Exact finite volume expectation values of local operators in excited states

TL;DR

This paper proposes an exact spectral expansion formula for finite volume expectation values of local operators in excited states of integrable quantum field theories, extending previous conjectures to more complex scattering scenarios.

Contribution

It introduces a generalized conjecture for finite volume expectation values in excited states, incorporating bound states and bootstrap structures, and provides proof and numerical validation.

Findings

Conjecture proven for energy-momentum tensor trace.

Numerical evidence supports validity for most states and volumes.

Expansion fails at small volumes with complex TBA singularities.

Abstract

We present a conjecture for the exact expression of finite volume expectation values in excited states in integrable quantum field theories, which is an extension of an earlier conjecture to the case of general diagonal factorized scattering with bound states and a nontrivial bootstrap structure. The conjectured expression is a spectral expansion which uses the exact form factors and the excited state thermodynamic Bethe Ansatz as building blocks. The conjecture is proven for the case of the trace of the energy-moment tensor. Concerning its validity for more general operators, we provide numerical evidence using the truncated conformal space approach. It is found that the expansion fails to be well-defined for small values of the volume in cases when the singularity structure of the TBA equations undergoes a non-trivial rearrangement under some critical value of the volume. Despite these shortcomings, the conjectured expression is expected to be valid for all volumes for most of the excited states, and as an expansion above the critical volume for the rest.