Transfer matrix computation of critical polynomials for two-dimensional Potts models

Abstract

Abstract.In our previous work [1] we have shown that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial P B (q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic twodimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = e K − 1 of P B (q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, P B (q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of P B (q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible.We present results for the critical polynomial on the (4, 8 2 ), kagome, and (3, 12 2 ) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures v c obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain v c (4, 82 ) = 3.742 489 (4), v c (kagome) = 1.876 459 7 (2), and v c (3, 122 ) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.

The three-state Potts antiferromagnet on plane quadrangulations

“…seven) quadrangulations of self-dual (resp. non-self-dual) type using several complementary techniques: MC simulations, TM computations, and the method of critical polynomials (CP) [28,35,36,70,71]. These numerical results agree, without exception, with conjecture 1.1.…”

“…The location of the phase transitions for the quadrangulations in figures 2 and 3 can also be studied by the method of critical polynomials [28,35,36,70,71]. These polynomials P B (q, v) can in principle be computed for any lattice generated by the tessellation of the plane by some finite basis B.…”

Phase diagram of the triangular-lattice Potts antiferromagnet

“…The existence of the BK phase has been verified for all Archimedean lattices in [39,40], and its extent in the (Q, v) plane has been accurately estimated. It was found that in most, but not all [38], cases Q c = 4.…”

Bond percolation thresholds on Archimedean lattices from critical polynomial roots

“…In addition to the interesting open problems related to the critical polynomial method, it is yet to be determined how widely applicable it is. The definition (3) is easily generalized to the Q-state Potts model and gives excellent estimates for any Q [31,40], even in the imaginary temperature regime [41]. The eigenvalue method presented here was also adapted to compute the growth constant of self-avoiding walks and was able finally to rule out a longstanding conjecture [39].…”