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

Non-universal exponents and marginal operators via a Monte Carlo renormalisation group

Jorge Chahine, J. R. Drugowich de Felício, Nestor Caticha

Published May 21, 1989• 18 citations
Publisher

Abstract

The authors study the critical exponents, eigenoperators and the phase diagram of the N=2 Ashkin-Teller (AT) model in two dimensions using a Monte Carlo renormalisation group. They pay special attention to the marginal operator responsible for the non-universal behaviour. The crossover exponent is measured and used to numerically calculate the thermal exponent of the eight-vertex model and of the AT model near the Potts-four-point, giving better results than a direct determination, since the crossover operator is not affected by the additional AT marginal operators.

DOI
10.1088/0305-4470/22/10/018

Related papers

Critical exponents and marginality of the four-state Potts model: Monte Carlo renormalization group

Robert H. Swendsen, David Andelman, A. Nihat Berker

The thermal critical exponent of the four-state Potts model in two dimensions is evaluated as ${\ensuremath{\nu}}^{\ensuremath{-}1}=1.49\ifmmode\pm\else\textpm\fi{}0.01$, using the Monte Carlo renormalization-group method. The presence of a marginal scaling direction is also indicated. These results confirm previous conjectures, universality, and logarithmic corrections. A purely Potts Hamiltonian is considered. A four-point interaction is used to control the chemical potential of effective vacancies, without the introduction of explicit vacancy states.

Physical review. B, Condensed matter•Dec 1, 1981•30 citations

A renormalisation group investigation of the Ashkin-Teller-Potts model

S E Ashley

The Ashkin-Teller-Potts model is examined using two renormalisation group methods: (a) the cluster expansion; and (b) the lower bound variational method. The critical surface is found to be in agreement with that conjectured by Wu and Lin (1974). The authors find a very strong indication of a line of fixed points along which they obtain a continuous variation of the specific heat exponent with the four-spin coupling. A comparison is made with the eight-vertex model.

Journal of Physics A Mathematical and General•Oct 1, 1978•11 citations

Critical relaxation of the eight-vertex model

M. Aydin, M. Cemal Yalabık

The dynamical critical exponent z is obtained through a Monte Carlo renormalisation group calculation for the eight-vertex model with non-conserved dynamics. The value of z at several points along the critical line is evaluated. No appreciable variation in z relative to the Ising value is observed within the accuracy of the work.

Journal of Physics A Mathematical and General•Feb 21, 1984•2 citations

Critical properties of the Ashkin-Teller model from the mean-field renormalisation group approach

J. A. Plascak, F. C. Sá Barreto

By using small size clusters, which carry all the basic symmetries of the Ashkin-Teller model, the authors obtain within the mean-field renormalisation group approach the complete phase diagram of the model. Estimates of the thermal critical exponent are also obtained.

Journal of Physics A Mathematical and General•Aug 1, 1986•35 citations

Phase diagram and critical properties of the (1+1)-dimensional Ashkin-Teller model

Ferenc Iglói, Jenő Sólyom

The (1+1)-dimensional Ashkin-Teller model is studied by the renormalisation group method and by the finite lattice extrapolation technique. The possible phases and their boundaries in an extended space of couplings are determined from the degeneracy structure of the low-lying levels in the thermodynamic limit. The critical behaviour is investigated for ferromagnetic couplings using increasing block sizes in the renormalisation transformation. The line of fixed points and its bifurcation at the Potts fixed point is studied.

Journal of Physics A Mathematical and General•May 11, 1984•22 citations

Universality and non-universality in the Ashkin-Teller model

Alessandro Giuliani

The Ashkin-Teller model can be formulated as a pair of 2D Ising models, interacting via a four-spin interaction. I consider the case of weak anisotropy (slight a-symmetry between the two Ising layers) and weak coupling. I show that the system admits two critical temperatures whose difference varies continuously with the strength of the coupling, scaling with an anomalous exponent as one let the a-symmetry parameter go to zero. The specific heat diverges logarithmically at the critical points (as for Ising) but the constant in front of the logarithm is renormalized by an anomalous critical exponent. The logarithmic divergence of the specific heat dominates only in an exponentially small interval around the critical temperatures and outside it is modified into an anomalous power law behaviour. The proof is based on an exact mapping of Ashkin-Teller into a model of (1+1)D interacting fermions and on the implementation of constructive Renormalization group methods on the fermionic system. This PhD thesis includes: (1) a review of the exact solution of 2D Ising in terms of Grassmann functional integrals; (2) a review of the Grassmann representation for a class of interacting Ising models, including Ashkin-Teller and the 8 vertex model; (3) a detailed discussion of the multiscale analysis of the Ashkin-Teller model, based on fermionic Renormalization Group methods, including the study of the flow of the effective coupling constants (in particular a proof of vanishing of the beta function for this and similar Luttinger-like models, based on modified approximate Ward identities, is included). This thesis is based on joint work with V. Mastropietro.

arXiv:cond-mat/0412176•Dec 7, 2004•0 citations
The authors study the critical exponents, eigenoperators and the phase diagram of the N=2 Ashkin-Teller (AT) model in two dimensions using a Monte Carlo renormalisation group. They pay special attention to the marginal operator responsible for the non-universal behaviour. The crossover exponent is measured and used to numerically calculate the thermal exponent of the eight-vertex model and of the AT model near the Potts-four-point, giving better results than a direct determination, since the crossover operator is not affected by the additional AT marginal operators.
The thermal critical exponent of the four-state Potts model in two dimensions is evaluated as ${\ensuremath{\nu}}^{\ensuremath{-}1}=1.49\ifmmode\pm\else\textpm\fi{}0.01$, using the Monte Carlo renormalization-group method. The presence of a marginal scaling direction is also indicated. These results confirm previous conjectures, universality, and logarithmic corrections. A purely Potts Hamiltonian is considered. A four-point interaction is used to control the chemical potential of effective vacancies, without the introduction of explicit vacancy states.
The (1+1)-dimensional Ashkin-Teller model is studied by the renormalisation group method and by the finite lattice extrapolation technique. The possible phases and their boundaries in an extended space of couplings are determined from the degeneracy structure of the low-lying levels in the thermodynamic limit. The critical behaviour is investigated for ferromagnetic couplings using increasing block sizes in the renormalisation transformation. The line of fixed points and its bifurcation at the Potts fixed point is studied.
The Ashkin-Teller model can be formulated as a pair of 2D Ising models, interacting via a four-spin interaction. I consider the case of weak anisotropy (slight a-symmetry between the two Ising layers) and weak coupling. I show that the system admits two critical temperatures whose difference varies continuously with the strength of the coupling, scaling with an anomalous exponent as one let the a-symmetry parameter go to zero. The specific heat diverges logarithmically at the critical points (as for Ising) but the constant in front of the logarithm is renormalized by an anomalous critical exponent. The logarithmic divergence of the specific heat dominates only in an exponentially small interval around the critical temperatures and outside it is modified into an anomalous power law behaviour. The proof is based on an exact mapping of Ashkin-Teller into a model of (1+1)D interacting fermions and on the implementation of constructive Renormalization group methods on the fermionic system. This PhD thesis includes: (1) a review of the exact solution of 2D Ising in terms of Grassmann functional integrals; (2) a review of the Grassmann representation for a class of interacting Ising models, including Ashkin-Teller and the 8 vertex model; (3) a detailed discussion of the multiscale analysis of the Ashkin-Teller model, based on fermionic Renormalization Group methods, including the study of the flow of the effective coupling constants (in particular a proof of vanishing of the beta function for this and similar Luttinger-like models, based on modified approximate Ward identities, is included). This thesis is based on joint work with V. Mastropietro.