Generalised Gibbs Ensembles from the ODE/IM Correspondence: the Lee--Yang Model

Abstract

We introduce a systematic scheme for constructing the ODE/IM counterpart of generalised Gibbs ensembles, and apply it to the chiral sector of the Lee--Yang conformal field theory. We encode the chemical potentials in a deformation of the cubic oscillator, fixed order by order by requiring that the large-energy WKB period reproduces exactly the driving term of the corresponding thermodynamic Bethe ansatz equation; the subleading WKB periods then yield, as predictions, the deformed eigenvalues of the local conserved charges. We work out the spin-5 ensemble in detail for two states of the model: the ground state, in the module of the primary $\phi$, dual to the deformed cubic oscillator, and the primary of the identity module, dual to the same oscillator supplemented by an apparent singularity, a double pole whose position is fixed by a trivial-monodromy condition. In both cases, the Stokes multiplier obeys the Lee--Yang Y-system, and the deformed eigenvalues match, to the orders checked, an independent conformal-field-theory computation. The Borel resummation of the resulting series for the ground-state energy reproduces the numerical TBA solution over many orders of magnitude in the chemical potential. The scheme relies only on general features of the ODE/IM correspondence and applies to generalised Gibbs ensembles in other integrable models.