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

## Abstract

We showed, In our previous work, 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 two-dimensional 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 symmetriesmore »

- Authors:

- LPTENS, Ecole Normale Superieure, Paris (France); Univ. Pierre et Marie Curie, Paris (France)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1240068

- Report Number(s):
- LLNL-JRNL-610232

Journal ID: ISSN 1751-8121

- Grant/Contract Number:
- AC52-07NA27344

- Resource 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 1751-8121

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE

### Citation Formats

```
Jacobsen, Jesper Lykke, and Scullard, Christian R. Transfer matrix computation of critical polynomials for two-dimensional Potts models. United States: N. p., 2013.
Web. doi:10.1088/1751-8113/46/7/075001.
```

```
Jacobsen, Jesper Lykke, & Scullard, Christian R. Transfer matrix computation of critical polynomials for two-dimensional Potts models. United States. doi:10.1088/1751-8113/46/7/075001.
```

```
Jacobsen, Jesper Lykke, and Scullard, Christian R. Mon .
"Transfer matrix computation of critical polynomials for two-dimensional Potts models". United States. doi:10.1088/1751-8113/46/7/075001. https://www.osti.gov/servlets/purl/1240068.
```

```
@article{osti_1240068,
```

title = {Transfer matrix computation of critical polynomials for two-dimensional Potts models},

author = {Jacobsen, Jesper Lykke and Scullard, Christian R.},

abstractNote = {We showed, In our previous work, that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional 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 = eK — 1 of PB(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, PB(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of PB(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, 82), kagome, and (3, 122) 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 vc 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 vc(4, 82) = 3.742 489 (4), vc(kagome) = 1.876 459 7 (2), and vc(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.},

doi = {10.1088/1751-8113/46/7/075001},

journal = {Journal of Physics. A, Mathematical and Theoretical (Online)},

number = 7,

volume = 46,

place = {United States},

year = {2013},

month = {2}

}

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