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Otsuka H, Mori K, Okabe Y, Nomura K. Level spectroscopy of the square-lattice three-state Potts model with a ferromagnetic next-nearest-neighbor coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:046103. [PMID: 16383464 DOI: 10.1103/physreve.72.046103] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2004] [Revised: 06/22/2005] [Indexed: 05/05/2023]
Abstract
We study the square-lattice three-state Potts model with the ferromagnetic next-nearest-neighbor coupling at finite temperature. Using the level-spectroscopy method, we numerically analyze the excitation spectrum of the transfer matrices and precisely determine the global phase diagram. Then we find that, contrary to a previous result based on the finite-size scaling, the massless region continues up to the decoupling point with Z3 x Z3 criticality in the antiferromagnetic region. We also check the universal relations among excitation levels to provide the reliability of our result.
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Affiliation(s)
- Hiromi Otsuka
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397 Japan
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2
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de Queiroz SLA. Finite-size investigation of scaling corrections in the square-lattice three-state Potts antiferromagnet. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:056104. [PMID: 12059644 DOI: 10.1103/physreve.65.056104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/01/2002] [Indexed: 05/23/2023]
Abstract
We investigate the finite-temperature corrections to scaling in the three-state square-lattice Potts antiferromagnet, close to the critical point at T=0. Numerical diagonalization of the transfer matrix on semi-infinite strips of width L sites, 4 < or = L < or = 14, yields finite-size estimates of the corresponding scaled gaps, which are extrapolated to L --> infinity. Owing to the characteristics of the quantities under study, we argue that the natural variable to consider is x identical with L e(-2 beta). For the extrapolated scaled gaps we show that square-root corrections, in the variable x, are present, and provide estimates for the numerical values of the amplitudes of the first- and second-order correction terms, for both the first and second scaled gaps. We also calculate the third scaled gap of the transfer matrix spectrum at T=0, and find an extrapolated value of the decay-of-correlations exponent, eta(3)=2.00(1). This is at odds with earlier predictions, to the effect that the third relevant operator in the problem would give eta(P(stagg))=3, corresponding to the staggered polarization.
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Affiliation(s)
- S L A de Queiroz
- Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21945-970 Rio de Janeiro RJ, Brazil.
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Kim SY, Creswick RJ. Density of states, Potts zeros, and Fisher zeros of the Q-state Potts model for continuous Q. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:066107. [PMID: 11415173 DOI: 10.1103/physreve.63.066107] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2000] [Revised: 01/16/2001] [Indexed: 05/23/2023]
Abstract
The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a-1 (a=e(betaJ)) and the coefficients of this polynomial, Phi(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute Phi(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z(CP)=Z(Q,a(c)(Q)), where a(c)(Q)=1+square root[Q]. We find a set of zeros in the complex w=square root[Q] plane that map to (or close to) the Beraha numbers for real positive Q. We also identify Q(c)(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/square root[Q] plane lies on the unit circle. By finite-size scaling we find that 1/Q(c)(L)-->0 as L-->infinity. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q(c)(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Baxter's conjecture Q(AF)(c)(a)=(1-a)(a+3). We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that Q(FM)(c)(a)=(a-1)(2). We show that the edge singularity in the complex Q plane approaches Q(c) as Q(c)(L) approximately Q(c)+AL(-y(q)), and determine the scaling exponent y(q) for several values of Q. Finally, by finite-size scaling of the Fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y(t) as a function of Q in the range 2</=Q</=3. Using data for lattices of size 3</=L</=8 we find that y(t) is a smooth function of Q and is well fitted by y(t)=(1+Au+Bu2)/(C+Du) where u=-(2/pi)cos(-1)(squareroot[Q]/2). For Q=3 we find y(t) approximately 0.6; however if we include lattices up to L=12 we find y(t) approximately 0.50(8) in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. 96, 461 (1999)].
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Affiliation(s)
- S Y Kim
- Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA.
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4
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Chang SC, Shrock R. Ground-state entropy of the potts antiferromagnet with next-nearest-neighbor spin-spin couplings on strips of the square lattice. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:4650-4664. [PMID: 11089004 DOI: 10.1103/physreve.62.4650] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/1999] [Revised: 05/15/2000] [Indexed: 05/23/2023]
Abstract
We present exact calculations of the zero-temperature partition function (chromatic polynomial) and W(q), the exponent of the ground-state entropy, for the q-state Potts antiferromagnet with next-nearest-neighbor spin-spin couplings on square lattice strips, of width L(y)=3 and L(y)=4 vertices and arbitrarily great length Lx vertices, with both free and periodic boundary conditions. The resultant values of W for a range of physical q values are compared with each other and with the values for the full two-dimensional lattice. These results give insight into the effect of such nonnearest-neighbor couplings on the ground-state entropy. We show that the q=2 (Ising) and q=4 Potts antiferromagnets have zero-temperature critical points on the Lx-->infinity limits of the strips that we study. With the generalization of q from Z+ to C, we determine the analytic structure of W(q) in the q plane for the various cases.
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Affiliation(s)
- SC Chang
- C. N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, New York 11794-3840, USA
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Moore C, Nordahl MG, Minar N, Shalizi CR. Vortex dynamics and entropic forces in antiferromagnets and antiferromagnetic Potts models. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:5344-51. [PMID: 11970403 DOI: 10.1103/physreve.60.5344] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/1999] [Revised: 07/27/1999] [Indexed: 04/18/2023]
Abstract
It is well known in models with an interface representation, such as the dimer model, the triangular Ising antiferromagnet, the six-vertex ice model, and the three-state antiferromagnetic Potts model on the square lattice, that topological defects of opposite charge are attracted with an entropically-driven Coulomb force. We examine the Potts model in detail, show explicitly how this force is felt through local fields, and calculate the defects' mobility. We then take two approaches to measuring this force numerically. First, we quench a random initial state to zero temperature and measure the density of defects rho(t) as a function of time. While this gives some evidence for a local force, we compare it with a free diffusion experiment, and show that the asymptotic decay of rho(t) depends on the initial distribution of defects rather than the forces between them. Second, we set up initial conditions with a single pair of vortices, and measure the force between them as a function of distance. This gives reasonable agreement with theory, although finite-size effects and a lack of ergodicity play a significant role.
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Affiliation(s)
- C Moore
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA.
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Temesvari T. Interface method for the antiferromagnetic three-state Potts model on a square lattice. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/15/11/010] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Abstract
The critical temperature of the antiferromagnetic
q
-state Potts model on the square lattice is located, and the critical free energy and internal energy are evaluated. As with the ferromagnetic model, the transition is continuous for
q
≼4, and its first-order (i. e. has latent heat) for
q
>4. However, only for
q
≼3 can the critical temperature be real. For the isotropic model the criticality condition is exp(
J
/
k
T
) = -1 + (4-
q
)
½
.
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Matveev V, Shrock R. Complex-temperature singularities in Potts models on the square lattice. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:6174-6185. [PMID: 9965837 DOI: 10.1103/physreve.54.6174] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Ferreira SJ, Sokal AD. Antiferromagnetic Potts models on the square lattice. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:6727-6730. [PMID: 9977213 DOI: 10.1103/physrevb.51.6727] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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11
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Potts model on infinitely ramified Sierpinski-gasket-type fractals and algebraic order at antiferromagnetic phases. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:11642-11656. [PMID: 10003053 DOI: 10.1103/physrevb.46.11642] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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12
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Qin Y, Yang ZR. Diamond-type hierarchical lattices for the Potts antiferromagnet. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:8576-8582. [PMID: 9996489 DOI: 10.1103/physrevb.43.8576] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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13
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Wang JS, Swendsen RH, Koteck R. Three-state antiferromagnetic Potts models: A Monte Carlo study. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 42:2465-2474. [PMID: 9995696 DOI: 10.1103/physrevb.42.2465] [Citation(s) in RCA: 71] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Wang JS, Swendsen RH, Kotecký R. Antiferromagnetic Potts models. PHYSICAL REVIEW LETTERS 1989; 63:109-112. [PMID: 10040782 DOI: 10.1103/physrevlett.63.109] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Banavar JR, Bray AJ. Heisenberg and Potts spin glasses: A renormalization-group study. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 38:2564-2569. [PMID: 9946566 DOI: 10.1103/physrevb.38.2564] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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16
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Koteck R. Long-range order for antiferromagnetic Potts models. PHYSICAL REVIEW. B, CONDENSED MATTER 1985; 31:3088-3092. [PMID: 9936165 DOI: 10.1103/physrevb.31.3088] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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