Conformal partition functions of critical percolation fromD₃thermodynamic Bethe Ansatz equations

Abstract

Using the planar Temperley-Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as LM(2, 3) in the Yang-Baxter integrable family of logarithmic minimal models LM(p, p ′ ). We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y -system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D 3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c = 0 and conformal weights ∆ 1,s for s ∈ Z 1 with ∆ r,s = (3r − 2s) 2 − 1 /24. For the periodic case, the finite-size corrections agree with the conformal weights ∆ 0,s , ∆ 1,s with s ∈ 1 2 Z 0 . These are obtained analytically using Rogers dilogarithm identities. We incorporate all finite excitations by formulating empirical selection rules for the patterns of zeros of all the eigenvalues of the standard modules. We thus obtain the conformal partition functions on the cylinder and the modular invariant partition function (MIPF) on the torus. By applying q-binomial and q-Narayana identities, it is shown that our refined finitized characters on the strip agree with those of Pearce, Rasmussen and Zuber. For percolation on the torus, the MIPF is a non-diagonal sesquilinear form in affine u(1) characters given by the u(1) partition function Z 2,3 (q) = Z

Extended T-systems, Q matrices and T-Q relations for $ \newcommand{\e}{{\rm e}} s\ell(2)$ models at roots of unity

“…In this way the patterns of zeros of the eigenvalues Q j,± (u) and Q ± (u) can be studied, for small system sizes N, without worrying about missing some eigenvalues [46] or dealing with unphysical solutions [44,45] to the Bethe ansatz equations. Indeed, this opens up the possibility that the patterns of zeros of Q j,± (u), and Q ± (u) can be completely classified in the same way that the patterns of zeros of T j (u) with j = 1, 2 are classified [65] for critical percolation.…”

“…Another advantage of the current approach is that all the T-systems considered here share a common universal D-type Y-system that closes finitely [65] and, at least in principle, can be solved analytically for the conformal spectra using the nonlinear integral equation and dilogarithm techniques of [13]. The Y-system is universal [66] in the sense that it holds for all boundary conditions and all topologies.…”

Yang–Baxter solution of dimers as a free-fermion six-vertex model

“…Here, we diagonalize the transfer matrices using inversion identities, to make clear the relation to critical dense polymers. The methods we use, based on Yang-Baxter integrability, functional equations and physical combinatorics, are more general, and can also be applied to percolation [45] and to the six-vertex model at other roots of unity.…”

Two-point boundary correlation functions of dense loop models

“…Recent results have set the door ajar to obtaining corner free energies in this way [5][6][7][8]. The integrable toolbox can furthermore be employed to compute finite-size corrections, either via the Bethe ansatz technique [9][10][11] or the approach using functional relations and Y -systems [12][13][14][15].…”