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

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.