Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras

Abstract

In previous work with Scullard, we have defined a graph polynomial PB(q, T) that gives access to the critical temperature Tc of the q-state Potts model defined on a general two-dimensional lattice It depends on a basis B, containing n × m unit cells of and the relevant root Tc(n, m) of PB(q, T) was observed to converge quickly to Tc in the limit Moreover, in exactly solvable cases there is no finite-size dependence at all. In this paper we show how to reformulate this method as an eigenvalue problem within the periodic Temperley–Lieb (TL) algebra. This corresponds to taking first, so that the bases B are semi-infinite cylinders of circumference n. The limit implies faster convergence in n, while maintaining the n-independence in exactly solvable cases. In this setup, Tc(n) is determined by equating the largest eigenvalues of two topologically distinct sectors of the transfer matrix. Crucially, these two sectors determine the same critical exponent in the continuum limit, and the observed fast convergence is thus corroborated by results of conformal field theory. We obtain similar results for the dense and dilute phases of the O(N) loop model, using now a transfer matrix within the dilute periodic TL algebra. Compared with our previous study, the eigenvalue formulation allows us to double the size n for which Tc(n) can be obtained, using the same computational effort. We study in details three significant cases: (i) bond percolation on the kagome lattice, up to nmax = 14; (ii) site percolation on the square lattice, to nmax = 21; and (iii) self-avoiding polygons on the square lattice, to nmax = 19. Convergence properties of Tc(n) and extrapolation schemes are studied in details for the first two cases. This leads to rather accurate values for the percolation thresholds: pc = 0.524 404 999 167 439(4) for bond percolation on the kagome lattice, and pc = 0.592 746 050 792 10(2) for site percolation on the square lattice.