1
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Devre HY, Berker AN. First-order to second-order phase transition changeover and latent heats of q-state Potts models in d=2,3 from a simple Migdal-Kadanoff adaptation. Phys Rev E 2022; 105:054124. [PMID: 35706195 DOI: 10.1103/physreve.105.054124] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2022] [Accepted: 04/26/2022] [Indexed: 06/15/2023]
Abstract
The changeover from first-order to second-order phase transitions in q-state Potts models is obtained at q_{c}=2 in spatial dimension d=3 and essentially at q_{c}=4 in d=2, using a physically intuited simple adaptation of the Migdal-Kadanoff renormalization-group transformation. This simple procedure yields the latent heats at the first-order phase transitions. In both d=2 and 3, the calculated phase transition temperatures, respectively compared with the exact self-duality and Monte Carlo results, are dramatically improved. The method, when applied to a slab of finite thickness, yields dimensional crossover.
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Affiliation(s)
- H Yağız Devre
- Üsküdar American Academy, Üsküdar, Istanbul 34664, Turkey
| | - A Nihat Berker
- Faculty of Engineering and Natural Sciences, Kadir Has University, Cibali, Istanbul 34083, Turkey
- TÜBITAK Research Institute for Fundamental Sciences, Gebze, Kocaeli 41470, Turkey
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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2
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Gürleyen SE, Berker AN. Asymmetric phase diagrams, algebraically ordered Berezinskii-Kosterlitz-Thouless phase, and peninsular Potts flow structure in long-range spin glasses. Phys Rev E 2022; 105:024122. [PMID: 35291165 DOI: 10.1103/physreve.105.024122] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2021] [Accepted: 02/02/2022] [Indexed: 11/07/2022]
Abstract
The Ising spin-glass model on the three-dimensional (d=3) hierarchical lattice with long-range ferromagnetic or spin-glass interactions is studied by the exact renormalization-group solution of the hierarchical lattice. The chaotic characteristics of the spin-glass phases are extracted in the form of our calculated, in this case continuously varying, Lyapunov exponents. Ferromagnetic long-range interactions break the usual symmetry of the spin-glass phase diagram. This phase-diagram symmetry breaking is dramatic, as it is underpinned by renormalization-group peninsular flows of the Potts multicritical type. A Berezinskii-Kosterlitz-Thouless (BKT) phase with algebraic order and a BKT-spin-glass phase transition with continuously varying critical exponents are seen. Similarly, for spin-glass long-range interactions, the Potts mechanism is also seen, by the mutual annihilation of stable and unstable fixed distributions causing the abrupt change of the phase diagram. On one side of this abrupt change, two distinct spin-glass phases, with finite (chaotic) and infinite (chaotic) coupling asymptotic behaviors are seen with a spin-glass to spin-glass phase transition.
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Affiliation(s)
- S Efe Gürleyen
- Department of Physics, Istanbul Technical University, Maslak, Istanbul 34469, Turkey
| | - A Nihat Berker
- Faculty of Engineering and Natural Sciences, Kadir Has University, Cibali, Istanbul 34083, Turkey.,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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3
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Chang SC, Shrock R. Exact results for average cluster numbers in bond percolation on infinite-length lattice strips. Phys Rev E 2021; 104:044107. [PMID: 34781558 DOI: 10.1103/physreve.104.044107] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2021] [Accepted: 09/15/2021] [Indexed: 11/07/2022]
Abstract
We calculate exact analytic expressions for the average cluster numbers 〈k〉_{Λ_{s}} on infinite-length strips Λ_{s}, with various widths, of several different lattices, as functions of the bond occupation probability p. It is proved that these expressions are rational functions of p. As special cases of our results, we obtain exact values of 〈k〉_{Λ_{s}} and derivatives of 〈k〉_{Λ_{s}} with respect to p, evaluated at the critical percolation probabilities p_{c,Λ} for the corresponding infinite two-dimensional lattices Λ. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in 〈k〉_{Λ_{s}} determine the radii of convergence of series expansions for small p and for p near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.
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Affiliation(s)
- Shu-Chiuan Chang
- Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
| | - Robert Shrock
- C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
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4
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Najafi MN, Cheraghalizadeh J, Herrmann HJ. Elastic backbone phase transition in the Ising model. Phys Rev E 2019; 100:042132. [PMID: 31770915 DOI: 10.1103/physreve.100.042132] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2019] [Indexed: 11/07/2022]
Abstract
The two-dimensional (zero magnetic field) Ising model is known to undergo a second-order paraferromagnetic phase transition, which is accompanied by a correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In this paper we uncover that there exists also a second temperature T_{eb}<T_{c} at which the elastic backbone of FK clusters undergoes a second-order phase transition to a dense phase. The corresponding universality class, which is characterized by determining various percolation exponents, is shown to be completely different from directed percolation, which leads us to propose a new anisotropic universality class with β=0.54±0.02, ν_{||}=1.86±0.01, ν_{⊥}=1.21±0.04, and d_{f}=1.53±0.03. All tested hyperscaling relations are shown to be valid.
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Affiliation(s)
- M N Najafi
- Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran.,Computational Physics, IfB, ETH Zurich, Stefano-Franscini-Platz 3, CH-8093 Zurich, Switzerland
| | - J Cheraghalizadeh
- Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
| | - H J Herrmann
- Computational Physics, IfB, ETH Zurich, Stefano-Franscini-Platz 3, CH-8093 Zurich, Switzerland.,Departamento de Física, Universidade Federal do Ceara, 60451-970 Fortaleza, Brazil.,ESPCI, CNRS UMR 7636, Laboratoire PMMH, 75005 Paris, France
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5
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Mertens S, Moore C. Percolation thresholds and Fisher exponents in hypercubic lattices. Phys Rev E 2018; 98:022120. [PMID: 30253462 DOI: 10.1103/physreve.98.022120] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2018] [Indexed: 06/08/2023]
Abstract
We use invasion percolation to compute highly accurate numerical values for bond and site percolation thresholds p_{c} on the hypercubic lattice Z^{d} for d=4,...,13. We also compute the Fisher exponent τ governing the cluster size distribution at criticality. Our results support the claim that the mean-field value τ=5/2 holds for d≥6, with logarithmic corrections to power-law scaling at d=6.
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Affiliation(s)
- Stephan Mertens
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
- Institut für Physik, Universität Magdeburg, Universitätsplatz 2, 39016 Magdeburg, Germany
| | - Cristopher Moore
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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6
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Yi H. First- and second-order quantum phase transitions of a q-state Potts model in fractal lattices. Phys Rev E 2018; 96:062105. [PMID: 29347356 DOI: 10.1103/physreve.96.062105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2017] [Indexed: 11/07/2022]
Abstract
Quantum phase transitions of a q-state Potts model in fractal lattices are studied using a continuous-time quantum Monte Carlo simulation technique. For small values of q, the transition is found to be second order and critical exponents of the quantum critical point are calculated. The dynamic critical exponent z is found to be greater than one for all fractals studied, which is in contrast to integer-dimensional regular lattices. When q is greater than a certain value q_{c}, the phase transition becomes first order, where q_{c} depends on the lattice. Further analysis shows that the characteristics of phase transitions are more sensitive to the average number of nearest neighbors than the Hausdorff dimension or the order of ramification.
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Affiliation(s)
- Hangmo Yi
- Department of Physics, Soongsil University, Seoul 06978, Korea and Institute for Integrative Basic Sciences, Soongsil University, Seoul 06978, Korea
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7
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Lajkó P, Iglói F. Entanglement entropy of the Q≥4 quantum Potts chain. Phys Rev E 2017; 95:012105. [PMID: 28208455 DOI: 10.1103/physreve.95.012105] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2016] [Indexed: 11/07/2022]
Abstract
The entanglement entropy S is an indicator of quantum correlations in the ground state of a many-body quantum system. At a second-order quantum phase-transition point in one dimension S generally has a logarithmic singularity. Here we consider quantum spin chains with a first-order quantum phase transition, the prototype being the Q-state quantum Potts chain for Q>4 and calculate S across the transition point. According to numerical, density matrix renormalization group results at the first-order quantum phase transition point S shows a jump, which is expected to vanish for Q→4^{+}. This jump is calculated in leading order as ΔS=lnQ[1-4/Q-2/(QlnQ)+O(1/Q^{2})].
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Affiliation(s)
- Péter Lajkó
- Department of Physics, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
| | - Ferenc Iglói
- Wigner Research Centre, Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, Hungary.,Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary
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8
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Hu H, Ziff RM, Deng Y. No-Enclave Percolation Corresponds to Holes in the Cluster Backbone. PHYSICAL REVIEW LETTERS 2016; 117:185701. [PMID: 27835010 DOI: 10.1103/physrevlett.117.185701] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2016] [Indexed: 06/06/2023]
Abstract
The no-enclave percolation (NEP) model introduced recently by Sheinman et al. can be mapped to a problem of holes within a standard percolation backbone, and numerical measurements of such holes give the same size-distribution exponent τ=1.82(1) as found for the NEP model. An argument is given that τ=1+d_{B}/2≈1.822 for backbone holes, where d_{B} is the backbone dimension. On the other hand, a model of simple holes within a percolation cluster yields τ=1+d_{f}/2=187/96≈1.948, where d_{f} is the fractal dimension of the cluster, and this value is consistent with the experimental results of gel collapse of Sheinman et al., which give τ=1.91(6). This suggests that the gel clusters are of the universality class of percolation cluster holes. Both models give a discontinuous maximum hole size at p_{c}, signifying explosive percolation behavior.
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Affiliation(s)
- Hao Hu
- Hefei National Laboratory for Physical Sciences at Microscale, Department of Modern Physics, University of Science and Technology of China, Hefei 230027, China
| | - Robert M Ziff
- Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA
| | - Youjin Deng
- Hefei National Laboratory for Physical Sciences at Microscale, Department of Modern Physics, University of Science and Technology of China, Hefei 230027, China
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9
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Najafi MN, Tavana A. Universality class of the structural phase transition in the normal phase of cuprate superconductors. Phys Rev E 2016; 94:022110. [PMID: 27627249 DOI: 10.1103/physreve.94.022110] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2016] [Indexed: 06/06/2023]
Abstract
The tetragonal-orthorhombic structural phase transition of oxygen atoms in the basal plane of YBa_{2}Cu_{3}O_{6+δ} high-T_{C} cuprate superconductors is studied numerically. By mapping the system onto the asymmetric next-nearest-neighbor Ising model, we characterize this phase transition. Results indicate the degrees of critical behavior. We show that this phase transition occurs at the temperature T_{C}≃0.148eV in the thermodynamic limit. By analyzing the critical exponents, it is found that this universality class displays some common features, with the two-dimensional three-state Potts model universality class, although the possibility of other universality classes cannot be ruled out. Conformal invariance at T=T_{c} is investigated using the Schramm-Loewner evolution (SLE) technique, and it is found that the SLE diffusivity parameter for this system is 3.34±0.01.
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Affiliation(s)
- M N Najafi
- Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
| | - A Tavana
- Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
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10
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Liu XW, Deng Y, Jacobsen JL. Recursive percolation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:010103. [PMID: 26274102 DOI: 10.1103/physreve.92.010103] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2014] [Indexed: 06/04/2023]
Abstract
We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and find compelling numerical evidence that it can be repeated recursively any number n of generations. In two dimensions, we determine the percolation thresholds up to n=5. The corresponding critical clusters become more and more compact as n increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any n. This family of exponents differs from previously known universality classes, and cannot be accommodated by existing analytical methods. We confirm that recursive percolation is well defined also in three dimensions.
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Affiliation(s)
- Xuan-Wen Liu
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Youjin Deng
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jesper Lykke Jacobsen
- Laboratoire de Physique Théorique, École Normale Supérieure, 24 rue Lhomond, 75231 Paris, France
- Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France
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11
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Sasidevan V. Continuum percolation of overlapping disks with a distribution of radii having a power-law tail. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022140. [PMID: 24032808 DOI: 10.1103/physreve.88.022140] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2013] [Revised: 06/05/2013] [Indexed: 06/02/2023]
Abstract
We study the continuum percolation problem of overlapping disks with a distribution of radii having a power-law tail; the probability that a given disk has a radius between R and R+dR is proportional to R(-(a+1)), where a>2. We show that in the low-density nonpercolating phase, the two-point function shows a power-law decay with distance, even at arbitrarily low densities of the disks, unlike the exponential decay in the usual percolation problem. As in the problem of fluids with long-range interaction, we argue that in our problem, the critical exponents take their short-range values for a>3-η(sr) whereas they depend on a for a<3-η(sr) where η(sr) is the anomalous dimension for the usual percolation problem. The mean-field regime obtained in the fluid problem corresponds to the fully covered regime, a≤2, in the percolation problem. We propose an approximate renormalization scheme to determine the correlation length exponent ν and the percolation threshold. We carry out Monte Carlo simulations and determine the exponent ν as a function of a. The determined values of ν show that it is independent of the parameter a for a>3-η(sr) and is equal to that for the lattice percolation problem, whereas ν varies with a for 2<a<3-η(sr). We also determine the percolation threshold of the system as a function of the parameter a.
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Affiliation(s)
- V Sasidevan
- Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India
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12
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Noh JD, Lee HK, Park H. Scaling of cluster heterogeneity in percolation transitions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:010101. [PMID: 21867102 DOI: 10.1103/physreve.84.010101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2011] [Indexed: 05/31/2023]
Abstract
We investigate a critical scaling law for the cluster heterogeneity H in site and bond percolations in d-dimensional lattices with d = 2,...,6. The cluster heterogeneity is defined as the number of distinct cluster sizes. As an occupation probability p increases, the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase, and back to a monodisperse one in the supercritical phase. We show analytically that H diverges algebraically, approaching the percolation critical point p(c) as H |p-p(c)|(-1/σ) with the critical exponent σ associated with the characteristic cluster size. Interestingly, its finite-size-scaling behavior is governed by a new exponent ν H = 1+d (f)/(d)ν, where d(f) is the fractal dimension of the critical percolating cluster and ν is the correlation length exponent. The corresponding scaling variable defines a singular path to the critical point. All results are confirmed by numerical simulations.
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Affiliation(s)
- Jae Dong Noh
- Department of Physics, University of Seoul, Seoul 130-743, Republic of Korea and School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea
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13
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Kim Y, Koh K, Roll MF, Laine RM, Matzger AJ. Porous Networks Assembled from Octaphenylsilsesquioxane Building Blocks. Macromolecules 2010. [DOI: 10.1021/ma101597h] [Citation(s) in RCA: 65] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
| | | | | | - Richard M. Laine
- Department of Material Science and Engineering
- Macromolecular Science and Engineering
| | - Adam J. Matzger
- Department of Chemistry
- Macromolecular Science and Engineering
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14
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Huang X, Gong S, Zhong F, Fan S. Finite-time scaling via linear driving: application to the two-dimensional Potts model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041139. [PMID: 20481709 DOI: 10.1103/physreve.81.041139] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2009] [Revised: 03/25/2010] [Indexed: 05/29/2023]
Abstract
We apply finite-time scaling to the q-state Potts model with q=3 and 4 on two-dimensional lattices to determine its critical properties. This consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields. From the finite-time scaling of the order parameter, the coercivity, and the hysteresis area and its derivative, we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature. The static critical exponents obtained in general and the magnetic exponent delta in particular agree reasonably with the conjectured ones. The dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short-time dynamic results for q=4. Our results also suggest an alternative way to characterize the weak universality.
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Affiliation(s)
- Xianzhi Huang
- State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Zhongshan University, Guangzhou 510275, People's Republic of China
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15
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Fernandez LA, Gordillo-Guerrero A, Martin-Mayor V, Ruiz-Lorenzo JJ. Microcanonical finite-size scaling in second-order phase transitions with diverging specific heat. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:051105. [PMID: 20364945 DOI: 10.1103/physreve.80.051105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2009] [Revised: 09/23/2009] [Indexed: 05/29/2023]
Abstract
A microcanonical finite-size ansatz in terms of quantities measurable in a finite lattice allows extending phenomenological renormalization (the so-called quotients method) to the microcanonical ensemble. The ansatz is tested numerically in two models where the canonical specific heat diverges at criticality, thus implying Fisher renormalization of the critical exponents: the three-dimensional ferromagnetic Ising model and the two-dimensional four-state Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows simulating systems as large as L=1024 (Potts) or L=128 (Ising). The quotients method provides accurate determinations of the anomalous dimension, eta, and of the (Fisher-renormalized) thermal nu exponent. While in the Ising model the numerical agreement with our theoretical expectations is very good, in the Potts case, we need to carefully incorporate logarithmic corrections to the microcanonical ansatz in order to rationalize our data.
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Affiliation(s)
- L A Fernandez
- Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain
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16
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Kenna R, Johnston DA, Janke W. Self-consistent scaling theory for logarithmic-correction exponents. PHYSICAL REVIEW LETTERS 2006; 97:155702. [PMID: 17155337 DOI: 10.1103/physrevlett.97.155702] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2006] [Indexed: 05/12/2023]
Abstract
Multiplicative logarithmic corrections frequently characterize critical behavior in statistical physics. Here, a recently proposed theory relating the exponents of such terms is extended to account for circumstances which often occur when the leading specific-heat critical exponent vanishes. Also, the theory is widened to encompass the correlation function. The new relations are then confronted with results from the literature, and some new predictions for logarithmic corrections in certain models are made.
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Affiliation(s)
- R Kenna
- Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, England
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17
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Kenna R, Johnston DA, Janke W. Scaling relations for logarithmic corrections. PHYSICAL REVIEW LETTERS 2006; 96:115701. [PMID: 16605842 DOI: 10.1103/physrevlett.96.115701] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2005] [Indexed: 05/08/2023]
Abstract
Multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical-mechanical systems. Here, a Lee-Yang zero approach is used to systematically analyze the exponents of such logarithms and to propose scaling relations between them. These proposed relations are then confronted with a variety of results from the literature.
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Affiliation(s)
- R Kenna
- Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, England, United Kingdom
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18
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Scaling properties of percolation clusters. ACTA ACUST UNITED AC 2005. [DOI: 10.1007/bfb0012540] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register]
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19
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Jacobsen JL, Zinn-Justin P. Monochromatic path crossing exponents and graph connectivity in two-dimensional percolation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:055102. [PMID: 12513543 DOI: 10.1103/physreve.66.055102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2002] [Indexed: 05/24/2023]
Abstract
We consider the fractal dimensions d(k) of the k-connected part of percolation clusters in two dimensions, generalizing the cluster (k=1) and backbone (k=2) dimensions. The codimensions x(k)=2-d(k) describe the asymptotic decay of the probabilities P(r,R) approximately (r/R)(x(k)) that an annulus of radii r<<1 and R>>1 is traversed by k disjoint paths, all living on the percolation clusters. Using a transfer matrix approach, we obtain numerical results for x(k), k<or=6. They are well fitted by the ansatz x(k)=1 / 12k(2)+1 / 48k+(1-k)C, with C=0.0181+/-0.0006.
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Affiliation(s)
- Jesper Lykke Jacobsen
- Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France.
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20
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Jacobsen JL, Zinn-Justin P. A transfer matrix for the backbone exponent of two-dimensional percolation. ACTA ACUST UNITED AC 2002. [DOI: 10.1088/0305-4470/35/9/304] [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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22
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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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Shnidman Y, Domany E. Destruction of a first-order transition by dimensional crossover. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0022-3719/14/26/001] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Sahimi M, Hughes BD, Scriven LE, Davis HT. Critical exponent of percolation conductivity by finite-size scaling. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0022-3719/16/16/004] [Citation(s) in RCA: 86] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Sahimi M, Jerauld GR. Random walks on percolation clusters at the percolation threshold. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0022-3719/16/29/005] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Adler J. Conductivity exponents from the analysis of series expansions for random resistor networks. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/18/2/020] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Fukugita M, Mino H, Okawa M, Ukawa A. Resolving the order of phase transitions in Monte Carlo simulations. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/23/11/009] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Kertesz J. Speculation on a scaling law for superconductor-resistor mixture exponent s in a percolation system. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/16/13/006] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Sakamoto S, Yonezawa F, Aoki K, Nose S, Hori M. Percolation in two-dimensional quasicrystals by Monte Carlo simulations. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/22/14/010] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Pokorny M, Newman CM, Meiron D. The trapping transition in dynamic (invasion) and static percolation. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/23/8/016] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Adler J, Privman V. Two-dimensional percolation: logarithmic corrections to the critical behaviour from series expansions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/14/11/008] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Gould H, Holl K. Diffusivity and radius of random animals, percolation clusters and compact clusters. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/14/11/005] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Burkhardt TW, Guim I. Finite-size scaling of the quantum Ising chain with periodic, free, and antiperiodic boundary conditions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/18/1/006] [Citation(s) in RCA: 60] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Wilkinson D, Barsony M. Monte Carlo study of invasion percolation clusters in two and three dimensions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/17/3/007] [Citation(s) in RCA: 70] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Sahimi M, Hughes BD, Scriven LE, Davis HT. On Polya random walks, lattice Green functions, and the bond percolation threshold. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/16/2/004] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Essam JW, Bhatti FM. Series expansion evidence supporting the Alexander-Orbach conjecture in two dimensions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/18/18/019] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Gawlinski ET, Stanley HE. Continuum percolation in two dimensions: Monte Carlo tests of scaling and universality for non-interacting discs. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/14/8/007] [Citation(s) in RCA: 223] [Impact Index Per Article: 8.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Nienhuis B. Analytical calculation of two leading exponents of the dilute Potts model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/15/1/028] [Citation(s) in RCA: 221] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Styer DF, Edwards MD, Andrews EA. The size function in two-dimensional bond percolation: a series analysis. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/21/23/009] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Nightingale P, Blote H. The relation between amplitudes and critical exponents in finite-size scaling. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/16/17/005] [Citation(s) in RCA: 103] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Nightingale MP, Blote HWJ. Linear defects in two dimensional systems: a finite size investigation. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/15/1/007] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Blote HWJ, Nightingale MP, Derrida B. Critical exponents of two-dimensional Potts and bond percolation models. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/14/2/005] [Citation(s) in RCA: 79] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Adler J, Privman V. Series expansion analysis of corrections to scaling in the three-state Potts model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/15/8/008] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Harris AB, Lubensky TC. Field theoretic approaches to biconnectedness in percolating systems. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/16/11/005] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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