Drude weight in non solvable quantum spin chains

TL;DR

This paper proves that in weakly interacting 1D quantum spin and fermionic systems, the Drude weight satisfies a universal relation with the Fermi velocity and susceptibility, confirming a key aspect of Luttinger liquid theory.

Contribution

It rigorously establishes the universal Luttinger liquid relation for non-solvable quantum spin chains using Renormalization Group methods.

Findings

Drude weight D relates to Fermi velocity v_s and susceptibility κ via a universal relation.

The proof applies to any weakly interacting 1D quantum system, regardless of integrability.

Completes the proof of the Luttinger liquid conjecture for these systems.

Abstract

For a quantum spin chain or 1D fermionic system, we prove that the Drude weight D verifies the universal Luttinger liquid relation $v_s^2=D/\kappa$, where $\kappa$ is the susceptibility and $v_s$ is the Fermi velocity. This result is proved by rigorous Renormalization Group methods and is true for any weakly interacting system, regardless its integrablity. This paper, combined with a previous paper of the same authors, completes the proof of the Luttinger liquid conjecture for such systems.