Abstract. Reported are two applications of the functional relations (T -system) among a commuting family of row-to-row transfer matrices proposed in the previous paper Part I.For a general simple Lie algebra X r , we determine the correlation lengths of the associated massive vertex models in the anti-ferroelectric regime and central charges of the RSOS models in two critical regimes. The results reproduce known values or even generalize them, demonstrating the efficiency of the T -system.

T- and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov–Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analog of L-operators in KP hierarchy, Stokes phenomena in 1D Schrödinger problem, AdS/CFT correspondence, Toda field equations on discrete spacetime, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ansätze, quantum transfer matrix method and so forth. This review is a collection of short reviews on these topics which can be read more or less independently.

The thermodynamics of solvable isotropic chains with arbitrary spins is addressed by the recently developed quantum transfer matrix (QTM) approach. The set of nonlinear equations which exactly characterize the free energy is derived by respecting the physical excitations at T = 0, spinons and RSOS kinks. We argue the implication of the present formulation to spinon character formula of level k = 2S SU(2) WZWN

We study the analytic Bethe ansatz in solvable vertex models associated with the Yangian Y (X r ) or its quantum affine analogue U q (X (1) r ) for X r = B r , C r and D r . Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations of Y (X r ). Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying the T -system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semistandard Young tableaux.