# Lower bounds for the conductivities of correlated quantum systems

## TL;DR

This paper derives lower bounds for electrical, spin, and heat conductivities in correlated quantum systems with small perturbations, using the memory matrix formalism, and discusses conditions for these bounds to be exact.

## Contribution

It introduces a systematic method to obtain and improve lower bounds for conductivities in correlated quantum systems with conservation laws and perturbations.

## Key findings

- Lower bounds can be derived for conductivities in systems with conservation laws.
- The bounds can be systematically improved using the memory matrix formalism.
- Conditions under which the bounds become exact are discussed.

## Abstract

We show how one can obtain a lower bound for the electrical, spin or heat conductivity of correlated quantum systems described by Hamiltonians of the form H = H0 + g H1. Here H0 is an interacting Hamiltonian characterized by conservation laws which lead to an infinite conductivity for g=0. The small perturbation g H1, however, renders the conductivity finite at finite temperatures. For example, H0 could be a continuum field theory, where momentum is conserved, or an integrable one-dimensional model while H1 might describe the effects of weak disorder. In the limit g to 0, we derive lower bounds for the relevant conductivities and show how they can be improved systematically using the memory matrix formalism. Furthermore, we discuss various applications and investigate under what conditions our lower bound may become exact.