Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

Jacobsen, Jesper Lykke; Salas, José Valero

doi:10.1007/s10955-005-8077-8

Cited by 36 publications

(152 citation statements)

References 92 publications

(424 reference statements)

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“…While this is in line with the general expectations outlined above, the possible connexion between P B (q, v) and the studies [34,35] larger bases might lead to the formation of vertical rays at other Beraha numbers than B 4 and B 6 .…”

Section: Square Latticesupporting

confidence: 89%

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

Jacobsen¹,

Scullard²

2013

J. Phys. A: Math. Theor.

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109

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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.

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Section: Square Latticesupporting

confidence: 89%

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

Jacobsen¹,

Scullard²

2013

J. Phys. A: Math. Theor.

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109

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“…For the square lattice we have only found phases with j ≤ 5. This should be compared with the arbitrarily high values of S z taken when approaching (Q, x) = (4, −1) from within the BK phase in the generic case [7,34].…”

Section: Discussionmentioning

confidence: 99%

Complex-temperature phase diagram of Potts and RSOS models

2006

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We study the phase diagram of Q-state Potts models, for Q = 4 cos 2 (π/p) a Beraha number (p > 2 integer), in the complex-temperature plane. The models are defined on L × N strips of the square or triangular lattice, with boundary conditions on the Potts spins that are periodic in the longitudinal (N ) direction and free or fixed in the transverse (L) direction. The relevant partition functions can then be computed as sums over partition functions of an A p−1 type RSOS model, thus making contact with the theory of quantum groups. We compute the accumulation sets, as N → ∞, of partition function zeros for p = 4, 5, 6, ∞ and L = 2, 3, 4 and study selected features for p > 6 and/or L > 4. This information enables us to formulate several conjectures about the thermodynamic limit, L → ∞, of these accumulation sets. The resulting phase diagrams are quite different from those of the generic case (irrational p). For free transverse boundary conditions, the partition function zeros are found to be dense in large parts of the complex plane, even for the Ising model (p = 4). We show how this feature is modified by taking fixed transverse boundary conditions.

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“…Shrock [16] gives a good introduction to their earlier work on chromatic polynomials, while the recent work [8] contains many other references. For the triangular lattice, which is the focus of this paper, the full Tutte polynomial for strips of this lattice has been computed and studied for widths up to 5 by Chang, Jacobsen, Salas & Shrock [7], and the chromatic polynomial for widths up to 10 by Jacobsen, Salas & Sokal [12].…”

Section: Double-ended Lattice Graphsmentioning

confidence: 99%

“…When Jacobsen, Salas & Sokal [12] examined the cylindrical triangular lattice 4 P × n F without end-graphs, they obtained a 2 × 2 transfer matrix, with only λ 2 and λ 3 as eigenvalues. The reason for this is that rather than using colour partitions to index the rows and columns of the transfer matrix, they exploited planarity by using only "noncrossing non-nearest neighbour" partitions.…”

Section: Notesmentioning

confidence: 99%