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We propose an exact summation method to compute thermodynamic observables inintegrable quantum field theories. The key idea is to use the matrix-treetheorem to write the Gaudin determinants that appear in the cluster expansionas a sum over graphs. For theories with a diagonal S-matrix, this method ismore powerful than the standard Thermodynamic Bethe Ansatz (TBA) technique asit is exact to all orders of powers in inverse volume. We have obtained usingthis method the TBA equation, the excited state energies in finite volume, theLeclair-Mussardo formula for one point functions, the finite-temperatureboundary entropy and cumulants of conserved charges in Generalized GibbsEnsembles. Moreover, the graph expansion can also be regarded as an alternativeto algebraic manipulations involving Gaudin determinants. We have applied thisidea to derive the equations of state and other transport properties inGeneralized Hydrodynamics. For theories with a non-diagonal S-matrix, thedescription of a complete set of states is more involved and it is not knownhow a cluster expansion can be implemented. It is nevertheless possible toapply the direct summation method in reverse and interpret known TBA equationswith complex strings in terms of diagrams.

A diagrammatic approach towards the thermodynamics of integrable systems