Anharmonic oscillators, the Thermodynamic Bethe Ansatz, and nonlinear integral equations

Zengo Tsuboi

We derive traditional thermodynamic Bethe ansatz (TBA) equations for the sl ( r + 1) Uimin–Sutherland model from the T -system of the quantum transfer matrix. These TBA equations are identical to the those from the string hypothesis. Next we derive a new family of nonlinear integral equations (NLIEs). In particular, a subset of these NLIEs forms a system of NLIEs which contains only a finite number of unknown functions. For r = 1, this subset of NLIEs reduces to Takahashi's NLIE for the XXX spin chain. A relation between the traditional TBA equations and our new NLIEs is clarified. Based on our new NLIEs, we also calculate the high-temperature expansion of the free energy.

Olalla A. Castro-Alvaredo, Andreas Fring

Craig A. Tracy, Harold Widom

We summarize some recent connections between a class of nonlinear integral equations related to the thermodynamic Bethe Ansatz and a class of linear integral equations related to the Painlevé equations.

K. S. Viswanathan

George P. Flessas, K. P. Das

Kazumitsu Sakai, Zengo Tsuboi

The thermodynamic Bethe ansatz is applied to a quantum integrable spin chain associated with the Lie superalgebra osp (1|2). Using the string hypothesis, we derive a set of infinite number of nonlinear integral equations (thermodynamic Bethe ansatz equation), which characterize the free energy. The low temperature limit of the free energy is also discussed.