Gauge invariance of excitonic linear and nonlinear optical response

TL;DR

This paper rigorously demonstrates the equivalence of four different theoretical approaches to calculating excitonic optical responses in semiconductors, emphasizing the importance of correct interaction Hamiltonians and applying the theory to hexagonal boron nitride monolayers.

Contribution

It establishes the conditions under which four common methods for excitonic optical response calculations are equivalent, clarifying gauge and current evaluation choices.

Findings

All four methods yield equivalent linear and nonlinear responses when using the correct interaction Hamiltonian.

The correct velocity gauge interaction involves a series of commutators with the unperturbed Hamiltonian and position operators.

Application to hexagonal boron nitride monolayers shows consistent optical response results across methods.

Abstract

We study the equivalence of four different approaches to calculate the excitonic linear and nonlinear optical response of multiband semiconductors. These four methods derive from two choices of gauge, i.e. length and velocity gauges, and two ways of computing the current density, i.e. direct evaluation and evaluation via the time-derivative of the polarization density. The linear and quadratic response functions are obtained for all methods by employing a perturbative density matrix approach within the mean-field approximation. The equivalence of all four methods is shown rigorously, when a correct interaction Hamiltonian is employed for the velocity gauge approaches. The correct interaction is written as a series of commutators containing the unperturbed Hamiltonian and position operators, which becomes equivalent to the conventional velocity gauge interaction in the limit of infinite Coulomb screening and infinitely many bands. As a case study, the theory is applied to hexagonal boron nitride monolayers, and the linear and nonlinear optical response found in different approaches are compared.