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

Four-coloring model on the square lattice: A critical ground state

Jané Kondev, Christopher L. Henley

Published Sep 1, 1995• 65 citations
Publisher

Abstract

We study critical properties of the four-coloring model, which is given by the equal-weighted ensemble of all possible edge colorings of the square lattice with four different colors. We map the four-coloring model onto an interface model for which we propose an effective Gaussian field theory, which allows us to calculate correlation functions of operators in the coloring model. The critical exponents are given by the stiffness of the interface, which we calculate exactly using recent results on the statistical topography of rough interfaces. Our numerical exponents, from Monte Carlo simulations of the four-coloring model, are in excellent agreement with the analytical calculations. These results support the conjecture that the scaling limit of the four-coloring model is given by the Su(4${)}_{\mathit{k}=1}$ Wess-Zumino-Witten model. Moreover, we show that our effective field theory is the free-field representation of the Su(4${)}_{\mathit{k}=1}$ Wess-Zumino-Witten model. Finally, we discuss connections to loop models, and some predictions of finite temperature properties of a particular Potts model for which the four-coloring model is the ground state.

DOI
10.1103/physrevb.52.6628

Related papers

A constrained Potts antiferromagnet model with an interface representation

J K Burton, Christopher L. Henley

We define a 4-state Potts model ensemble on the square lattice, with the constraints that neighbouring spins must have different values, and that no plaquette may contain all four states. The spin configurations may be mapped into those of a two-dimensional interface in a 2 + 5 dimensional space. If this interface is in a Gaussian rough phase (as is the case for most other models with such a mapping) then the spin correlations are critical and their exponents can be related to the stiffness governing the interface fluctuations. Results of our Monte Carlo simulations show height fluctuations with an anomalous dependence on the wavevector, intermediate between the behaviours expected in a rough and in a smooth phase; we argue that the smooth phase (which would imply long-range spin order) is the best interpretation.

arXiv:cond-mat/9708171•Journal of Physics A Mathematical and General•Dec 21, 1997•62 citations

Zero-temperature phase transitions of an antiferromagnetic Ising model of general spin on a triangular lattice

Chen Zeng, Christopher L. Henley

We map the ground-state ensemble of antiferromagnetic Ising model of spin-S on a triangular lattice to an interface model whose entropic fluctuations are proposed to be described by an effective Gaussian free energy, which enables us to calculate the critical exponents of various operators in terms of the stiffness constant of the interface. Monte Carlo simulations for the ground-state ensemble utilizing this interfacial representation are performed to study both the dynamical and the static properties of the model. This method yields more accurate numerical results for the critical exponents. By varying the spin magnitude in the model, we find that the model exhibits three phases with a Kosterlitz-Thouless phase transition at ${\mathrm{S}}_{\mathrm{KT}}$2 and a locking phase transition at ${\mathrm{S}}_{\mathrm{L}}$\ensuremath{\leqslant}3. The phase diagram at finite temperatures is also discussed.

arXiv:cond-mat/9609007•Physical review. B, Condensed matter•Jun 1, 1997•61 citations

Ising antiferromagnet in the critical magnetic field on square lattice with free boundary conditions

Seung‐Yeon Kim

The properties of the antiferromagnetic Ising model in the critical magnetic field are investigated by evaluating exactly its all possible ground states on L × L square lattice (L = 4∼16) with free boundary conditions for the first time. From the exact integer values for the number of ground states Ωc(L) on L × L square lattice, the exact ground-state entropy (in unit of the Boltzmann constant kB) per volume sc(L) = L−2lnΩc(L) and the exact ground-state magnetization mc(L;T = 0) = L−2〈Σiσi〉 are calculated in the critical magnetic field, where σi is the magnetic spin on a lattice site i. The values of the ground-state entropy sc(L) and the ground-state magnetization mc(L), in the limit L → ∞, are accurately estimated to be sc/kB = 0.40750±0.00108 and mc = 0.54696±0.00065. In addition, for the first time, we estimate the values of the scaling exponents ωs and ωm, controlling the finite-size scaling behaviors of the ground-state entropy and the ground-state magnetization as sc−sc(L)∼L−ωs and mc−mc(L)∼L−ωm respectively. Our results show clearly that ωs = ωm = 1.

AIP conference proceedings•Jan 1, 2012•0 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

Mott transition in the square-lattice SU(4) fermionic Hubbard model with a {\pi}-flux gauge field

Zhichao Zhou, Congjun Wu, Yu Wang

Inspired by recent ultracold atom experiments in both the synthetic gauge field and the SU(2N) symmetry, by employing the large-scale projector quantum Monte Carlo simulations, we investigate the ground state properties of the square-lattice SU(4) fermionic Hubbard model with a {\pi}-flux gauge field. In the noninteracting limit, the ground state is the gapless Dirac semimetal. With the increase of repulsive interaction, we show that, in the presence of a {\pi}-flux gauge field per plaquette, a Mott transition occurs from the semimetal to the valence bond solid phase, accompanied by the breaking of $Z_4$ discrete symmetry. The phase transition point and the critical exponent have also been estimated by the finite size scaling of numerical data. Interestingly, in the strong coupling limit we have analyzed the role of ring exchange terms for the order formation of the ground state by perturbation theory.

APS March Meeting Abstracts•Dec 22, 2017•0 citations

Critical exponents of the four-state Potts model

B Hu

The two-dimensional four-state Potts model at finite temperature can be transformed, via the transfer matrix, into a one-dimensional quantum mechanical model at zero temperature. The duality invariant renormalisation group introduced by Fernandez-Pacheco (1979) is then employed to study the ground-state critical properties of this model. The fixed point is located at exactly the self-dual critical point K*=1. The thermal exponent is calculated to be y T =1.3219. It is in excellent agreement with the recent series value of Ditzian and Kadanoff (1979) (y T =1.33). Although it is not inconsistent with den Nijs's (1979) conjectured exact value of 3 / 2 , the difference is nevertheless substantial.

Journal of Physics A Mathematical and General•Sep 1, 1980•15 citations
We study critical properties of the four-coloring model, which is given by the equal-weighted ensemble of all possible edge colorings of the square lattice with four different colors. We map the four-coloring model onto an interface model for which we propose an effective Gaussian field theory, which allows us to calculate correlation functions of operators in the coloring model. The critical exponents are given by the stiffness of the interface, which we calculate exactly using recent results on the statistical topography of rough interfaces. Our numerical exponents, from Monte Carlo simulations of the four-coloring model, are in excellent agreement with the analytical calculations. These results support the conjecture that the scaling limit of the four-coloring model is given by the Su(4)k=1{)}_{\mathit{k}=1})k=1​ Wess-Zumino-Witten model. Moreover, we show that our effective field theory is the free-field representation of the Su(4)k=1{)}_{\mathit{k}=1})k=1​ Wess-Zumino-Witten model. Finally, we discuss connections to loop models, and some predictions of finite temperature properties of a particular Potts model for which the four-coloring model is the ground state.
We define a 4-state Potts model ensemble on the square lattice, with the constraints that neighbouring spins must have different values, and that no plaquette may contain all four states. The spin configurations may be mapped into those of a two-dimensional interface in a 2 + 5 dimensional space. If this interface is in a Gaussian rough phase (as is the case for most other models with such a mapping) then the spin correlations are critical and their exponents can be related to the stiffness governing the interface fluctuations. Results of our Monte Carlo simulations show height fluctuations with an anomalous dependence on the wavevector, intermediate between the behaviours expected in a rough and in a smooth phase; we argue that the smooth phase (which would imply long-range spin order) is the best interpretation.
We map the ground-state ensemble of antiferromagnetic Ising model of spin-S on a triangular lattice to an interface model whose entropic fluctuations are proposed to be described by an effective Gaussian free energy, which enables us to calculate the critical exponents of various operators in terms of the stiffness constant of the interface. Monte Carlo simulations for the ground-state ensemble utilizing this interfacial representation are performed to study both the dynamical and the static properties of the model. This method yields more accurate numerical results for the critical exponents. By varying the spin magnitude in the model, we find that the model exhibits three phases with a Kosterlitz-Thouless phase transition at ${\mathrm{S}}_{\mathrm{KT}}$2 and a locking phase transition at SL{\mathrm{S}}_{\mathrm{L}}SL​\ensuremath{\leqslant}3. The phase diagram at finite temperatures is also discussed.
The properties of the antiferromagnetic Ising model in the critical magnetic field are investigated by evaluating exactly its all possible ground states on L × L square lattice (L = 4∼16) with free boundary conditions for the first time. From the exact integer values for the number of ground states Ωc(L) on L × L square lattice, the exact ground-state entropy (in unit of the Boltzmann constant kB) per volume sc(L) = L−2lnΩc(L) and the exact ground-state magnetization mc(L;T = 0) = L−2〈Σiσi〉 are calculated in the critical magnetic field, where σi is the magnetic spin on a lattice site i. The values of the ground-state entropy sc(L) and the ground-state magnetization mc(L), in the limit L → ∞, are accurately estimated to be sc/kB = 0.40750±0.00108 and mc = 0.54696±0.00065. In addition, for the first time, we estimate the values of the scaling exponents ωs and ωm, controlling the finite-size scaling behaviors of the ground-state entropy and the ground-state magnetization as sc−sc(L)∼L−ωs and mc−mc(L)∼L−ωm respectively. Our results show clearly that ωs = ωm = 1.
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.
Inspired by recent ultracold atom experiments in both the synthetic gauge field and the SU(2N) symmetry, by employing the large-scale projector quantum Monte Carlo simulations, we investigate the ground state properties of the square-lattice SU(4) fermionic Hubbard model with a {\pi}-flux gauge field. In the noninteracting limit, the ground state is the gapless Dirac semimetal. With the increase of repulsive interaction, we show that, in the presence of a {\pi}-flux gauge field per plaquette, a Mott transition occurs from the semimetal to the valence bond solid phase, accompanied by the breaking of Z4Z_4Z4​ discrete symmetry. The phase transition point and the critical exponent have also been estimated by the finite size scaling of numerical data. Interestingly, in the strong coupling limit we have analyzed the role of ring exchange terms for the order formation of the ground state by perturbation theory.
The two-dimensional four-state Potts model at finite temperature can be transformed, via the transfer matrix, into a one-dimensional quantum mechanical model at zero temperature. The duality invariant renormalisation group introduced by Fernandez-Pacheco (1979) is then employed to study the ground-state critical properties of this model. The fixed point is located at exactly the self-dual critical point K*=1. The thermal exponent is calculated to be y T =1.3219. It is in excellent agreement with the recent series value of Ditzian and Kadanoff (1979) (y T =1.33). Although it is not inconsistent with den Nijs's (1979) conjectured exact value of 3 / 2 , the difference is nevertheless substantial.