Alexander G. Abanov

Thermal Hall Effect and Geometry with Torsion.

We formulate a geometric framework that allows us to study momentum and energy transport in nonrelativistic systems. It amounts to a coupling of the nonrelativistic system to the Newton-Cartan (NC) geometry with torsion. The approach generalizes the classic Luttingers formulation of thermal transport. In particular, we clarify the geometric meaning of the fields conjugated to energy and energy current. These fields describe the geometric background with nonvanishing temporal torsion. We use the developed formalism to construct the equilibrium partition function of a nonrelativistic system coupled to the NC geometry in 2+1 dimensions and to derive various thermodynamic relations.

Density-curvature response and gravitational anomaly.

We study constraints imposed by the Galilean invariance on linear electromagnetic and elastic responses of two-dimensional gapped systems in a background magnetic field. Exact relations between response functions following from the Ward identities are derived. In addition to the viscosity-conductivity relations known in the literature, we find new relations between the density-curvature response and the thermal Hall response.

Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field

Recent interest to the Hall viscosity in the theory of Fractional Quantum Hall effect (FQHE) and the interest to the interplay of defects and mechanical stresses with electromagnetic properties of materials motivates studies of gravitational, electromagnetic and mixed responses in condensed matter physics. Gravitational field in condensed matter systems can be understood either as a way to represent deformational strains present in the material under consideration or as a technical tool allowing to extract correlation functions involving stress tensor components. It is always important to have a simple model system for which such responses can be calculated exactly. For the quantum Hall effect one can consider two-dimensional electron gas in a constant magnetic field (2DEGM) as such a model. When the density of fermions is commensurate with magnetic field the integer number of Landau levels is filled and one expects local and computable response to weak external fields. This model is as important starting point of analysis for quantum Hall systems as a free electron gas for the theory of metals. However, while some electromagnetic responses for 2DEGM can be found in literature we were not able to find the complete results for mixed and gravitational linear responses. The goal of this paper is to compute these responses providing the analogue of Lindhard 1 function, both e/m and gravitational, for 2DEGM. We compute the effective action encoding linear responses in the presence of external inhomogeneous, time-dependent, slowly changing electromagnetic and gravitational fields. We compare and find an agreement of the obtained responses with known e/m responses 2–5 and with known results for Hall viscosity at integer fillings 6,7 . In addition we find the stress, charge and current densities induced by spatial curvature. Another point of comparison is given by phenomenological hydrodynamic models for FQHE 8–13 and Ward identities following from the exact local Galilean symmetry (also known as non-relativistic diffeomorphism) of the model 14,15 . The paper is organized as follows. In Section II we describe the model as a non-relativistic quantum field theory and present our results in terms of the effective action. In Section III we extract the electromagnetic responses from the effective action. We demonstrate some peculiar physical effects such as charge accumulation/depletion in the presence of conic singularity in metric, non-dissipative current perpendicular to a gradient of curvature and report higher gradient corrections to Hall conductivity. Our main results are presented in the Section IV where we discuss the gravitational responses. We present higher gradient and dynamic corrections to Hall viscosity. We leave the in-depth discussion of dynamic responses and their relation to the local Galilean invariance for a separate publication.

Quantum hydrodynamics, the quantum benjamin-ono equation, and the Calogero model.

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.

Theta-terms in nonlinear sigma-models

We trace the origin of θ-terms in nonlinear σ-models as a nonperturbative anomaly of current algebras. The nonlinear σ-models emerge as a low energy limit of fermionic σ-models. The latter describe Dirac fermions coupled to chiral bosonic fields. We discuss the geometric phases in three hierarchies of fermionic σ-models in space-time dimension (d+1) with chiral bosonic fields taking values on d-, d+1-, and d+2-dimensional spheres. The geometric phases in the first two hierarchies are θ-terms. We emphasize a relation between θ-terms and quantum numbers of solitons.

Integrable hydrodynamics of Calogero-Sutherland model : bidirectional Benjamin-Ono equation

We develop a hydrodynamic description of the classical Calogero-Sutherland liquid: a Calogero-Sutherland model with an infinite number of particles and a non-vanishing density of particles. The hydrodynamic equations, being written for the density and velocity fields of the liquid, are shown to be a bidirectional analogue of Benjamin-Ono equation. The latter is known to describe internal waves of deep stratified fluids. We show that the bidirectional Benjamin-Ono equation appears as a real reduction of the modified KP hierarchy. We derive the Chiral Non-linear Equation which appears as a chiral reduction of the bidirectional equation. The conventional Benjamin-Ono equation is a degeneration of the Chiral Non-Linear Equation at large density. We construct multi-phase solutions of the bidirectional Benjamin-Ono equations and of the Chiral Non-Linear equations.

Observation of Shock Waves in a Strongly Interacting Fermi Gas

We study collisions between two strongly interacting atomic Fermi gas clouds. We observe exotic nonlinear hydrodynamic behavior, distinguished by the formation of a very sharp and stable density peak as the clouds collide and subsequent evolution into a boxlike shape. We model the nonlinear dynamics of these collisions by using quasi-1D hydrodynamic equations. Our simulations of the time-dependent density profiles agree very well with the data and provide clear evidence of shock wave formation in this universal quantum hydrodynamic system.

Framing anomaly in the effective theory of the fractional quantum hall effect

We consider the geometric part of the effective action for the fractional quantum Hall effect (FQHE). It is shown that accounting for the framing anomaly of the quantum Chern-Simons theory is essential to obtain the correct gravitational linear response functions. In the lowest order in gradients, the linear response generating functional includes Chern-Simons, Wen-Zee, and gravitational Chern-Simons terms. The latter term has a contribution from the framing anomaly which fixes the value of thermal Hall conductivity and contributes to the Hall viscosity of the FQH states on a sphere. We also discuss the effects of the framing anomaly on linear responses for non-Abelian FQH states.

Asymptotics of Toeplitz Determinants and the Emptiness Formation Probability for the XY Spin Chain

We study an asymptotic behaviour of a special correlator known as the emptiness formation probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field. This correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models. For the XY spin chain, the correlator can be expressed as the determinant of a Toeplitz matrix and its asymptotical behaviours for n → ∞ throughout the phase diagram are obtained using known theorems and conjectures on Toeplitz determinants. We find that the decay is exponential everywhere in the phase diagram of the XY model except on the critical lines, i.e. where the spectrum is gapless. In these cases, a power-law prefactor with a universal exponent arises in addition to an exponential or Gaussian decay. The latter Gaussian behaviour holds on the critical line corresponding to the isotropic XY model, while at the critical value of the magnetic field the EFP decays exponentially. At small anisotropy one has a crossover from the Gaussian to the exponential behaviour. We study this crossover using the bosonization approach.

Nonlinear quantum shock waves in fractional quantum Hall edge states.

Using the Calogero model as an example, we show that the transport in interacting non-dissipative electronic systems is essentially non-linear. Non-linear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for non-linear systems, propagating wave packets are unstable. At finite time shock wave singularities develop, the wave packet collapses, and oscillatory features arise. They evolve into regularly structured localized pulses carrying a fractionally quantized charge - {\it soliton trains}. We briefly discuss perspectives of observation of Quantum Shock Waves in edge states of Fractional Quantum Hall Effect and a direct measurement of the fractional charge.