Integrability and scattering of the boson field theory on a lattice

TL;DR

This paper explores the integrability of the free boson field theory on a 2D lattice by applying exactly solvable model techniques, deriving the S-matrix, and revealing its connection to Yang-Baxter integrability.

Contribution

It introduces a novel application of integrable model methods to free boson lattice theories, deriving the S-matrix and conserved quantities.

Findings

Boltzmann weights satisfy Yang-Baxter equation

Diagonalization of transfer matrix and conserved quantities

Construction of factorized scattering S-matrix models

Abstract

A free boson on a lattice is the simplest field theory one can think of. Its partition function can be easily computed in momentum space. However, this straightforward solution hides its integrability properties. Here, we use the methods of exactly solvable models, that are currently applied to spin systems, to a massless and massive free boson on a 2D lattice. The Boltzmann weights of the model are shown to satisfy the Yang-Baxter equation with a uniformization given by trigonometric functions in the massless case, and Jacobi elliptic functions in the massive case. We diagonalize the row-to-row transfer matrix, derive the conserved quantities, and implement the quantum inverse scattering method. Finally, we construct two factorized scattering $S$ matrix models for continuous degrees of freedom using trigonometric and elliptic functions. These results place the free boson model in 2D in the same position as the rest of the models that are exactly solvable \`a la Yang-Baxter, offering possible applications in quantum computation.