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Parisen Toldin F. Finite-size scaling at fixed renormalization-group invariant. Phys Rev E 2022; 105:034137. [PMID: 35428125 DOI: 10.1103/physreve.105.034137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2022] [Accepted: 03/07/2022] [Indexed: 06/14/2023]
Abstract
Finite-size scaling at fixed renormalization-group invariant is a powerful and flexible technique to analyze Monte Carlo data at a critical point. It consists in fixing a given renormalization-group invariant quantity to a given value, thereby trading its statistical fluctuations with those of a parameter driving the transition. One remarkable feature is the observed significant improvement of statistical accuracy of various quantities, as compared to a standard analysis. We review the method, discussing in detail its implementation, the error analysis, and a previously introduced covariance-based optimization. Comprehensive benchmarks on the Ising model in two and three dimensions show large gains in the statistical accuracy, which are due to cross-correlations between observables. As an application, we compute an accurate estimate of the inverse critical temperature of the improved O(2) ϕ^{4} model on a three-dimensional cubic lattice.
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Affiliation(s)
- Francesco Parisen Toldin
- Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
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2
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Parisen Toldin F. Boundary Critical Behavior of the Three-Dimensional Heisenberg Universality Class. PHYSICAL REVIEW LETTERS 2021; 126:135701. [PMID: 33861124 DOI: 10.1103/physrevlett.126.135701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2020] [Accepted: 02/19/2021] [Indexed: 06/12/2023]
Abstract
We study the boundary critical behavior of the three-dimensional Heisenberg universality class, in the presence of a bidimensional surface. By means of high-precision Monte Carlo simulations of an improved lattice model, where leading bulk scaling corrections are suppressed, we prove the existence of a special phase transition, with unusual exponents, and of an extraordinary phase with logarithmically decaying correlations. These findings contrast with naïve arguments on the bulk-surface phase diagram, and allow us to explain some recent puzzling results on the boundary critical behavior of quantum spin models.
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Affiliation(s)
- Francesco Parisen Toldin
- Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
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Dohm V. Crossover from low-temperature to high-temperature fluctuations: Universal and nonuniversal Casimir forces of isotropic and anisotropic systems. Phys Rev E 2018; 97:062128. [PMID: 30011477 DOI: 10.1103/physreve.97.062128] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2017] [Indexed: 11/07/2022]
Abstract
We study the crossover from low-temperature to high-temperature fluctuations including Goldstone-dominated and critical fluctuations in confined isotropic and weakly anisotropic O(n)-symmetric systems on the basis of a finite-size renormalization-group approach at fixed dimension d introduced previously [V. Dohm, Phys. Rev. Lett. 110, 107207 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.107207]. Our theory is formulated within the φ^{4} lattice model in a d-dimensional block geometry with periodic boundary conditions. We calculate the finite-size scaling functions F^{ex} and X of the excess free-energy density and the thermodynamic Casimir force, respectively, for 1≤n≤∞, 2<d<4. Exact results are derived for n→∞. Applications are given for L_{∥}^{d-1}×L slab geometry with an aspect ratio ρ=L/L_{∥}>0 and for film geometry (ρ=0). Good overall agreement is found with Monte Carlo (MC) data for isotropic spin models with n=1,2,3. For ρ=0, the low-temperature limits of F^{ex} and X vanish for n=1, whereas they are finite for n≥2. For ρ>0 and n=1, we find a finite low-temperature limit of F^{ex}, which deviates from that of the Ising model. We attribute this deviation to the nonuniversal difference between the φ^{4} model with continuous variables and the Ising model with discrete variables. For n≥2 and ρ>0, a logarithmic divergence of F^{ex} in the low-temperature limit is predicted, in excellent agreement with MC data. For 2≤n≤∞ and ρ<ρ_{0}=0.8567 the Goldstone modes generate a negative low-temperature Casimir force that vanishes for ρ=ρ_{0} and becomes positive for ρ>ρ_{0}. For anisotropic systems a unified hypothesis of multiparameter universality is introduced for both bulk and confined systems. The dependence of their scaling functions on d(d+1)/2-1 microscopic anisotropy parameters implies a substantial reduction of the predictive power of the theory for anisotropic systems as compared to isotropic systems. An exact representation is derived for the nonuniversal large-distance behavior of the bulk correlation function of anisotropic systems and quantitative predictions are made. The validity of multiparameter universality is proven analytically for the d=2,n=1 universality class. A nonuniversal anisotropy-dependent minimum of the Casimir force scaling function X is found. Both the sign and magnitude of X and the shift of the film critical temperature are affected by the lattice anisotropy.
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Affiliation(s)
- Volker Dohm
- Institute for Theoretical Physics, RWTH Aachen University, D-52056 Aachen, Germany
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Nagy S. Critical exponents of theO(N)model in the infrared limit from functional renormalization. Int J Clin Exp Med 2012. [DOI: 10.1103/physrevd.86.085020] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Parisen Toldin F. Improvement of Monte Carlo estimates with covariance-optimized finite-size scaling at fixed phenomenological coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:025703. [PMID: 21929057 DOI: 10.1103/physreve.84.025703] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2011] [Revised: 06/20/2011] [Indexed: 05/31/2023]
Abstract
In the finite-size scaling analysis of Monte Carlo data, instead of computing the observables at fixed Hamiltonian parameters, one may choose to keep a renormalization-group invariant quantity, also called phenomenological coupling, fixed at a given value. Within this scheme of finite-size scaling, we exploit the statistical covariance between the observables in a Monte Carlo simulation in order to reduce the statistical errors of the quantities involved in the computation of the critical exponents. This method is general and does not require additional computational time. This approach is demonstrated in the Ising model in two and three dimensions, where large gain factors in CPU time are obtained.
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Affiliation(s)
- Francesco Parisen Toldin
- Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany.
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Collura M. Off-equilibrium relaxational dynamics with an improved Ising Hamiltonian. JOURNAL OF STATISTICAL MECHANICS: THEORY AND EXPERIMENT 2010; 2010:P12036. [DOI: 10.1088/1742-5468/2010/12/p12036] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/02/2023]
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Weigel M, Janke W. Error estimation and reduction with cross correlations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:066701. [PMID: 20866538 DOI: 10.1103/physreve.81.066701] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2010] [Indexed: 05/29/2023]
Abstract
Besides the well-known effect of autocorrelations in time series of Monte Carlo simulation data resulting from the underlying Markov process, using the same data pool for computing various estimates entails additional cross correlations. This effect, if not properly taken into account, leads to systematically wrong error estimates for combined quantities. Using a straightforward recipe of data analysis employing the jackknife or similar resampling techniques, such problems can be avoided. In addition, a covariance analysis allows for the formulation of optimal estimators with often significantly reduced variance as compared to more conventional averages.
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Affiliation(s)
- Martin Weigel
- Theoretische Physik, Universität des Saarlandes, D-66041 Saarbrücken, Germany.
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Balog I, Uzelac K. Equilibriumlike invaded cluster algorithm: critical exponents and dynamical properties. Phys Rev E 2010; 81:041111. [PMID: 20481681 DOI: 10.1103/physreve.81.041111] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2010] [Indexed: 11/07/2022]
Abstract
We present a detailed study of the equilibrium-like invaded cluster algorithm, recently proposed as an extension of the invaded cluster algorithm, designed to drive the system to criticality while still preserving the equilibrium ensemble. We perform extensive simulations on two special cases of the Potts model and examine the precision of critical exponents by including the leading corrections. We show that both thermal and magnetic critical exponents can be obtained with high accuracy compared to the best available results. The choice of the auxiliary parameters of the algorithm is discussed in context of dynamical properties. We also discuss the relation to the Li-Sokal bound for the dynamical exponent z.
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Affiliation(s)
- I Balog
- Institute of Physics, P.O. Box 304, Bijenicka cesta 46, HR-10001 Zagreb, Croatia.
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Weigel M, Janke W. Cross correlations in scaling analyses of phase transitions. PHYSICAL REVIEW LETTERS 2009; 102:100601. [PMID: 19392099 DOI: 10.1103/physrevlett.102.100601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2008] [Indexed: 05/27/2023]
Abstract
Thermal or finite-size scaling analyses of importance sampling Monte Carlo computer simulations in the vicinity of phase transition points often combine different estimates for the same quantity, such as a critical exponent, with the intent to reduce statistical fluctuations. We point out that the origin of such estimates in the same time series often results in pronounced cross correlations which are usually ignored even in high-precision studies, generically leading to significant underestimation of statistical fluctuations. We suggest to use a simple extension of the conventional analysis taking correlation effects into account, which leads to improved estimators with often substantially reduced statistical fluctuations at almost no extra cost in terms of computation time.
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Affiliation(s)
- Martin Weigel
- Theoretische Physik, Universität des Saarlandes, D-66041 Saarbrücken, Germany.
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Hasenbusch M, Toldin FP, Pelissetto A, Vicari E. Universal dependence on disorder of two-dimensional randomly diluted and random-bond +/-J Ising models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:011110. [PMID: 18763922 DOI: 10.1103/physreve.78.011110] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2008] [Indexed: 05/26/2023]
Abstract
We consider the two-dimensional randomly site diluted Ising model and the random-bond +/-J Ising model (also called the Edwards-Anderson model), and study their critical behavior at the paramagnetic-ferromagnetic transition. The critical behavior of thermodynamic quantities can be derived from a set of renormalization-group equations, in which disorder is a marginally irrelevant perturbation at the two-dimensional Ising fixed point. We discuss their solutions, focusing in particular on the universality of the logarithmic corrections arising from the presence of disorder. Then, we present a finite-size scaling analysis of high-statistics Monte Carlo simulations. The numerical results confirm the renormalization-group predictions, and in particular the universality of the logarithmic corrections to the Ising behavior due to quenched dilution.
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Affiliation(s)
- Martin Hasenbusch
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, Leipzig, Germany
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Calabrese P, Martín-Mayor V, Pelissetto A, Vicari E. Three-dimensional randomly dilute Ising model: Monte Carlo results. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:036136. [PMID: 14524861 DOI: 10.1103/physreve.68.036136] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2003] [Indexed: 05/24/2023]
Abstract
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
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Affiliation(s)
- Pasquale Calabrese
- Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy.
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Campostrini M, Pelissetto A, Rossi P, Vicari E. 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:066127. [PMID: 12188803 DOI: 10.1103/physreve.65.066127] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2002] [Indexed: 05/23/2023]
Abstract
25th-order high-temperature series are computed for a general nearest-neighbor three-dimensional Ising model with arbitrary potential on the simple cubic lattice. In particular, we consider three improved potentials characterized by suppressed leading scaling corrections. Critical exponents are extracted from high-temperature series specialized to improved potentials, obtaining gamma=1.2373(2), nu=0.63012(16), alpha=0.1096(5), eta=0.036 39(15), beta=0.326 53(10), and delta=4.78 93(8). Moreover, biased analyses of the 25th-order series of the standard Ising model provide the estimate Delta=0.52(3) for the exponent associated with the leading scaling corrections. By the same technique, we study the small-magnetization expansion of the Helmholtz free energy. The results are then applied to the construction of parametric representations of the critical equation of state, using a systematic approach based on a global stationarity condition. Accurate estimates of several universal amplitude ratios are also presented.
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Affiliation(s)
- Massimo Campostrini
- Dipartimento di Fisica dell'Università di Pisa and INFN, I-00185 Rome, Italy.
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Calabrese P, Pelissetto A, Vicari E. Critical structure factors of bilinear fields in O(N) vector models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046115. [PMID: 12005934 DOI: 10.1103/physreve.65.046115] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2001] [Indexed: 05/23/2023]
Abstract
We compute the two-point correlation functions of general quadratic operators in the high-temperature phase of the three-dimensional O(N) vector model by using field-theoretical methods. In particular, we study the small- and large-momentum behavior of the corresponding scaling functions, and give general interpolation formulas based on a dispersive approach. Moreover, we determine the crossover exponent phi(T) associated with the traceless tensorial quadratic field, by computing and analyzing its six-loop perturbative expansion in fixed dimension. We find phi(T)=1.184(12), phi(T)=1.271(21), and phi(T)=1.40(4) for N=2,3,5, respectively.
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Affiliation(s)
- Pasquale Calabrese
- Scuola Normale Superiore and INFN, Sezione di Pisa, I-56100 Pisa, Italy.
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Hasenbusch M. Eliminating leading corrections to scaling in the three-dimensionalO(N)-symmetric ϕ4model:N= 3 and 4. ACTA ACUST UNITED AC 2001. [DOI: 10.1088/0305-4470/34/40/302] [Citation(s) in RCA: 106] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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16
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Calabrese P, Caselle M, Celi A, Pelissetto A, Vicari E. Non-analyticity of the Callan-Symanzik β-function of two-dimensional O(N) models. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0305-4470/33/46/301] [Citation(s) in RCA: 42] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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17
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Itakura M. Monte Carlo renormalization-group analysis of the lattice phi 4 model in D = 3,4. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:5924-5929. [PMID: 11031655 DOI: 10.1103/physreve.61.5924] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
We present a simple, sophisticated method to capture renormalization-group flow in Monte Carlo simulation, which provides important information of critical phenomena. We applied the method to the D = 3,4 lattice phi 4 model and obtained a renormalization flow diagram that well reproduces theoretically predicted behavior of the continuum phi 4 model. We also show that the method can be easily applied to much more complicated models, such as frustrated spin models.
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Affiliation(s)
- M Itakura
- Center for Promotion of Computational Science and Engineering, Japan Atomic Energy Research Institute, Tokyo, Japan
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Campostrini M, Pelissetto A, Rossi P, Vicari E. Improved high-temperature expansion and critical equation of state of three-dimensional Ising-like systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3526-63. [PMID: 11970188 DOI: 10.1103/physreve.60.3526] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/1999] [Indexed: 04/18/2023]
Abstract
High-temperature series are computed for a generalized three-dimensional Ising model with arbitrary potential. Three specific "improved" potentials (suppressing leading scaling corrections) are selected by Monte Carlo computation. Critical exponents are extracted from high-temperature series specialized to improved potentials, achieving high accuracy; our best estimates are gamma=1.2371(4), nu=0.630 02(23), alpha=0.1099(7), eta=0.0364(4), beta=0.326 48(18). By the same technique, the coefficients of the small-field expansion for the effective potential (Helmholtz free energy) are computed. These results are applied to the construction of parametric representations of the critical equation of state. A systematic approximation scheme, based on a global stationarity condition, is introduced (the lowest-order approximation reproduces the linear parametric model). This scheme is used for an accurate determination of universal ratios of amplitudes. A comparison with other theoretical and experimental determinations of universal quantities is presented.
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Affiliation(s)
- M Campostrini
- Dipartimento di Fisica dell'Università di Pisa and INFN, I-56126 Pisa, Italy.
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Hasenbusch M, Török T. High-precision Monte Carlo study of the 3DXY-universality class. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/36/301] [Citation(s) in RCA: 95] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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