Free massive fermions inside the quantum discrete sine-Gordon model

TL;DR

This paper extends the quantum discrete sine-Gordon model to include free massive fermions, establishing a correspondence between quantum lattice equations and fermionic models, with implications for integrable systems.

Contribution

It introduces a new approach to quantum light cone lattice equations, linking them to free massive fermions and providing a novel compatibility equation for the algebra of observables.

Findings

Derived a compatibility equation for roots of central elements.

Established a one-to-one correspondence with free massive fermions.

Extended the notion of space shifts for sine-Gordon type models.

Abstract

We extend the notion of space shifts introduced by L. D. Faddeev and A. Yu. Volkov (Phys. Lett. B 315 (1993)) for certain quantum light cone lattice equations of sine-Gordon type at root of unity. As a result we obtain a compatibility equation for the roots of central elements within the algebra of observables (also called current algebra). The equation which is obtained by exponentiating these roots is exactly the evolution equation for the "classical background" as described in V. Bazhanov, A. Bobenko, N. Reshetikhin (Comm. Math. Phys. 175 (1996)). As an application for the introduced constructions, a one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice as constructed in Destri and de Vega (Nucl. Phys. B 290 (1987)) is derived.