Airy Functions in the Thermodynamic Bethe Ansatz

Thermodynamic Bethe ansatz equations are coupled non-linear integral equations which appear frequently when solving integrable models. Those associated with models with N=2 supersymmetry can be related to differential equations, among them Painlevé III and the Toda hierarchy. In the simplest such case the massless limit of these non-linear integral equations can be solved in terms of the Airy function. This is the only known closed-form solution of thermodynamic Bethe ansatz equations, outside of free or classical models. This turns out to give the spectral determinant of the Schrodinger equation in a linear potential. A great deal of interesting mathematical physics has arisen from the study of integrable models of statistical mechanics and field theory. One interesting area is known as the thermodynamic Bethe ansatz (TBA), which has proven a useful tool for computing the free energy of an integrable 1+1 dimensional system [1]. One ends up with a set of coupled non-linear integral equations, the “TBA equations”. One completely-unexpected result was a correspondence between a limit of these integral equations and some very well-studied non-linear differential equations, namely the Toda hierarchy [2]. The purpose of this paper is to extend these results further, and show that in at least one case there is a closed-form but non-trivial solution of the integral equations. Not only is it interesting that such complicated equations have a simple solution in terms of the Airy function, but proving it requires some utilizing some very intricate results involving the Painlevé III differential equation [2, 3, 4]. Moreover, it turns out to be related to the spectral determinant of the Schrodinger equation in a linear potential [5, 6, 7]. The TBA integral equations are generically of the form ǫa(θ) = ma cosh θ − ∑ b ∫ dθ 2π φab(θ − θ) ln(1 + ebb ′)). (1) Physically, T ǫa(θ) is the energy for creating a particle of type a and rapidity θ in a thermal bath at temperature T . The ma are the particle masses over temperature, while the μa are their chemical potentials over temperature. The kernels φab are a result of the interactions between particles. This and all unlabelled integrals in this paper run from −∞ to ∞. The free energy per unit length is