Statistical mechanics approach to the electric polarization and dielectric constant of band insulators

TL;DR

This paper introduces a new theoretical framework for calculating the free energy, electric polarization, and dielectric constant of band insulators under electric fields, valid at both zero and finite temperatures.

Contribution

It presents a novel analytic method combining Wannier-Stark spectra and modified statistical mechanics to compute dielectric properties of band insulators.

Findings

Recovered known polarization results at finite temperature

Derived a general formula for electric susceptibility

Validated the theory on one-dimensional models

Abstract

We develop a theory for the analytic computation of the free energy of band insulators in the presence of a uniform and constant electric field. The two key ingredients are a perturbation-like expression of the Wannier-Stark energy spectrum of electrons and a modified statistical mechanics approach involving a local chemical potential in order to deal with the unbounded spectrum and impose the physically relevant electronic filling. At first order in the field, we recover the result of King-Smith, Vanderbilt and Resta for the electric polarization in terms of a Zak phase - albeit at finite temperature - and, at second order, deduce a general formula for the electric susceptibility, or equivalently for the dielectric constant. Advantages of our method are the validity of the formalism both at zero and finite temperature and the easy computation of higher order derivatives of the free energy. We verify our findings on two different one-dimensional tight-binding models.