One-point functions in finite volume/temperature: a case study

TL;DR

This paper combines numerical and analytical methods to accurately compute finite volume and temperature expectation values of local operators in integrable quantum field theories, revealing insights into singularities and finite size corrections.

Contribution

It demonstrates that the truncated conformal space approach, combined with a renormalization group, can be extended to low energies and matched with low-temperature expansions for precise expectation value calculations.

Findings

Validated the consistency between numerical and analytical methods.

Identified and analyzed singularities in finite volume matrix elements.

Highlighted the importance of exponential finite size corrections for a complete description.

Abstract

We consider finite volume (or equivalently, finite temperature) expectation values of local operators in integrable quantum field theories using a combination of numerical and analytical approaches. It is shown that the truncated conformal space approach, when supplemented with a recently proposed renormalization group, can be sufficiently extended to the low-energy regime that it can be matched with high precision by the low-temperature expansion proposed by Leclair and Mussardo. Besides verifying the consistency of the two descriptions, their combination leads to an evaluation of expectation values which is valid to a very high precision for all volume/temperature scales. As a side result of the investigation, we also discuss some unexpected singularities in the framework recently proposed by Pozsgay and Tak\'acs for the description of matrix elements of local operators in finite volume, and show that while some of these singularities are resolved by the inclusion of the class of exponential finite size corrections known as \mu-terms, these latter corrections themselves lead to the appearance of new singularities. We point out that a fully consistent description of finite volume matrix elements is expected to be free of singularities, and therefore a more complete and systematic understanding of exponential finite size corrections is necessary.