalcogenides, the organic superconductors, and Lithium Purple Bronze, as robust platforms for TRI superconductivity, and (4) benchmark reliable methods to detect the Majorana-Kramers pairs in TRI topolgical systems in experiments.

In the context of both class D and DIII systems I will study the relevance of lattice point group symmetries (e.g., reflection, crystal rotation, mirror) to the protection of MFs (the so-called “symmetry-protected topological states”), particularly, in the presence of disorder which breaks any such spatial symmetry and yet the MFs remain un-split. This curious result, that multiple MFs persist even when their wave-functions overlap, despite the absence of any spatial or non-spatial symmetry protecting them, was revealed in our recent (soon-to-be-published) work. This problem will be approached in detail using the symmetry of the numerically solved BdG wave-functions (understanding why the wave-function overlaps are zero), and symmetry/group theory (specifically, looking for an as-yet unidentified hidden symmetry responsible for the protection of the MFs).

My research interests also include possible realization of Weyl semi-metal and Weyl superconductor phases in non-centrosymmetric systems with spin-orbit interaction. In principle, the Weyl phase allows a chiral magnetic effect (electric current in response to a parallel magnetic field in the absence of an electric field!) which may have interesting technological applications. Moreover the non-trivial Berry curvatures near Weyl points can give rise to gyrotropic conductivity (which has its origin in the breakdown of space inversion symmetry in the non-centrosymmetric systems) whose real and imaginary parts can give rise to Faraday and Kerr rotations in polarization of reflected and transmitted light. These effects will be investigated in the framework of an axionic field theory for Weyl fermions (which I recently developed with co-author Goswami) in the continuum limit and by numerical calculations on lattices.

A part of my research will also involve developing the appropriate quantum computation architecture. Specifically, I will devise methods to initialize the non-Abelian qubits and design the quantum gates, the elementary building blocks for TQC. I will also address the qubit measuring techniques in such systems, which will be required for carrying out the final state read-out in a putative quantum computer. It is important to acknowledge that, even after realizing an experimentally verifiable non-Abelian system, implementing the TQC architecture can be quite challenging. Nevertheless, my broader direction of research, which emphasizes the basic principles responsible for the formation of the non-Abelian states in various physical platforms, will enhance our understanding of quantum topological phases of matter. My research will help us understand in what systems and under what conditions the interesting non-Abelian quantum phases (in classes D, DIII, and BDI), with or without an additional lattice point group symmetry protecting the MFs, can be realized and tested experimentally, TQC being a revolutionary long-term application.

#### Cuprate High Temperature Superconductors

For cuprate high temperature superconductivity, in my Ph.D work I helped develop the theory and phenomenology of the *d*-density wave state (DDW, a version of the staggered flux state) applicable to the underdoped regime of the cuprates. Recently I have been able to explain many new exciting experimental results in the framework of the DDW theory of the pseudogap phase. These include pronounced Nernst and other thermoelectric effects, diamagnetism, aspects of the neutron scattering, and the polar Kerr effect (an indication of broken time reversal symmetry) above in the underdoped regime of the cuprates. I plan on exploring aspects of the DDW state such as the collective mode spectrum and the neutron scattering signatures in the pseudogap regime of the cuprates. The collective mode spectrum of the DDW state should take into account the current fluctuations on the lattice and this would necessitate mapping the problem on a suitable vertex model such as the six vertex model. The statistical mechanical six vertex model has multiple ordered phases in the full phase diagram and the DDW or the staggered flux phase is only one of them realized under certain conditions on the vertex energies. It follows naturally that the fluctuation spectrum above the DDW state should contain fluctuations of the other possible ordered states of the vertex models which are nearby in energy to the DDW state. In recent work I have shown that this implies the amplitude fluctuation spectrum of the DDW state to contain not only an inelastic peak at the ordering wave vector () of the DDW state but also a second peak at the wave vector (0,0) corresponding to a putative ordered state with orbital *ferromagnetic* order. Such multiple collective mode peaks at two different wave vectors from a single underlying ordered state may have important implications for the magnetic excitation spectrum in the pseudogap regime.

In recent experiments it has been seen that the inelastic magnetic scattering spectrum in the pseudogap phase of the cuprates has multiple low energy peaks the origin of which remains mysterious. An important goal of my future research will be to calculate the inelastic neutron scattering spectrum following from the collective mode spectrum of the staggered flux state and compare them with neutron scattering experiments.

Additionally, I will also consider other candidate phases for the pseudogap regime which are close in energy with the DDW state. These include the electronic nematic state and the loop current state within the CuO unit cell *without* broken translational symmetry. In the short term I will focus on the magnetic excitation spectrum of these states that follows from their collective mode spectrum and compare with the neutron scattering experiments. These states will also be investigated with regard to their Fermi surface topology, the recently discovered quantum oscillation signatures of the Fermi surfaces, and diamagnetic response/Nernst effect, as well as their transport signatures in the pseudogap phase.

The exciting new experiments involving polar Kerr effect and Faraday rotation in a restricted range of parameter space in the underdoped cuprates will be investigated in terms of finite gyroptropic conductivity in the nematic state (if somehow accompanied by space inversion) and also the newly discovered checkerboard charge density wave state with wave vector close to , . We will calculate the gyrotropic conductivity directly using the Kubo formula but retaining terms linear in momentum, whose real and imaginary components will be connected with the Faraday rotation and the polar Kerr effect, respectively.

#### Ultra Cold Atom Physics

Recent experimental breakthrough at NIST realizing spin-orbit coupling for ultra-cold atoms has opened exciting new possibilities for the observation of novel topological superfluid phases in cold atomic systems. In these systems, with collaborators Zhang and Dassarma, I wrote the very first paper on using artificially generated spin-orbit coupling, Zeeman field, and s-wave superfluid interactions to create topological superfluid states and MFs (this paper,*Physical Review Letters 101, 160401 (2008)*, in fact had the genesis of the idea that was later applied to the condensed matter systems). In the short term I will develop concrete schemes for realizing TS states and the associated topological phase transitions using experimental techniques which have already been realized. These include spin-orbit coupling in 1D systems developed in the Spieleman group at NIST and the photoemission spectroscopy developed in the Jin group at JILA.