Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus

AbstractUsing results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x★=limL→∞ξ(L)/L and the first four magnetization moment ratios V2n=〈 $$M$$ 2n〉/〈 $$M$$ 2〉n. As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*2n. We confirm these predictions by a high-precision Monte Carlo simulation.

[1]  J. Zuber,et al.  Critical Ising correlation functions in the plane and on the torus , 1987 .

[2]  Guo,et al.  Hyperuniversality and the renormalization group for finite systems. , 1987, Physical review. B, Condensed matter.

[3]  McCullough,et al.  Higher-order corrections for the quadratic Ising lattice susceptibility. , 1988, Physical review. B, Condensed matter.

[4]  Charles M. Newman,et al.  Normal fluctuations and the FKG inequalities , 1980 .

[5]  Low-temperature effective potential of the Ising model , 1998, cond-mat/9805317.

[6]  Eleventh-order calculation of Ising-limit Green's functions for scalar quantum field theory in arbitrary space-time dimension D. , 1994, Physical review. D, Particles and fields.

[7]  Finite Size Scaling and “perfect” actions: the three dimensional Ising model , 1998, hep-lat/9805022.

[8]  Multigrid Monte Carlo simulation via XY embedding. II. Two-dimensional SU(3) principal chiral model , 1996, hep-lat/9610021.

[9]  G. Parisi,et al.  Ising exponents in the two-dimensional site-diluted Ising model , 1997, cond-mat/9707179.

[10]  Janke,et al.  Critical exponents of the classical three-dimensional Heisenberg model: A single-cluster Monte Carlo study. , 1993, Physical review. B, Condensed matter.

[11]  Hungary.,et al.  Comparison of the O ( 3 ) Bootstrap σ-Model with the Lattice Regularization at Low Energies , 1999 .

[12]  Perturbative renormalization group, exact results, and high-temperature series to order 21 for the N-vector spin models on the square lattice. , 1996, Physical review. B, Condensed matter.

[13]  B. Nickel.  On the singularity structure of the 2D Ising model susceptibility , 1999 .

[14]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[15]  H. Blote,et al.  Universal ratio of magnetization moments in two-dimensional Ising models , 1993 .

[16]  Li,et al.  Rigorous lower bound on the dynamic critical exponents of the Swendsen-Wang algorithm. , 1989, Physical review letters.

[17]  Critical exponents of the N-vector model , 1998, cond-mat/9803240.

[18]  Universal amplitude combinations for self-avoiding walks, polygons and trails , 1993, cond-mat/9303035.

[19]  N. Kawashima,et al.  Renormalized coupling constant in the Ising model , 1996 .

[20]  A. Sokal,et al.  The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk , 1988 .

[21]  M. Fisher,et al.  Universal critical amplitudes in finite-size scaling , 1984 .

[22]  B. Widom,et al.  Book Review:The Critical Point. A Historical Introduction to the Modern Theory of Critical Phenomena. Cyril Domb, Taylor and Francis, London, 1996 , 1998 .

[23]  A. Pelissetto,et al.  The effective potential in three-dimensional O(N) models , 1998, cond-mat/9801098.

[24]  Universal amplitude ratios in the two-dimensional $q$-state Potts model and percolation from quantum field theory , 1997, hep-th/9712111.

[25]  Kim,et al.  Studying the continuum limit of the Ising model. , 1993, Physical Review D, Particles and fields.

[26]  U. Wolff,et al.  A Numerical method to compute the running coupling in asymptotically free theories , 1991 .

[27]  Lai,et al.  Renormalized coupling constants and related amplitude ratios for Ising systems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[28]  Low-temperature series for the correlation length in the d = 3 Ising model , 1994, hep-lat/9407023.

[29]  C. L. Henley,et al.  A constrained Potts antiferromagnet model with an interface representation , 1997 .

[30]  David P. Landau,et al.  Phase transitions and critical phenomena , 1989, Computing in Science & Engineering.

[31]  3D Ising Model:The Scaling Equation of State , 1996, hep-th/9610223.

[32]  S. Shlosman.  Signs of the Ising model Ursell functions , 1986 .

[33]  C. Tracy,et al.  Spin spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region , 1976 .

[34]  Series studies of the Potts model: III. The 3-state model on the simple cubic lattice , 1993, hep-lat/9312083.

[35]  Critical renormalized coupling constants in the symmetric phase of the Ising models , 1999, cond-mat/9905138.

[36]  Jean Zinn-Justin,et al.  Critical exponents from field theory , 1980 .

[37]  G. A. Baker.  The Markov property method applied to Ising model calculations , 1994 .

[38]  Probability distribution of the order parameter for the three-dimensional ising-model universality class: A high-precision monte carlo study , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[39]  J. Bricmont.  The Gaussian inequality for multicomponent rotators , 1977 .

[40]  Off-shell dynamics of the O(3) NLS model beyond Monte Carlo and perturbation theory , 1996, hep-th/9612039.

[41]  M. Hasenbusch,et al.  Critical behaviour of the 3D XY-model: a Monte Carlo study , 1993 .

[42]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[43]  C. Itzykson,et al.  Conformal invariance and applications to statistical mechanics , 1998 .

[44]  Alan D. Sokal,et al.  Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study , 1998 .

[45]  N vector spin models on the sc and the bcc lattices: A Study of the critical behavior of the susceptibility and of the correlation length by high temperature series extended to order beta(21) , 1997, hep-lat/9703018.

[46]  UNIVERSAL AMPLITUDES IN THE FINITE-SIZE SCALING OF THREE-DIMENSIONAL SPIN MODELS , 1998, cond-mat/9809253.

[47]  Universal critical coupling constants for the three-dimensional n-vector model from field theory. , 1998, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[48]  M. Nijs,et al.  Critical fan in the antiferromagnetic three-state Potts model , 1982 .

[49]  Universal amplitude ratios in the two-dimensional Ising model 1 Work supported by the European Union , 1997, hep-th/9710019.

[50]  M. Fisher,et al.  Nonlinear scaling fields and corrections to scaling near criticality , 1983 .

[51]  Erik Luijten,et al.  Ising universality in three dimensions: a Monte Carlo study , 1995, cond-mat/9509016.

[52]  Susceptibility amplitude ratios in the two-dimensional Potts model and percolation , 1999, cond-mat/9908453.

[53]  Calculation of universal amplitude ratios in three-loop order , 1996, cond-mat/9606091.

[54]  Alan D. Sokal,et al.  Logarithmic Corrections and Finite-Size Scaling in the Two-Dimensional 4-State Potts Model , 1996 .

[55]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[56]  C. Domb,et al.  The Critical Point: A Historical Introduction To The Modern Theory Of Critical Phenomena , 1996 .

[57]  C. Naón.  COMMENT: Checking Cardy's formula for the Baxter model , 1989 .

[58]  Alan D. Sokal,et al.  Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks , 1994 .

[59]  A. D. Sokal,et al.  Dynamic critical behavior of a Swendsen-Wang-Type algorithm for the Ashkin-Teller model , 1996 .

[60]  Meyer,et al.  Monte Carlo renormalization of the 3D Ising model: Analyticity and convergence. , 1996, Physical Review Letters.

[61]  Andrea J. Liu,et al.  The three-dimensional Ising model revisited numerically , 1989 .

[62]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[63]  A. Sokal,et al.  Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. , 1988, Physical review. D, Particles and fields.

[64]  Pierre Mathieu,et al.  Conformal Field Theory , 1999 .

[65]  L. Schulman.  In: Finite size scaling and numerical simulation of statistical systems , 1990 .

[66]  Universal amplitude ratios in the 2D four-state Potts model , 1999, cond-mat/9902146.

[67]  C. Fortuin,et al.  On the random-cluster model II. The percolation model , 1972 .

[68]  G. Duerinckx.  On the one-point Friedrichs model , 1984 .

[69]  Four-point renormalized coupling constant and Callan-Symanzik beta-function in O(N) models , 1997, cond-mat/9711078.

[70]  Series studies of the Potts model. I. The simple cubic Ising model , 1992, hep-lat/9212032.

[71]  Kim.  Application of finite size scaling to Monte Carlo simulations. , 1993, Physical review letters.

[72]  M. Fisher,et al.  Universal surface-tension and critical-isotherm amplitude ratios in three dimensions , 1996 .

[73]  Ferreira,et al.  Extrapolating Monte Carlo simulations to infinite volume: Finite-size scaling at xi /L >> 1. , 1995, Physical review letters.

[74]  The Three-State Square-Lattice Potts Antiferromagnet at Zero Temperature , 1998, cond-mat/9801079.

[75]  Universal amplitude ratios in the three-dimensional Ising model , 1997, hep-lat/9701007.

[76]  C. Newman.  A general central limit theorem for FKG systems , 1983 .

[77]  C. Merrifield.  Elementary Treatise on Elliptic Functions , 1877, Nature.

[78]  Renormalized couplings and scaling correction amplitudes in the N -vector spin models on the sc and the bcc lattices , 1998, hep-lat/9805025.

[79]  by Arch. Rat. Mech. Anal. , 2022 .

[80]  High-temperature series analyses of the classical Heisenberg and XY models , 1993, hep-lat/9305005.

[81]  Low temperature expansion for the Ising model. , 1992, Physical review letters.

[82]  J. Cardy,et al.  Central charge and universal combinations of amplitudes in two-dimensional theories away from criticality. , 1988, Physical review letters.