Transfer matrix computation of critical polynomials for twodimensional Potts models
We showed, In our previous work, that critical manifolds of the qstate 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 contractiondeletion 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 contractiondeletion. We discuss in detail the role of the symmetries and the embedding of B. The criticalmore »
 Authors:

^{[1]};
^{[2]}
 LPTENS, Ecole Normale Superieure, Paris (France); Univ. Pierre et Marie Curie, Paris (France)
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 OSTI Identifier:
 1240068
 Report Number(s):
 LLNLJRNL610232
Journal ID: ISSN 17518121
 Grant/Contract Number:
 AC5207NA27344
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Physics. A, Mathematical and Theoretical (Online)
 Additional Journal Information:
 Journal Name: Journal of Physics. A, Mathematical and Theoretical (Online); Journal Volume: 46; Journal Issue: 7; Journal ID: ISSN 17518121
 Publisher:
 IOP Publishing
 Research Org:
 Lawrence Livermore National Laboratory (LLNL), Livermore, CA
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE