Nonlinear Integral Equations and high temperature expansion for the $U_{q}(\hat{sl}(r+1|s+1))$ Perk-Schultz Model

We propose a system of nonlinear integral equations (NLIE) which gives the free energy of the $U_{q}(widehat{sl}(r+1|s+1))$ Perk-Schultz model. In contrast with traditional thermodynamic Bethe ansatz equations, our NLIE contain only r+s+1 unknown functions. In deriving the NLIE, the quantum (supersymmetric) Jacobi-Trudi and Giambelli formula and a duality for an auxiliary function play important roles. By using our NLIE, we also calculate the high temperature expansion of the free energy. General formulae of the coefficients with respect to arbitrarily rank r+s+1, chemical potentials $\{\mu_{a}\}$ and q have been written down in terms of characters up to the order of 5. In particular for specific values of the parameters, we have calculated the high temperature expansion of the specific heat up to the order of 40.