Thermal transport in a noncommutative hydrodynamics

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hermal Transport in a Noncommutative Hydrodynamics1 M. Geracie* and D. T. Son** Kadanoff Center for Theoretical Physics, University of Chicago 60637, Illinois, Chicago, USA *email: [email protected] **email: [email protected] Received October 1, 2014

Abstract—We find the hydrodynamic equations of a system of particles constrained to be in the lowest Lan dau level. We interpret the hydrodynamic theory as a Hamiltonian system with the Poisson brackets between the hydrodynamic variables determined from the noncommutativity of space. We argue that the most general hydrodynamic theory can be obtained from this Hamiltonian system by allowing the Righi–Leduc coeffi cient to be an arbitrary function of thermodynamic variables. We compute the Righi–Leduc coefficient at high temperatures and show that it satisfies the requirements of particle–hole symmetry, which we outline. Contribution for the JETP special issue in honor of V.A.Rubakov’s 60th birthday DOI: 10.1134/S1063776115030061 1

1. INTRODUCTION

Interacting electrons in very high magnetic fields show extremely rich behaviors, the most wellknown of which is the fractional quantum Hall (FQH) effect [1, 2]. In the most interesting limit, all the physics occurs in the lowest Landau level (LLL) and originates from the interactions. In this paper, we study the finitetemperature dynamics of electrons in a magnetic field so high that all particles are constrained to be on the LLL. This problem is a finitetemperature counterpart of the FQH problem. While many quantum phenomena are smeared out by the temperature, the hydrodynamic theory, which takes hold at distances and time scales much larger than the mean free path/time, is expected to be universal. We assume that the system is clean, without impurities, and the only relaxation mecha nism is the interactions between particles. This regime is particularly relevant for the proposed realizations of the FQH regime in cold atomic gases [3–5]. The main outcome of our investigation is the set of hydrody namic equations (Eqs. (8), (18), and (30)) which describes the longwavelength dynamics of the system, the identification of the kinetic coefficients, and the computation of the thermal Hall coefficient in the hightemperature regime (Eq. (34)). Previous studies of transport in high magnetic fields include [6–9]. In particular, in [9], a general approach based on conservation laws is developed for the hydrodynamics of a system in a quantizing mag netic field. This is the approach that we follow in this 1 The article is published in the original.

paper. But we concentrate here on the LLL limit (the zero mass limit), which should be a regular limit when the particle carries a magnetic moment corresponding to the gyromagnetic factor g = 2. This allows us to con sider the response of the system to variations of the magnetic field, as well as to discuss the particle–hole symmetry of the hydrodynamic equations. An important concept in our discussion is that a particle in the LLL effectively lives on a noncommuta tive

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