Problems with using separated variables for computing expectation values for higher ranks

TL;DR

This paper investigates the challenges in computing expectation values in a classical integrable model with a non-hyperelliptic spectral curve, highlighting the complexity of integrals that do not factorize, which may also impact quantum cases.

Contribution

It identifies the problem of non-factorizable integrals in classical models and discusses potential implications for quantum integrable systems.

Findings

Complex integrals arise in expectation value calculations.

These integrals do not factorize, complicating computations.

Implications for quantum models are suggested.

Abstract

We consider the simplest classical integrable model corresponding to a non-hyperelliptic spectral curve. We show that a certain complicated integral occurs when computing the average of observables in this model. This integral does not factorise. Since similar problems should also exist in the quantum case, we think that a serious question arises of how to deal with these integrals.