Restricted triangular ice beyond the solvable manifolds: chiral ice and the critical region of dominant chiral vertices

Abstract

We study two restrictions of Baxter's symmetric twenty-vertex model on the triangular lattice, in which the three classes of vertices allowed by the ice rule carry the weights $a$, $b$ and $c$: chiral ice, in which only the twelve chiral vertices are allowed, and the model that excludes only the two alternating vertices. We show that neither model belongs to any of the known exactly solvable families of the twenty-vertex model, and we characterise chiral ice as the ice model in which every vertex has exactly one through line, a property that imposes a parity constraint along every closed lattice line. We compute the residual entropies of both models with transfer matrices on cylinders of up to twelve vertices in circumference, extrapolated with the finite-size correction predicted by conformal invariance, which we adapt to the triangular geometry and verify at two exactly solved points. The number of upward arrows crossing the cylinder is conserved, and the cost of changing it shows that both models are critical, with the long-distance behaviour of a free massless field. Chiral ice is the extreme point of a region of the phase diagram in which ordering had been suggested; along two paths through that region we find the same critical behaviour at every point, with no feature at the proposed boundary of the region. Along Baxter's exactly solved curve we obtain the corresponding exponents in closed form.