2015/07/10 by Jesper Lykke Jacobsen, Jacobsen, Jesper Lykke · 1 citation
Mathematics · Physics and Astronomy · #Opinion Dynamics and Social Influence #Quantum many-body systems #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1507.03027
33 pages, 7 figures, 5 tables
arxiv created 2015/07/10 · arxiv updated 2015/07/19
In previous work with Scullard, we defined a graph polynomial PB(q,T) that gives access to the critical temperature Tc of the q-state Potts model on a general two-dimensional lattice L. It depends on a basis B, containing n x m unit cells of L, and the relevant root of PB(q,T) was observed to converge quickly to Tc in the limit n,m to infinity. Moreover, in exactly solvable cases there is no finite-size dependence at all. We reformulate this method as an eigenvalue problem within the periodic Temperley-Lieb algebra. This corresponds to taking m to infinity first, so 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 Temperley-Lieb algebra. 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 n = 14; (ii) site percolation on the square lattice, to n = 21; and (iii) self-avoiding polygons on the square lattice, to n = 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 = 0.524404999167439(4) for bond percolation on the kagome lattice, and pc = 0.59274605079210(2) for site percolation on the square lattice.