Zero frequency divergence and gauge phase factor in the optical response theory

TL;DR

This paper identifies the importance of the gauge phase factor in optical response theory to resolve zero frequency divergence issues and achieve gauge-invariant results, demonstrated through models of trans-polyacetylene.

Contribution

It reveals that properly including the gauge phase factor in wavefunctions is essential for consistent optical susceptibility calculations across different gauges.

Findings

Proper treatment of gauge phase factor resolves ZFD in optical susceptibilities.

Demonstrates gauge invariance in SSH and TLM models when phase factor is included.

Explains why dipole-dipole correlation is preferable over current-current correlation in practice.

Abstract

The static current-current correlation leads to the definitional zero frequency divergence (ZFD) in the optical susceptibilities. Previous computations have shown nonequivalent results between two gauges (${\bf p\cdot A}$ and ${\bf E \cdot r}$) under the exact same unperturbed wave functions. We reveal that those problems are caused by the improper treatment of the time-dependent gauge phase factor in the optical response theory. The gauge phase factor, which is conventionally ignored by the theory, is important in solving ZFD and obtaining the equivalent results between these two gauges. The Hamiltonians with these two gauges are not necessary equivalent unless the gauge phase factor is properly considered in the wavefunctions. Both Su-Shrieffer-Heeger (SSH) and Takayama-Lin-Liu-Maki (TLM) models of trans-polyacetylene serve as our illustrative examples to study the linear susceptibility $\chi^{(1)}$ through both current-current and dipole-dipole correlations. Previous improper results of the $\chi^{(1)}$ calculations and distribution functions with both gauges are discussed. The importance of gauge phase factor to solve the ZFD problem is emphasized based on SSH and TLM models. As a conclusion, the reason why dipole-dipole correlation favors over current-current correlation in the practical computations is explained.