arXiv subjects

R. Shankar

Publications and source records attributed to R. Shankar.

At least 37 records · Page 2Linked to original sources

arXiv2013

Quarter-filled Kitaev-Hubbard Model: A Quantum Hall State in an Optical Lattice

We analyze the Physics of cold atoms in honeycomb optical lattices with on-site repulsion and spin-orbit couplings that break time reversal symmetry. Such systems, at half filling and large on-site repulsion, have been proposed as a possible realization of the Kitaev model. The spin-orbit couplings break the spin degeneracy and, if strong-enough, lead to four non-overlapping bands in the non-interacting limit. These bands carry non-zero Chern number and therefore the non-interacting system has non-zero angular momentum and chiral edge states at 1/4 and 3/4 filling. We have investigated the effect of interactions using the variational cluster perturbation theory and conclude that the chiral edge states exist in finite range of interaction and hopping parameter space.

cond-mat.str-el↗

arXiv2013

Topological transitions in a model with Particle-Hole symmetry, Pancharatnam-Berry Curvature and Dirac Points

We study the topology and geometry of a fermionic model on the honeycomb lattice with spin-dependent hopping which breaks the time-reversal and charge-conjugation symmetries but preserves their composition. We show that in such a case the Zak phases are topological invariants that characterize the semi-metallic state at half-filling and determine the edge state structure. As the strength of the spin-dependent hopping varies, the model shows several Lifshitz transitions corresponding to creation and merging of multiple Dirac points. We discuss the possible realization of this model in cold atom systems and propose experimental signals of detecting the Dirac points and Pancharathnam-Berry curvature of the bands.

cond-mat.str-el↗

arXiv2013

Topological aspects of an exactly solvable spin chain

We analyse a spin-1/2 chain with two-spin interactions which shown to exactly solvable by Lieb, Schultz and Mattis. We show that the model can be viewed as a generalised Kitaev model that is analytically solvable for all defect sectors. We present an alternate proof that the defect free sector is the ground state, which is valid for a larger parameter range. We show that the defect sectors have degenerate ground states corresponding to unpaired Majorana fermion modes and that the degeneracy is topologically protected against disorder in the spin-spin couplings. The unpaired Majorana fermions can be manipulated by tuning the model parameters and can hence be used for topological quantum computation.

cond-mat.str-el↗

arXiv2013

A stable Algebraic Spin Liquid in a Hubbard model

We show the existence of a stable Algebraic Spin Liquid (ASL) phase in a Hubbard model defined on a honeycomb lattice with spin-dependent hopping that breaks time-reversal symmetry. The effective spin model is the Kitaev model for large on-site repulsion. The gaplessness of the emergent Majorana fermions is protected by the time reversal (TR) invariance of this model. We prove that the effective spin model is TR invariant in the entire Mott phase thus ensuring the stability of the ASL. The model can be physically realized in cold atom systems and we propose experimental signals of the ASL.

cond-mat.str-el↗

arXiv2012

Dipolar bogolons: from superfluids to Pfaffians

We study the structure of Bogoliubov quasiparticles, 'bogolons,' the fermionic excitations of paired superfluids that arise from fermion (BCS) pairing, including neutral superfluids, superconductors, and paired quantum Hall states. The naive construction of a stationary quasiparticle in which the deformation of the pair field is neglected leads to a contradiction: it carries a net electrical current even though it does not move. However, treating the pair field self-consistently resolves this problem: In a neutral superfluid, a dipolar current pattern is associated with the quasiparticle for which the total current vanishes. When Maxwell electrodynamics is included, as appropriate to a superconductor, this pattern is confined over a penetration depth. For paired quantum Hall states of composite fermions, the Maxwell term is replaced by a Chern-Simons term, which leads to a dipolar charge distribution and consequently to a dipolar current pattern.

cond-mat.str-el↗

arXiv2012

Topological Surface States and Dirac point tuning in ternary Bi2Te2Se class of topological insulators

Using angle-resolved photoemission spectroscopy, we report electronic structure for representative members of ternary topological insulators. We show that several members of this family, such as Bi2Se2Te, Bi2Te2Se, and GeBi2Te4, exhibit a singly degenerate Dirac-like surface state, while Bi2Se2S is a fully gapped insulator with no measurable surface state. One of these compounds, Bi2Se2Te, shows tunable surface state dispersion upon its electronic alloying with Sb (SbxBi2-xSe2Te series). Other members of the ternary family such as GeBi2Te4 and BiTe1.5S1.5 show an in-gap surface Dirac point, the former of which has been predicted to show nonzero weak topological invariants such as (1;111); thus belonging to a different topological class than BiTe1.5S1.5. The measured band structure presented here will be a valuable guide for interpreting transport, thermoelectric, and thermopower measurements on these compounds. The unique surface band topology observed in these compounds contributes towards identifying designer materials with desired flexibility needed for thermoelectric and spintronic device fabrication.

cond-mat.mes-hall↗

arXiv2012

Hamiltonian theory of fractionally filled Chern bands

There is convincing numerical evidence that fractional quantum Hall (FQH)-like ground states arise in fractionally filled Chern bands (FCB). Here we show that the Hamiltonian theory of Composite Fermions (CF) can be as useful in describing the FCB as it was in describing the FQHE in the continuum. We are able to introduce CFs into the FCB problem even though there is no external magnetic field by following a two-stage process. First we construct an algebraically exact mapping which expresses the electron density projected to the Chern band, $ρ_{\tiny FCB}$, as a sum of Girvin-MacDonald-Platzman density operators, $ρ_{\tiny GMP}$, that obey the Magnetic Translation Algebra. Next, following our Hamiltonian treatment of the FQH problem, we rewrite the GMP operators in terms of CF variables which reproduce the same algebra. This naturally produces a unique Hartree-Fock ground state for the CFs, which can be used as a springboard for computing gaps, response functions, temperature-dependent phenomena, and the influence of disorder. We give two concrete examples, one of which has no analog in the continuum FQHE with $ν= {1 \over 5}$ and $σ_{xy}={2\over 5}$. Our approach can be easily extended to fractionally filled, strongly interacting two-dimensional time-reversal-invariant topological insulators.

cond-mat.mes-hall↗

arXiv2012

Observation of Topological Order in the TlBiSe2 class : Probing the "spin" and "phase" on topological insulator surfaces

A topological insulator is characterized by spin-momentum locking on its boundary. The spin momentum locking on the surface of a three dimensional topological insulator leads to the existence of a non-trivial Berry's phase which leads to exotic transport phenomena on topological surfaces. Using spin-sensitive probes (Mott polarimetry), we observe the spin-momentum coupling and uncover the chiral nature of surface electrons in TlBiSe2. We demonstrate that the surface electrons in TlBiSe2 collectively carry a quantum Berry's phase of $π$ and a definite chirality ($η$ = -1, left-handed) associated with its spin-texture or vortex-structure on the Fermi surface on both the top and the bottom surfaces. Our experimental results for the first time not only prove the existence of Z2 topological-order in the bulk but also reveal the existence of helical quasiparticle modes on the topological surface. Spin-texture calculations would be reported elsewhere.

cond-mat.mes-hall↗

arXiv2012

A semi-classical analysis of Dirac fermions in 2+1 dimensions

We investigate the semiclassical dynamics of massless Dirac fermions in 2+1 dimensions in the presence of external electromagnetic fields. By generalizing the $α$ matrices to the spin-$S$ matrices and doing a certain scaling, we formulate a $S\rightarrow\infty$ limit where the spin and the orbital degrees become classical. We solve for the classical trajectories for a free particle on a cylinder and a particle in a constant magnetic field. We compare the semiclassical spectrum, obtained by Bohr-Sommerfeld quantization with the exact quantum spectrum for low values of. For the free particle, the semiclassical spectrum is exact. For the particle in a constant magnetic field, the semiclassical spectrum reproduces all the qualitative features of the exact quantum spectrum at all $S$. The quantitative fit for $S=1/2$ is reasonably good.

cond-mat.mes-hall↗

arXiv2011

RVB gauge theory and the Topological degeneracy in the Honeycomb Kitaev model

We relate the Z$_2$ gauge theory formalism of the Kitaev model to the SU(2) gauge theory of the resonating valence bond (RVB) physics. Further, we reformulate a known Jordan-Wigner transformation of Kitaev model on a torus in a general way that shows that it can be thought of as a Z$_2$ gauge fixing procedure. The conserved quantities simplify in terms of the gauge invariant Jordan-Wigner fermions, enabling us to construct exact eigen states and calculate physical quantities. We calculate the fermionic spectrum for flux free sector for different gauge field configurations and show that the ground state is four-fold degenerate on a torus in thermodynamic limit. Further on a torus we construct four mutually anti-commuting operators which enable us to prove that all eigenstates of this model are four fold degenerate in thermodynamic limit.

cond-mat.str-el↗

arXiv2011

The last word in strong correlations

In the Fractional Quantum Hall Effect (FQHE), in the noninteracting limit, only a fraction $ν$ of the Lowest Landau Level (LLL) is occupied, producing a huge degeneracy. Interactions lift this degeneracy and mix in higher LL's. In the limit in which we ignore all but the LLL (i.e., let the inverse electron mass ${1 \over m}\to \infty$), the kinetic energy is an irrelevant constant and the ratio of potential to kinetic energy is essentially infinite, making this the most strongly correlated problem imaginable. I give a telegraphic review of the Hamiltonian Theory of the FQHE developed with Ganpathy Murthy that deals with this problem with some success. A nodding acquaintance with FQHE physics is presumed.

cond-mat.str-el↗

arXiv2011

Quarter-Filled Honeycomb Lattice with a Quantized Hall Conductance

We study a generic two-dimensional hopping model on a honeycomb lattice with strong spin-orbit coupling, without the requirement that the half-filled lattice be a Topological Insulator. For quarter-(or three-quarter) filling, we show that a state with a quantized Hall conductance generically arises in the presence of a Zeeman field of sufficient strength. We discuss the influence of Hubbard interactions and argue that spontaneous ferromagnetism (which breaks time-reversal) will occur, leading to a quantized anomalous Hall effect.

cond-mat.mes-hall↗

arXiv2011

Composite Fermions for Fractionally Filled Chern Bands

We address the question of whether fractionally filled bands with a nontrivial Chern index in zero external field could also exhibit a Fractional Quantum Hall Effect (FQHE). Numerical works suggest this is possible. Analytic treatments are complicated by a non-vanishing band dispersion and a non-constant Berry flux. We propose embedding the Chern band in an auxiliary lowest Landau level (LLL) and then using composite fermions. We find some states which have no analogue in the continuum, and dependent on the interplay between interactions and the lattice. The approach extends to two-dimensional time-reversal invariant topological insulators.

cond-mat.str-el↗

arXiv2011

Equality of bulk wave functions and edge correlations in topological superconductors: A spacetime derivation

For certain systems, the N-particle ground-state wavefunctions of the bulk happen to be exactly equal to the N-point space-time correlation functions at the edge, in the infrared limit. We show why this had to be so for a class of topological superconductors, beginning with the p+ip state in D=2+1. Varying the chemical potential as a function of Euclidean time between weak and strong pairing states is shown to extract the wavefunction. Then a Euclidean rotation that exchanges time and space and approximate Lorentz invariance lead to the edge connection. We illustrate straightforward extension to other dimensions (eg. 3He- B phase in D=3+1) and to correlated states like fractionalized topological superconductors.

cond-mat.str-el↗

arXiv2011

Spin correlations and phase diagram of the perturbed Kitaev model

We present a general classification of the perturbations to the Kitaev model on the basis of their effect on it's spin correlation functions. We derive a necessary and sufficient condition for the spin correlators to exhibit a long ranged power-law behavior in the presence of such perturbations. We substantiate our result by a study of the phase diagram of the Kitaev model augmented by a loop term and perturbed by an Ising term, within a RVB mean-field theory. We estimate the stability of the spin-liquid phase against such perturbations and show that this model exhibits both confinement-deconfinement transitions from spin liquid to antiferromagnetic/spin-chain/ferromagnetic phases as well as topological quantum phase transitions between gapped and gapless spin liquid phases.

cond-mat.str-el↗

arXiv2010

Spin-1 Kitaev model in one dimension

We study a one-dimensional version of the Kitaev model on a ring of size N, in which there is a spin S > 1/2 on each site and the Hamiltonian is J \sum_i S^x_i S^y_{i+1}. The cases where S is integer and half-odd-integer are qualitatively different. We show that there is a Z_2 valued conserved quantity W_n for each bond (n,n+1) of the system. For integer S, the Hilbert space can be decomposed into 2^N sectors, of unequal sizes. The number of states in most of the sectors grows as d^N, where d depends on the sector. The largest sector contains the ground state, and for this sector, for S=1, d =(\sqrt{5}+1)/2. We carry out exact diagonalization for small systems. The extrapolation of our results to large N indicates that the energy gap remains finite in this limit. In the ground state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term λ\sum_n W_n, and show that this has gapless excitations in the range λ^c_1 \leq λ\leq λ^c_2. We use the variational wave functions to study how the ground state energy and the defect density vary near the two critical points λ^c_1 and λ^c_2.

cond-mat.stat-mech↗

arXiv2010

Magnetic impurities in the honeycomb Kitaev model

We study the effect of coupling magnetic impurities to the honeycomb lattice spin-1/2 Kitaev model in its spin liquid phase. We show that a spin-S impurity coupled to the Kitaev model is associated with an unusual Kondo effect with an intermediate coupling unstable fixed point K_c J/S separating topologically distinct sectors of the Kitaev model. We also show that the massless spinons in the spin liquid mediate an interaction of the form S_{iα}^{2}S_{jβ}^{2}/R_{ij}^{3} between distant impurities unlike the usual dipolar RKKY interaction S_{iα}S_{jα}/R_{ij}^{3} noted in various 2D impurity problems with a pseudogapped density of states of the spin bath. Furthermore, this long-range interaction is possible only if the impurities (a) couple to more than one neighboring spin on the host lattice and (b) the impurity spin is not a spin-1/2.$

cond-mat.str-el↗

arXiv2010

The Renormalization Group and the Superconducting Susceptibility of a Fermi Liquid

A free Fermi gas has, famously, a superconducting susceptibility that diverges logarithmically at zero temperature. In this paper we ask whether this is still true for a Fermi liquid and find that the answer is that it does {\it not}. From the perspective of the renormalization group for interacting fermions, the question arises because a repulsive interaction in the Cooper channel is a marginally irrelevant operator at the Fermi liquid fixed point and thus is also expected to infect various physical quantities with logarithms. Somewhat surprisingly, at least from the renormalization group viewpoint, the result for the superconducting susceptibility is that two logarithms are not better than one. In the course of this investigation we derive a Callan-Symanzik equation for the repulsive Fermi liquid using the momentum-shell renormalization group, and use it to compute the long-wavelength behavior of the superconducting correlation function in the emergent low-energy theory. We expect this technique to be of broader interest.

cond-mat.str-el↗