Exact Derivation of Luttinger Liquid Relation in a One-Dimensional Two-Component Quantum System with Hyperbolic Interactions

TL;DR

This paper provides an exact derivation of the Luttinger liquid relation for a specific one-dimensional two-component quantum system with hyperbolic interactions, confirmed by numerical and conformal methods.

Contribution

It introduces an exact analytical approach to derive the Luttinger liquid relation in a two-component quantum model with hyperbolic interactions, extending applicability to related models.

Findings

Exact Luttinger liquid relation derived for the SC model

Confirmation via numerical and conformal methods

Exact susceptibility formula obtained

Abstract

We present an exact calculation of the Luttinger liquid relation for the one-dimensional, two-component SC model in the interaction strength range $-1<s<0$ by appropriately varying the limits of the integral Bethe Ansatz equations. The result is confirmed by numerical and conformal methods. By a related study of the transport properties of the SC model, we can give an exact formula for the susceptibility. Our method is applicable to a wide range of models such as, e.g., the Heisenberg-Ising chain.