Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Finite-size scaling analysis of generalized mean-field theories

Steffen D. Frischat, Reimer Kühn

Published May 21, 1995• 1 citation
Publisher

Abstract

We investigate families of generalized mean-field theories that can be formulated using the Peierls-Bogoliubov inequality. For test Hamiltonians describing mutually non-interacting subsystems of increasing size, the thermodynamics of these mean-field-type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean-field critical points. Finite-size scaling analysis of this mean-field critical behaviour allows us to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki (1987), which is thus given a clear interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls-Bogoliubov inequality. In the case of the 2D Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity in the thermodynamic functions.

DOI
10.1088/0305-4470/28/10/010

Related papers

Finite--Size Scaling Analysis of Generalized Mean--Field Theories

Steffen D. Frischat, Reimer Kühn

We investigate families of generalized mean--field theories that can be formulated using the Peierls--Bogoliubov inequality. For test--Hamiltonians describing mutually non--interacting subsystems of increasing size, the thermodynamics of these mean--field type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean--field critical points. Finite--size scaling analysis of this mean--field critical behaviour allows to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki, which is thus given a transparent interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls--Bogoliubov inequality. In the case of the 2--$d$ Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity of thermodynamic functions.

arXiv:cond-mat/9501002•Jan 3, 1995•0 citations

Coherent-Anomaly Method in Critical Phenomena. III. Mean-Field Transfer-Matrix Method in the 2D Ising Model

Xiao Hu, Makoto Katori, Masuo Suzuki

Two kinds of systematic mean-field transfer-matrix methods are formulated in the 2-dimensional Ising spin system, by introducing Weiss-like and Bethe-like approximations. All the critical exponents as well as the true critical point can be estimated in these methods following the CAM procedure. The numerical results of the above system are T c * ≃2.271 ( J / k B ), γ=γ'≃1.749, β≃0.131 and δ≃15.1. The specific heat is confirmd to be continuous and to have a logarithmic divergence at the true critical point, i.e., α=α'=0. Thus, the finite-degree-of-approximation scaling ansatz is shown to be correct and very powerful in practical estimations of the critical exponents as well as the true critical point.

Journal of the Physical Society of Japan•Nov 15, 1987•57 citations

Anomalous mean-field behavior of the fully connected Ising model

Louis Colonna-Romano, Harvey Gould, W. Klein

Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.

arXiv:1403.4854•Physical Review E•Oct 6, 2014•17 citations

New Method to Study Critical Phenomena–Mean-Field Finite-Size Scaling Theory

Masuo Suzuki, Makoto Katori

A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.

Journal of the Physical Society of Japan•Jan 15, 1986•45 citations

Determination of the critical exponents in dissipative phase transitions: Coherent anomaly approach

Jiasen Jin, Wenbin He, Fernando Iemini +4 more

We propose a generalization of the coherent anomaly method to extract the critical exponents of a phase transition occurring in the steady-state of an open quantum many-body system. The method, originally developed by Suzuki [J. Phys. Soc. Jpn. 55, 4205 (1986)] for equilibrium systems, is based on the scaling properties of the singularity in the response functions determined through cluster mean-field calculations. We apply this method to the dissipative transverse-field Ising model and the dissipative $XYZ$ model in two dimensions obtaining convergent results already with small clusters.

arXiv:2103.07255•Physical review. B./Physical review. B•Dec 1, 2021•7 citations

Extrapolation-CAM theory for critical exponents

Howard L Richards, Naomichi Hatano, M. A. Novotny

We propose and test a new method for generating canonical sequences for analysis by the coherent anomaly method (CAM) from non-mean-field approximations. By intentionally underestimating the rate of convergence of exact-diagonalization values for the mass or energy gaps of finite systems, we form families of sequences of gap estimates. The gap estimates cross zero with generically nonzero linear terms in their Taylor expansions, so that for each member of these sequences of estimates. Thus, the CAM can be used to determine . Our freedom in deciding exactly how to underestimate the convergence allows us to choose the sequence that displays the clearest coherent anomaly. We demonstrate this approach on the two-dimensional ferromagnetic Ising model, for which . We also use it on the three-dimensional ferromagnetic Ising model, finding , in good agreement with other estimates. Finally, we apply it to an antiferromagnetic spin-1 Heisenberg chain, finding at the phase transition between the Haldane phase and the dimerized phase, in agreement with the field-theoretic prediction . Although the specific systems used to test the extrapolation-CAM procedure involve finite system sizes, the method could be applied to other finite approximations, such as systematic variational approximations.

arXiv:cond-mat/9703222•Journal of Physics A Mathematical and General•Sep 21, 1997•0 citations
We investigate families of generalized mean-field theories that can be formulated using the Peierls-Bogoliubov inequality. For test Hamiltonians describing mutually non-interacting subsystems of increasing size, the thermodynamics of these mean-field-type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean-field critical points. Finite-size scaling analysis of this mean-field critical behaviour allows us to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki (1987), which is thus given a clear interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls-Bogoliubov inequality. In the case of the 2D Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity in the thermodynamic functions.
We investigate families of generalized mean--field theories that can be formulated using the Peierls--Bogoliubov inequality. For test--Hamiltonians describing mutually non--interacting subsystems of increasing size, the thermodynamics of these mean--field type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean--field critical points. Finite--size scaling analysis of this mean--field critical behaviour allows to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki, which is thus given a transparent interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls--Bogoliubov inequality. In the case of the 2--ddd Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity of thermodynamic functions.
Two kinds of systematic mean-field transfer-matrix methods are formulated in the 2-dimensional Ising spin system, by introducing Weiss-like and Bethe-like approximations. All the critical exponents as well as the true critical point can be estimated in these methods following the CAM procedure. The numerical results of the above system are T c * ≃2.271 ( J / k B ), γ=γ'≃1.749, β≃0.131 and δ≃15.1. The specific heat is confirmd to be continuous and to have a logarithmic divergence at the true critical point, i.e., α=α'=0. Thus, the finite-degree-of-approximation scaling ansatz is shown to be correct and very powerful in practical estimations of the critical exponents as well as the true critical point.
Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.
A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.
We propose a generalization of the coherent anomaly method to extract the critical exponents of a phase transition occurring in the steady-state of an open quantum many-body system. The method, originally developed by Suzuki [J. Phys. Soc. Jpn. 55, 4205 (1986)] for equilibrium systems, is based on the scaling properties of the singularity in the response functions determined through cluster mean-field calculations. We apply this method to the dissipative transverse-field Ising model and the dissipative XYZXYZXYZ model in two dimensions obtaining convergent results already with small clusters.
We propose and test a new method for generating canonical sequences for analysis by the coherent anomaly method (CAM) from non-mean-field approximations. By intentionally underestimating the rate of convergence of exact-diagonalization values for the mass or energy gaps of finite systems, we form families of sequences of gap estimates. The gap estimates cross zero with generically nonzero linear terms in their Taylor expansions, so that for each member of these sequences of estimates. Thus, the CAM can be used to determine . Our freedom in deciding exactly how to underestimate the convergence allows us to choose the sequence that displays the clearest coherent anomaly. We demonstrate this approach on the two-dimensional ferromagnetic Ising model, for which . We also use it on the three-dimensional ferromagnetic Ising model, finding , in good agreement with other estimates. Finally, we apply it to an antiferromagnetic spin-1 Heisenberg chain, finding at the phase transition between the Haldane phase and the dimerized phase, in agreement with the field-theoretic prediction . Although the specific systems used to test the extrapolation-CAM procedure involve finite system sizes, the method could be applied to other finite approximations, such as systematic variational approximations.