An exact solution of a 2*n generalisation (two states for horizontal arrows and n states for vertical arrows with total arrow number conservation) of the six-vertex model is given using the commuting transfer matrix method. In the symmetric case, the free energy is a sum of n-1 terms each of which is the free energy of a six-vertex model such that the phase transition into the frozen ferroelectric state of the 2*n model is identical to that in the zero-field six-vertex model. For the six-vertex model the commuting transfer matrix solution is extended to the case of arbitrary vertex weights and a derivation of the Baxter parametrisation is provided.
Abstract In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime
Δ<1 Δ<1 . As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when
a=b=1 a=b=1 and
c≥1 c≥1 , and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.