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

Australian Journal of Physics

Finite-Size Scaling and Effective Lagrangian Theory

Chris J. Hamer

Published Dec 20, 1997• 0 citations
Publisher

Abstract

A brief review is given of the ‘effective Lagrangian’ approach of Leutwyler, Hasenfratz and others, which describes the behaviour at low energies or temperatures, or large distances, in lattice systems which undergo a first-order transition involving spontaneous breakdown of a continuous symmetry. Universal predictions can be given, based on a continuum field theory of the massless Goldstone bosons generated by the breakdown of the symmetry, which control the behaviour in these regimes. In particular, the finite-size scaling behaviour can be predicted, in a way very similar to the predictions of conformal invariance for a two-dimensional system at a critical point. Conversely, measurements of the finite-size scaling behaviour can give estimates of the parameters of the Goldstone bosons. These points are illustrated using data for the O(2) Heisenberg spin model in (2+1) dimensions, and the XXZ antiferromagnet on a square lattice.

DOI
10.1071/p97001

Related papers

Finite-Size Scaling and Efiective Lagrangian Theory⁄

C. J. Hamer

A brief review is given of the ‘efiective Lagrangian ’ approach of Leutwyler, Hasenfratz and others, which describes the behaviour at low energies or temperatures, or large distances, in lattice systems which undergo a flrst-order transition involving spontaneous breakdown of a continuous symmetry. Universal predictions can be given, based on a continuum fleld theory of the massless Goldstone bosons generated by the breakdown of the symmetry, which control the behaviour in these regimes. In particular, the flnite-size scaling behaviour can be predicted, in a way very similar to the predictions of conformal invariance for a two-dimensional system at a critical point. Conversely, measurements of the flnite-size scaling behaviour can give estimates of the parameters of the Goldstone bosons. These points are illustrated using data for the O(2) Heisenberg spin model in (2+1) dimensions, and the XXZ antiferromagnet on a square lattice. 1.

•Jul 9, 2015•0 citations

Finite-size scaling for a first-order transition where a continuous symmetry is broken: The spin-flop transition in the three-dimensional XXZ Heisenberg antiferromagnet

Jiahao Xu, Shan-Ho Tsai, D. P. Landau +1 more

Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological "degeneracy" factor included. Predictions are compared with data from Monte Carlo simulations of the three-dimensional, XXZ Heisenberg antiferromagnet in a field in order to study the finite-size behavior on a L×L×L simple cubic lattice for the first-order "spin-flop" transition between the Ising-like antiferromagnetic state and the canted, XY-like state. Our theory predicts that for large linear dimension L the field dependence of all moments of the order parameters as well as the fourth-order cumulants exhibit universal intersections. Corrections to leading order should scale as the inverse volume. The values of these intersections at the spin-flop transition point can be expressed in terms of a factor q that characterizes the relative degeneracy of the ordered phases. Our theory yields q=π, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality can be invoked for first-order phase transitions.

arXiv:1901.10643•Physical review. E•Feb 13, 2019•13 citations

Finite-size scaling in the spin- model on a square lattice

Chris J. Hamer, Tobias Hövelborn, Michael Bachhuber

Energy eigenvalues and order parameters are calculated by exact diagonalization for the isotropic spin- model on square lattices of up to sites. The finite-size scaling behaviour is in excellent agreement with the effective Lagrangian predictions of Hasenfratz and Niedermayer (Hasenfratz P and Niedermayer F 1993 Z. Phys. B 92 91). Estimates are obtained for the bulk ground-state energy per site, the spontaneous magnetization, the spin-wave velocity and the spin-wave stiffness.

Journal of Physics A Mathematical and General•Jan 1, 1999•12 citations

Partition Function in One, Two, and Three Spatial Dimensions from Effective Lagrangian Field Theory

Christoph P. Hofmann

The systematic effective Lagrangian method was first formulated in the context of the strong interaction; chiral perturbation theory (CHPT) is the effective theory of quantum chromodynamics (QCD). It was then pointed out that the method can be transferred to the nonrelativistic domain—in particular, to describe the low-energy properties of ferromagnets. Interestingly, whereas for Lorentz-invariant systems the effective Lagrangian method fails in one spatial dimension (ds=1) , it perfectly works for nonrelativistic systems in ds=1 . In the present brief review, we give an outline of the method and then focus on the partition function for ferromagnetic spin chains, ferromagnetic films, and ferromagnetic crystals up to three loops in the perturbative expansion—an accuracy never achieved by conventional condensed matter methods. We then compare ferromagnets in ds=1 , 2, 3 with the behavior of QCD at low temperatures by considering the pressure and the order parameter. The two apparently very different systems (ferromagnets and QCD) are related from a universal point of view based on the spontaneously broken symmetry. In either case, the low-energy dynamics is described by an effective theory containing Goldstone bosons as basic degrees of freedom.

arXiv:1402.0806•ISRN Thermodynamics•Mar 11, 2014•2 citations

Finite-lattice methods in quantum Hamiltonian field theory. I. O(2) and O(3) Heisenberg models

Chris J. Hamer, Michael N. Barber

For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.

Journal of Physics A Mathematical and General•Jan 1, 1981•70 citations

Phase transitions and criticality: Finite-size effects, conformal covariance and other current problems

Michael Ellis Fisher

This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!

Journal of Magnetism and Magnetic Materials•Feb 1, 1986•3 citations
A brief review is given of the ‘effective Lagrangian’ approach of Leutwyler, Hasenfratz and others, which describes the behaviour at low energies or temperatures, or large distances, in lattice systems which undergo a first-order transition involving spontaneous breakdown of a continuous symmetry. Universal predictions can be given, based on a continuum field theory of the massless Goldstone bosons generated by the breakdown of the symmetry, which control the behaviour in these regimes. In particular, the finite-size scaling behaviour can be predicted, in a way very similar to the predictions of conformal invariance for a two-dimensional system at a critical point. Conversely, measurements of the finite-size scaling behaviour can give estimates of the parameters of the Goldstone bosons. These points are illustrated using data for the O(2) Heisenberg spin model in (2+1) dimensions, and the XXZ antiferromagnet on a square lattice.
A brief review is given of the ‘efiective Lagrangian ’ approach of Leutwyler, Hasenfratz and others, which describes the behaviour at low energies or temperatures, or large distances, in lattice systems which undergo a flrst-order transition involving spontaneous breakdown of a continuous symmetry. Universal predictions can be given, based on a continuum fleld theory of the massless Goldstone bosons generated by the breakdown of the symmetry, which control the behaviour in these regimes. In particular, the flnite-size scaling behaviour can be predicted, in a way very similar to the predictions of conformal invariance for a two-dimensional system at a critical point. Conversely, measurements of the flnite-size scaling behaviour can give estimates of the parameters of the Goldstone bosons. These points are illustrated using data for the O(2) Heisenberg spin model in (2+1) dimensions, and the XXZ antiferromagnet on a square lattice. 1.
Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological "degeneracy" factor included. Predictions are compared with data from Monte Carlo simulations of the three-dimensional, XXZ Heisenberg antiferromagnet in a field in order to study the finite-size behavior on a L×L×L simple cubic lattice for the first-order "spin-flop" transition between the Ising-like antiferromagnetic state and the canted, XY-like state. Our theory predicts that for large linear dimension L the field dependence of all moments of the order parameters as well as the fourth-order cumulants exhibit universal intersections. Corrections to leading order should scale as the inverse volume. The values of these intersections at the spin-flop transition point can be expressed in terms of a factor q that characterizes the relative degeneracy of the ordered phases. Our theory yields q=π, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality can be invoked for first-order phase transitions.
The systematic effective Lagrangian method was first formulated in the context of the strong interaction; chiral perturbation theory (CHPT) is the effective theory of quantum chromodynamics (QCD). It was then pointed out that the method can be transferred to the nonrelativistic domain—in particular, to describe the low-energy properties of ferromagnets. Interestingly, whereas for Lorentz-invariant systems the effective Lagrangian method fails in one spatial dimension (ds=1) , it perfectly works for nonrelativistic systems in ds=1 . In the present brief review, we give an outline of the method and then focus on the partition function for ferromagnetic spin chains, ferromagnetic films, and ferromagnetic crystals up to three loops in the perturbative expansion—an accuracy never achieved by conventional condensed matter methods. We then compare ferromagnets in ds=1 , 2, 3 with the behavior of QCD at low temperatures by considering the pressure and the order parameter. The two apparently very different systems (ferromagnets and QCD) are related from a universal point of view based on the spontaneously broken symmetry. In either case, the low-energy dynamics is described by an effective theory containing Goldstone bosons as basic degrees of freedom.
For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.
This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!