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

Physical review. B, Condensed matter

Exact exponents for infinitely many new multicritical points

David A. Huse

Published Oct 1, 1984• 252 citations
Publisher

Abstract

Andrews, Baxter, and Forrester have recently solved two infinite sequences of models that are generalizations of the Ising and hard-square models and exhibit multicriticality. The nature of these new two-dimensional multicritical points is examined in order to determine what universality classes they may represent. Appropriate order parameters are defined and their critical exponents reported. One infinite sequence of multicritical points are continuous melting transitions of $p\ifmmode\times\else\texttimes\fi{}1$ commensurate ordered phases for all integers $p\ensuremath{\ge}2$. The other sequence appears to consist of "generic" multicritical points terminating lines of $n$-phase coexistence, again for all integers $n\ensuremath{\ge}2$. The multicritical exponents in this latter sequence coincide with those found by Friedan, Qiu, and Shenker on the basis of assuming conformal invariance and unitarity.

DOI
10.1103/physrevb.30.3908

Related papers

Multicritical two-dimensional vertex models

Somendra M. Bhattacharjee, John J. Rajasekaran

We study the multicritical behavior of a class of two-dimensional ice-type vertex models on different lattices using renormalization-group theory. The models are classified by an integer m, with m=2 corresponding to the known square lattice case. For m>2, the specific-heat exponent is a=(m-2)/(m-1) with an upper critical dimensional confluent (lnt${)}^{1/2}$ divergence for m=3. The nature of the transition is similar to the mth-order multicritical point, yet the exponents are not those known from c1 conformal invariance. The models are anisotropic with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$=1 and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=12. A few special features of the models are discussed.

Physical Review A•Jul 1, 1992•7 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

Two-dimensional semi-infinite multicritical Ising models

Per Fröjdh

The order-parameter correlation functions of the Z 2 -invariant multicritical points in the unitary minimal series of conformal field theory are derived for a semi-infinite plane constrained to fixed or free boundary conditions. These yield the corresponding universal surface exponents which distinguish the behaviour between even- and odd-critical models. We also make explicit the interplay between duality and boundary conditions.

Journal of Physics A Mathematical and General•Jan 7, 1994•0 citations

Bi- and tetracritical phase diagrams in three dimensions

Amnon Aharony A, Ora Entin-wohlman A, Andrey Kudlis

The critical behavior of many physical systems involves two competing n1- and n2-component order-parameters, S1 and S2, respectively, with n = n1 + n2. Varying an external control parameter g, one encounters ordering of S1 below a critical (second-order) line for g < 0 and of S2 below another critical line for g > 0. These two ordered phases are separated by a first-order line, which meets the above critical lines at a bicritical point, or by an intermediate (mixed) phase, bounded by two critical lines, which meet the above critical lines at a tetracritical point. For n = 1 + 2 = 3, the critical behavior around the (bi- or tetra-) multicritical point either belongs to the universality class of a non-rotationally invariant (cubic or biconical) fixed point, or it has a fluctuation driven first-order transition. These asymptotic behaviors arise only very close to the transitions. We present accurate renormalization-group flow trajectories yielding the effective crossover exponents near multicriticality.

arXiv:2203.01168•Low Temperature Physics•May 24, 2022•4 citations

Multicritical scaling in Baxter's hard square lattice gas

David A. Huse

The scaling behaviour of the hard square lattice gas with diagonal interactions is examined on the basis of Baxter's (1980, 1981, 1982) recent exact solution. It is demonstrated that all the corrections to scaling found may be accounted for in terms of a single irrelevant scaling field that scales simply as a length. The lattice cut-off is proposed as the source of these corrections, although other sources cannot be ruled out. The melting of the 3*1 commensurate ordered phase in this model is examined in a more general context in which the line of Potts critical points found by Baxter appears to be a line of multicritical points on which the effective uniaxially chiral symmetry breaking field vanishes.

Journal of Physics A Mathematical and General•Dec 21, 1983•18 citations

Critical phenomena: Some recent theoretical developments (invited)

Michael Ellis Fisher

The aim of this contribution is to draw attention to some recent developments in the theory of critical phenomena and phase transitions that may be of interest to the magnetism community in that magnetic systems have provided a fertile testing ground for modern theoretical ideas concerning ordering and phase transitions while, on the other hand, many aspects of observed magnetic phase behavior have required sophisticated theory for their proper elucidation. The new works in question have been published or will appear shortly: accordingly, only an outline is presented here together with references to the main literature sources which, in turn, include fuller references to earlier work not explicitly mentioned. Four linked topics are addressed: (i) finite-size effects at first-order transitions; (ii) asymptotic degeneracy in transfer matrices, which is intimately related to long-range order; (iii) some consequences of the conformal covariance of fluctuations at a critical point in bulk and in finite systems, especially in two spatial dimensions; (iv) the discovery of ‘‘exactly soluble’’ two-dimensional models exhibiting two infinite classes of distinct critical and multicritical points.

Journal of Applied Physics•Apr 15, 1985•2 citations
Andrews, Baxter, and Forrester have recently solved two infinite sequences of models that are generalizations of the Ising and hard-square models and exhibit multicriticality. The nature of these new two-dimensional multicritical points is examined in order to determine what universality classes they may represent. Appropriate order parameters are defined and their critical exponents reported. One infinite sequence of multicritical points are continuous melting transitions of p\ifmmode×\else\texttimes\fi1p\ifmmode\times\else\texttimes\fi{}1p\ifmmode×\else\texttimes\fi1 commensurate ordered phases for all integers p\ensuremath≥2p\ensuremath{\ge}2p\ensuremath≥2. The other sequence appears to consist of "generic" multicritical points terminating lines of nnn-phase coexistence, again for all integers n\ensuremath≥2n\ensuremath{\ge}2n\ensuremath≥2. The multicritical exponents in this latter sequence coincide with those found by Friedan, Qiu, and Shenker on the basis of assuming conformal invariance and unitarity.
We study the multicritical behavior of a class of two-dimensional ice-type vertex models on different lattices using renormalization-group theory. The models are classified by an integer m, with m=2 corresponding to the known square lattice case. For m>2, the specific-heat exponent is a=(m-2)/(m-1) with an upper critical dimensional confluent (lnt)1/2{)}^{1/2})1/2 divergence for m=3. The nature of the transition is similar to the mth-order multicritical point, yet the exponents are not those known from c1 conformal invariance. The models are anisotropic with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$=1 and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=12. A few special features of the models are discussed.
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!
The critical behavior of many physical systems involves two competing n1- and n2-component order-parameters, S1 and S2, respectively, with n = n1 + n2. Varying an external control parameter g, one encounters ordering of S1 below a critical (second-order) line for g < 0 and of S2 below another critical line for g > 0. These two ordered phases are separated by a first-order line, which meets the above critical lines at a bicritical point, or by an intermediate (mixed) phase, bounded by two critical lines, which meet the above critical lines at a tetracritical point. For n = 1 + 2 = 3, the critical behavior around the (bi- or tetra-) multicritical point either belongs to the universality class of a non-rotationally invariant (cubic or biconical) fixed point, or it has a fluctuation driven first-order transition. These asymptotic behaviors arise only very close to the transitions. We present accurate renormalization-group flow trajectories yielding the effective crossover exponents near multicriticality.
The scaling behaviour of the hard square lattice gas with diagonal interactions is examined on the basis of Baxter's (1980, 1981, 1982) recent exact solution. It is demonstrated that all the corrections to scaling found may be accounted for in terms of a single irrelevant scaling field that scales simply as a length. The lattice cut-off is proposed as the source of these corrections, although other sources cannot be ruled out. The melting of the 3*1 commensurate ordered phase in this model is examined in a more general context in which the line of Potts critical points found by Baxter appears to be a line of multicritical points on which the effective uniaxially chiral symmetry breaking field vanishes.
The aim of this contribution is to draw attention to some recent developments in the theory of critical phenomena and phase transitions that may be of interest to the magnetism community in that magnetic systems have provided a fertile testing ground for modern theoretical ideas concerning ordering and phase transitions while, on the other hand, many aspects of observed magnetic phase behavior have required sophisticated theory for their proper elucidation. The new works in question have been published or will appear shortly: accordingly, only an outline is presented here together with references to the main literature sources which, in turn, include fuller references to earlier work not explicitly mentioned. Four linked topics are addressed: (i) finite-size effects at first-order transitions; (ii) asymptotic degeneracy in transfer matrices, which is intimately related to long-range order; (iii) some consequences of the conformal covariance of fluctuations at a critical point in bulk and in finite systems, especially in two spatial dimensions; (iv) the discovery of ‘‘exactly soluble’’ two-dimensional models exhibiting two infinite classes of distinct critical and multicritical points.