CONNECTIONS of the LIOUVILLE MODEL and XXZ SPIN CHAIN

TL;DR

This paper establishes a connection between the lattice Liouville model and the XXZ spin chain by deriving Bethe Ansatz equations, revealing their shared integrable structure and conformal field theory properties.

Contribution

It develops an integrable lattice Liouville theory framework using the algebraic Bethe Ansatz, linking it to the XXZ spin chain and conformal field theories.

Findings

Bethe Ansatz equations for lattice Liouville theory derived

Mapping of lattice Liouville Bethe states to XXZ spin chain states

Emergence of unitary conformal field theories at specific couplings

Abstract

The quantum theory of the Liouville model with imaginary field is considered using the quantum inverse scattering method. An integrable structure with nontrivial spectral parameter dependence is developed for lattice Liouville theory by scaling the $L$-matrix of lattice sine-Gordon theory. This $L$-matrix yields Bethe Ansatz equations for Liouville theory, by the methods of the algebraic Bethe Ansatz. Using the string picture of exited Bethe states, the lattice Liouville Bethe equations are mapped to the corresponding equations for the spin 1/2 XXZ chain. The well developed theory of finite size corrections in spin chains is used to deduce the conformal properties of the lattice Liouville Bethe states. The unitary series of conformal field theories emerge for Liouville couplings of the form $\gam = \pi\frac{\nu}{\nu+1}$, corresponding to root of unity XXZ anisotropies. The Bethe states give the full spectrum of the corresponding unitary conformal field theory, with the primary states in the \Kac table parameterized by a string length $K$, and the remnant of the chain length mod $(\nu+1)$.