A diagrammatic approach towards the thermodynamics of integrable systems

TL;DR

This paper introduces an exact diagrammatic summation method for calculating thermodynamic properties in integrable quantum field theories, surpassing traditional TBA techniques in diagonal S-matrix cases and providing new insights into non-diagonal cases.

Contribution

It develops a novel graph-based summation approach using the matrix-tree theorem to compute thermodynamic observables exactly in integrable systems, extending beyond standard TBA methods.

Findings

Derived the TBA equation using the graph expansion

Computed excited state energies and one-point functions in finite volume

Applied the method to generalized hydrodynamics equations

Abstract

We propose an exact summation method to compute thermodynamic observables in integrable quantum field theories. The key idea is to use the matrix-tree theorem to write the Gaudin determinants that appear in the cluster expansion as a sum over graphs. For theories with a diagonal S-matrix, this method is more powerful than the standard Thermodynamic Bethe Ansatz (TBA) technique as it is exact to all orders of powers in inverse volume. We have obtained using this method the TBA equation, the excited state energies in finite volume, the Leclair-Mussardo formula for one point functions, the finite-temperature boundary entropy and cumulants of conserved charges in Generalized Gibbs Ensembles. Moreover, the graph expansion can also be regarded as an alternative to algebraic manipulations involving Gaudin determinants. We have applied this idea to derive the equations of state and other transport properties in Generalized Hydrodynamics. For theories with a non-diagonal S-matrix, the description of a complete set of states is more involved and it is not known how a cluster expansion can be implemented. It is nevertheless possible to apply the direct summation method in reverse and interpret known TBA equations with complex strings in terms of diagrams.