Thermal Hall Effect and Geometry with Torsion

TL;DR

This work develops a geometric framework for non-relativistic thermal transport by coupling to Newton-Cartan geometry with torsion, extending Luttinger’s gravitational-potential idea. It provides explicit sources for momentum, energy density, and energy current and constructs a 2+1D equilibrium partition function that remains consistent with local symmetries, including temporal torsion to model temperature gradients. The authors derive magnetization currents and Streda-type relations that connect Hall and thermal responses to derivatives of magnetizations with respect to chemical potential, temperature, and external NC fields. The approach yields general, symmetry-aware thermodynamics for gapped non-relativistic systems and offers a versatile tool for condensed-matter hydrodynamics, with potential extensions to spin effects and non-Galilean contexts.

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