Thermal Hall Effect and Geometry with Torsion

TL;DR

This paper develops a geometric framework using Newton-Cartan geometry with torsion to analyze momentum and energy transport in non-relativistic systems, extending Luttinger's approach and clarifying the geometric nature of energy-related fields.

Contribution

It introduces a novel geometric formalism coupling non-relativistic systems to torsional Newton-Cartan geometry, enhancing the understanding of thermal transport and energy currents.

Findings

Constructed equilibrium partition function in 2+1 dimensions

Derived thermodynamic relations in non-relativistic systems

Clarified geometric interpretation of energy fields

Abstract

We formulate a geometric framework that allows to study momentum and energy transport in non-relativistic systems. It amounts to coupling of the non-relativistic system to the Newton-Cartan geometry with torsion. The approach generalizes the classic Luttinger's 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 non-vanishing temporal torsion. We use the developed formalism to construct the equilibrium partition function of a non-relativistic system coupled to the NC geometry in 2+1 dimensions and to derive various thermodynamic relations