Transport functions for hypercubic and Bethe lattices

TL;DR

This paper derives exact integral and differential equation expressions for transport functions on hypercubic and Bethe lattices, enabling efficient calculations of various transport coefficients in strongly correlated systems.

Contribution

It provides a unified method to compute transport functions for hypercubic and Bethe lattices, including new results for the Bethe lattice's Hall and Nernst coefficients.

Findings

Exact integral expressions for transport functions on hypercubic lattices.

Differential equations derived from sum rules for transport functions.

New analytical results for Bethe lattice Hall and Nernst coefficients.

Abstract

In calculations of transport quantities, such as the electrical conductivity, thermal conductivity, Seebeck, Peltier, Nernst, Ettingshausen, Righi-Leduc, or Hall coefficients, sums over the Brillouin zone of wave-vector derivatives of the dispersion relation commonly appear. When the self-energy depends only on frequency, as in single-site dynamical mean-field theory, it is advantageous to perform these sums once and for all. We show here that in the case of a hypercubic lattice in d dimensions, the sums needed for any of the transport coefficients can be expressed as integrals over powers of the energy weighted by the energy-dependent non-interacting density of states. It is also shown that our exact expressions for the transport functions can be obtained from differential equations that follow from sum rules. By substituting the Bethe lattice density of states, one can obtain the previously unknown transport function for the electrical or thermal Hall coefficients and for the Nernst coefficient of the Bethe lattice.