The critical boundary RSOS $\mathcal{M}(3,5)$ model

We consider the critical nonunitary minimal model $\mathcal{M}(3,5)$ with integrable boundaries and analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory using the thermodynamic Bethe ansatz (TBA) equations. Solving the TBA functional equation satisfied by the transfer matrices of the associated $A_4$ restricted solid-on-solid Forrester–Baxter lattice model in regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the $(r,s)=(1,1)$ sector and then determine their corresponding energies. We classify the excitations in terms of $(m,n)$ systems.