Confining electrons on a topological insulator surface using potentials and a magnetic field

We study the effects of extended and localized potentials and a magnetic field on the Dirac electrons residing at the surface of a three-dimensional topological insulator. We use a lattice model to numerically study the various states; we show how the potentials can be chosen in a way which effectively avoids the problem of fermion doubling on a lattice. We show that extended potentials of different shapes can give rise to states which propagate freely along the potential but decay exponentially away from it. For an infinitely long potential barrier, the dispersion and spin structure of these states are unusual and these can be varied continuously by changing the barrier strength. In the presence of a magnetic field applied perpendicular to the surface, these states become separated from the gapless surface states by a gap, thereby giving rise to a quasi-one-dimensional system. Similarly, a magnetic field along with a localized potential can give rise to exponentially localized states which are separated from the surface states by a gap and thereby form a zero-dimensional system. Finally, we show that a long barrier and an impurity potential can produce bound states which are localized at the impurity, and an L-shaped potential can have both bound states at the corner of the L and extended states which travel along the arms of the potential.

Edge states of a three-dimensional topological insulator

We use the bulk Hamiltonian for a three-dimensional topological insulator such as $rm Bi_2 Se_3$ to study the states which appear on its various surfaces and along the edge between two surfaces. We use both analytical methods based on the surface Hamiltonians (which are derived from the bulk Hamiltonian) and numerical methods based on a lattice discretization of the bulk Hamiltonian. We find that the application of a potential along an edge can give rise to states localized at that edge. These states have an unusual energy-momentum dispersion which can be controlled by applying a potential along the edge; in particular, the velocity of these states can be tuned to zero. The scattering across the edge is studied as a function of the edge potential. We show that a magnetic field in a particular direction can also give rise to zero energy states on certain edges. We point out possible experimental ways of looking for the various edge states.

Transport across a junction of topological insulators and a superconductor

We study transport across a line junction lying between two orthogonal topological insulator surfaces and a superconductor which can have either s-wave (spin-singlet) or p-wave (spin-triplet) pairing symmetry. We present a formalism for studying the effect of a general time-reversal invariant barrier at the junction and show that such a barrier can be completely described by three arbitrary parameters. We compute the charge and the spin conductance across such a junction and study their behaviors as a function of the bias voltage applied across the junction and the three parameters used to characterize the barrier. We find that the presence of topological insulators and a superconductor leads to both Dirac and Schrodinger-like features in charge and spin conductances. We discuss the effect of bound states on the superconducting side of the barrier on the conductance; in particular, we show that for triplet p-wave superconductors such a junction may be used to determine the spin state of its Cooper pairs. Our study reveals that there is a non-zero spin conductance for some particular spin states of the triplet Cooper pairs; this is an effect of the topological insulators which break the spin rotation symmetry. Finally, we find an unusual satellite peak (in addition to the usual zero bias peak) in the spin conductance for p-wave symmetry of the superconductor order parameter.

Non-equilibrium dynamics of quantum systems: order parameter evolution, defect generation, and qubit transfer

In this review, we study some aspects of the non-equilibrium dynamics of quantum systems. In particular, we consider the effect of varying a parameter in the Hamiltonian of a quantum system which takes it across a quantum critical point or line. We study both sudden and slow quenches in a variety of systems including one-dimensional ultracold atoms in an optical lattice, an infinite range ferromagnetic Ising model, and some exactly solvable spin models in one and two dimensions (such as the Kitaev model). We show that quenching leads to the formation of defects whose density has a power-law dependence on the quenching rate; the power depends on the dimensionalities of the system and of the critical surface and on some of the exponents associated with the critical point which is being crossed. We also study the effect of non-linear quenching; the power law of the defects then depends on the degree of non-linearity. Finally, we study some spin-1/2 models to discuss how a qubit can be transferred across a system.

Enhancement of tunneling density of states at a junction of three Luttinger liquid wires

We study the tunneling density of states (TDOS) for a junction of three Tomonaga-Luttinger liquid wires. We show that there are fixed points which allow for the enhancement of the TDOS, which is unusual for Luttinger liquids. The distance from the junction over which this enhancement occurs is of the order of x = v/(2 omega), where v is the plasmon velocity and omega is the bias frequency. Beyond this distance, the TDOS crosses over to the standard bulk value independent of the fixed point describing the junction. This finite range of distances opens up the possibility of experimentally probing the enhancement in each wire individually.

Spin-S Kitaev model: Classical Ground States, Order by Disorder and Exact Correlation Functions

In the first part of this paper, we study the spin-S Kitaev model using spin wave theory. We discover a remarkable geometry of the minimum energy surface in the N-spin space. The classical ground states, called Cartesian or CN-ground states, whose number grows exponentially with the number of spins N, form a set of points in the N-spin space. These points are connected by a network of flat valleys in the N-spin space, giving rise to a continuous family of classical ground states. Further, the CN-ground states have a correspondence with dimer coverings and with self avoiding walks on a honeycomb lattice. The zero point energy of our spin wave theory picks out a subset from a continuous family of classically degenerate states as the quantum ground states; the number of these states also grows exponentially with N. In the second part, we present some exact results. For arbitrary spin-S, we show that localized Z_2 flux excitations are present by constructing plaquette operators with eigenvalues pm 1 which commute with the Hamiltonian. This set of commuting plaquette operators leads to an exact vanishing of the spin-spin correlation functions, beyond nearest neighbor separation, found earlier for the spin-1/2 model [G. Baskaran, S. Mandal and R. Shankar, Phys. Rev. Lett. 98, 247201 (2007)]. We introduce a generalized Jordan-Wigner transformation for the case of general spin-S, and find a complete set of commuting link operators, similar to the spin-1/2 model, thereby making the Z_2 gauge structure more manifest. The Jordan-Wigner construction also leads, in a natural fashion, to Majorana fermion operators for half-integer spin cases and hard-core boson operators for integer spin cases, strongly suggesting the presence of Majorana fermion and boson excitations in the respective low energy sectors.

Scattering of electrons from an interacting region

We address the problem of transmission of electrons between two noninteracting leads through a region where they interact (quantum dot). We use a model of spinless electrons hopping on a one-dimensional lattice and with an interaction on a single bond. We show that all the two-particle scattering states can be found exactly. Comparisons are made with numerical results on the time evolution of a two-particle wave packet and several interesting features are found for scattering. For N particles the scattering state is obtained by perturbation theory. For a dot connected to Fermi seas at different chemical potentials, we find an expression for the change in the Landauer current resulting from the interactions on the dot. We end with some comments on the case of spin-1/2 electrons.

Quench dynamics and defect production in the Kitaev and extended Kitaev models

We study quench dynamics and defect production in the Kitaev and the extended Kitaev models. For the Kitaev model in one dimension, we show that in the limit of slow quench rate, the defect density n sim 1/sqrt{tau} where 1/tau is the quench rate. We also compute the defect correlation function by providing an exact calculation of all independent non-zero spin correlation functions of the model. In two dimensions, where the quench dynamics takes the system across a critical line, we elaborate on the results of earlier work [K. Sengupta, D. Sen and S. Mondal, Phys. Rev. Lett. 100, 077204 (2008)] to discuss the unconventional scaling of the defect density with the quench rate. In this context, we outline a general proof that for a d dimensional quantum model, where the quench takes the system through a d-m dimensional gapless (critical) surface characterized by correlation length exponent u and dynamical critical exponent z, the defect density n sim 1/tau^{m u /(z u +1)}. We also discuss the variation of the shape and the spatial extent of the defect correlation function with the change of both the rate of quench and the model parameters and compute the entropy generated during such a quench process. Finally, we study the defect scaling law, entropy generation and defect correlation function of the two-dimensional extended Kitaev model.

Exact results for quench dynamics and defect production in a two-dimensional model

We show that for a d-dimensional model in which a quench with a rate tau^{-1} takes the system across a d-m dimensional critical surface, the defect density scales as n sim 1/tau^{m u/(z u +1)}, where u and z are the correlation length and dynamical critical exponents characterizing the critical surface. We explicitly demonstrate that the Kitaev model provides an example of such a scaling with d=2 and m= u=z=1. We also provide the first example of an exact calculation of some multispin correlation functions for a two-dimensional model which can be used to determine the correlation between the defects. We suggest possible experiments to test our theory.