Bukhvostov-Lipatov model and quantum-classical duality

TL;DR

This paper establishes a detailed quantum-classical duality for the Bukhvostov-Lipatov model, linking quantum vacuum energy to classical sinh-Gordon solutions through inverse scattering and Bethe ansatz methods.

Contribution

It provides a complete derivation of the quantum-classical duality using inverse scattering, Bethe ansatz, and differential equations for the Bukhvostov-Lipatov model.

Findings

Bethe ansatz equations correspond to classical connection coefficients.

Derived non-linear integral equations for vacuum energy.

Confirmed duality between quantum and classical integrable models.

Abstract

The Bukhvostov-Lipatov model is an exactly soluble model of two interacting Dirac fermions in 1+1 dimensions. The model describes weakly interacting instantons and anti-instantons in the $O(3)$ non-linear sigma model. In our previous work [arXiv:1607.04839] we have proposed an exact formula for the vacuum energy of the Bukhvostov-Lipatov model in terms of special solutions of the classical sinh-Gordon equation, which can be viewed as an example of a remarkable duality between integrable quantum field theories and integrable classical field theories in two dimensions. Here we present a complete derivation of this duality based on the classical inverse scattering transform method, traditional Bethe ansatz techniques and analytic theory of ordinary differential equations. In particular, we show that the Bethe ansatz equations defining the vacuum state of the quantum theory also define connection coefficients of an auxiliary linear problem for the classical sinh-Gordon equation. Moreover, we also present details of the derivation of the non-linear integral equations determining the vacuum energy and other spectral characteristics of the model in the case when the vacuum state is filled by 2-string solutions of the Bethe ansatz equations.