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 Letters

Tricriticality of Interacting Hard Squares: Some Exact Results

David A. Huse

Published Oct 18, 1982• 54 citations
Publisher

Abstract

Baxter has solved a restricted class of square lattice gas models with nearest-neighbor exclusion (thus "hard squares") and next-nearest-neighbor interactions. Arguments are presented which demonstrate that the subspace spanned by his exact solution contains the line of tricritical points and the associated surface of first-order transitions of hard squares with attractive next-nearest-neighbor interactions. The tricritical exponents so identified confirm those obtained by Nienhuis for a dilute Ising model.

DOI
10.1103/physrevlett.49.1121

Related papers

Tricritical Properties of Antiferromagnetic Ising Model on the Square Lattice

A. Bobák, M. Borovský, T. Lučivjanský +1 more

The Ising square lattice model with nearest-neighbor (nn) interactions ($J_1$) is one of the few exactly solvable models [1]. Adding next-neareast- neighbor (nnn) interactions ($J_2$) or a magnetic field (or both) leads to the non solvability of the model and only some approximate solutions are possible. In this brief report we will review some results obtained within effective field theory. We will show that besides second-order transitions there are also lines of first-order transitions and the coordinates of tricritical points are calculated.

arXiv:1512.05929•Dec 18, 2015•0 citations

Hard squares with diagonal attractions

R. J. Baxter, Paul A. Pearce

The authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next-nearest neighbour (diagonal) interactions on the special surface corresponding to regimes III and IV of the generalised hard hexagon model. The interfacial tension, correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row-to-row and corner transfer matrices. The associated critical exponents are found to be mu =v= 5 / 4 , beta = 3 / 32 in regime III and mu '=v'= 5 / 2 , beta '= 1 / 4 in regime IV. In particular, their results confirm the recent proposal by Huse that regime III is the first-order coexistence surface (separating the disordered fluid phase from the square ordered solid phases) and that the regime III/regime IV boundary is a line of tricritical points.

Journal of Physics A Mathematical and General•Jul 11, 1983•62 citations

On Ising models

Hoede, C., Zandvliet, Henricus J.W.

For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments.

•Jun 1, 2006•0 citations

Tricritical relaxation of the hard squares model with attractive next-nearest-neighbour interactions

Enis Oḡuz

The tricritical relaxation of the hard squares model with attractive next-nearest-neighbour interactions is studied. The dynamic exponent z t of the model at its exactly known tricritical point is determined using the dynamic Monte Carlo renormalisation group method. The value of z t is found to be 2.18+or-0.11.

Journal of Physics A Mathematical and General•Jun 21, 1988•6 citations

Magnetic hard squares: exact solution

Paul A. Pearce

A three-state IRF model of a magnetic hard-square lattice gas with anisotropic interactions is solved exactly on special two-dimensional manifolds in the full five-dimensional thermodynamic space spanned by the (ferro- or antiferro-) magnetic interactions J, K, the (attractive or repulsive) lattice-gas interactions L, M and the activity z. The phase boundaries determined include multicritical lines and critical surfaces as well as first-order surfaces of three- and fivefold phase coexistence between the fluid and various (para, ferro, etc.) magnetic square-ordered solid phases. Analytic expressions are given for free energies, interfacial tensions, correlation lengths, sublattice densities and magnetisations. The critical behaviour of these quantities is studied and the associated critical exponents obtained.

Journal of Physics A Mathematical and General•Nov 11, 1985•21 citations

Lattice gas with nearest- and next-to-nearest-neighbor exclusion

Xiaomei Feng, Henk W. J. Blöte, Bernard Nienhuis

We investigate a hard-square lattice gas on the square lattice by means of transfer-matrix and Monte Carlo methods. The size of the hard squares is equal to two lattice constants, so the simultaneous occupation of nearest-neighbor sites as well as of next-to-nearest-neighbor sites is excluded. Near saturation of the particle density, this system is known to undergo a phase transition to one out of four partially ordered phases. We find that this transition displays strong finite-size corrections to scaling and that the correlation functions deviate from isotropy to rather large distances. In contrast with an earlier study, we find that the critical temperature exponent of the transition is not Ising-like.

Physical Review E•Jun 30, 2011•31 citations
Baxter has solved a restricted class of square lattice gas models with nearest-neighbor exclusion (thus "hard squares") and next-nearest-neighbor interactions. Arguments are presented which demonstrate that the subspace spanned by his exact solution contains the line of tricritical points and the associated surface of first-order transitions of hard squares with attractive next-nearest-neighbor interactions. The tricritical exponents so identified confirm those obtained by Nienhuis for a dilute Ising model.
The Ising square lattice model with nearest-neighbor (nn) interactions (J1J_1J1​) is one of the few exactly solvable models [1]. Adding next-neareast- neighbor (nnn) interactions (J2J_2J2​) or a magnetic field (or both) leads to the non solvability of the model and only some approximate solutions are possible. In this brief report we will review some results obtained within effective field theory. We will show that besides second-order transitions there are also lines of first-order transitions and the coordinates of tricritical points are calculated.
The authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next-nearest neighbour (diagonal) interactions on the special surface corresponding to regimes III and IV of the generalised hard hexagon model. The interfacial tension, correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row-to-row and corner transfer matrices. The associated critical exponents are found to be mu =v= 5 / 4 , beta = 3 / 32 in regime III and mu '=v'= 5 / 2 , beta '= 1 / 4 in regime IV. In particular, their results confirm the recent proposal by Huse that regime III is the first-order coexistence surface (separating the disordered fluid phase from the square ordered solid phases) and that the regime III/regime IV boundary is a line of tricritical points.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
A three-state IRF model of a magnetic hard-square lattice gas with anisotropic interactions is solved exactly on special two-dimensional manifolds in the full five-dimensional thermodynamic space spanned by the (ferro- or antiferro-) magnetic interactions J, K, the (attractive or repulsive) lattice-gas interactions L, M and the activity z. The phase boundaries determined include multicritical lines and critical surfaces as well as first-order surfaces of three- and fivefold phase coexistence between the fluid and various (para, ferro, etc.) magnetic square-ordered solid phases. Analytic expressions are given for free energies, interfacial tensions, correlation lengths, sublattice densities and magnetisations. The critical behaviour of these quantities is studied and the associated critical exponents obtained.
We investigate a hard-square lattice gas on the square lattice by means of transfer-matrix and Monte Carlo methods. The size of the hard squares is equal to two lattice constants, so the simultaneous occupation of nearest-neighbor sites as well as of next-to-nearest-neighbor sites is excluded. Near saturation of the particle density, this system is known to undergo a phase transition to one out of four partially ordered phases. We find that this transition displays strong finite-size corrections to scaling and that the correlation functions deviate from isotropy to rather large distances. In contrast with an earlier study, we find that the critical temperature exponent of the transition is not Ising-like.