Theory of perturbatively nonlinear quantum transport I: general formulation and structure of the retarded correlator

TL;DR

This paper develops a group-theoretical framework for analyzing the structure of retarded correlators in nonlinear quantum transport, connecting symmetry properties to physical effects like shift and injection currents.

Contribution

It introduces a formal decomposition of retarded correlators based on symmetry groups, establishing second order fluctuation-dissipation theorems and classifying tensor properties for crystal systems.

Findings

Decomposition of retarded correlators by irreducible representations

Establishment of second order fluctuation-dissipation theorems

Full point group classification for rank 3 and 4 tensors

Abstract

This article is the first of a trilogy that addresses various aspects of the perturbative response of general quantum systems, with possibly nontrivial ground state geometry, beyond linear order. Here, we use group theoretical considerations to investigate the structure of retarded correlators, and demonstrate how they decompose according to irreducible representations of a `time-reversal group' and relevant permutation groups, with the former probing dissipative and time-reversal properties, and the latter discerning configurational properties -- longitudinal, transverse and their generalizations. We establish second order fluctuation-dissipation and fluctuation-reaction theorems, and connect them to well-known second order transport effects such as the shift and injection currents. Exploiting the Schur-Weyl duality between irreducible representations of general linear groups and irreducible representations of permutation groups, we show how to decide which terms in the decomposition based on the latter can be supported by different crystals described via the 32 point groups, and perform the full point group classification for rank 3 and 4 polar and axial tensors. Our results provide a formal basis for the extraction of uniquely differing physical effects from retarded correlators. Applications to second order charge current responses in selected cases are given in part III.