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

First- and second-neighbor square-lattice Potts model: Renormalization-group treatment

P.M.C. de Oliveira, Constantino Tsallis, Georges Schwachheim

Published Mar 1, 1984• 15 citations
Publisher

Abstract

Within a real-space renormalization-group framework, we study the $q$-state Potts model in a square lattice, assuming a positive second-neighbor coupling constant and an arbitrary sign for the first-neighbor coupling constant. The $q$ evolution of the full phase diagram as well as the thermal and magnetic critical exponents are obtained. Whenever comparison with available results is possible, the agreement is either exact or very satisfactory.

DOI
10.1103/physrevb.29.2755

Related papers

Blume-Emery-Griffiths-Potts model in two dimensions: Phase diagram and critical properties from a position-space renormalization group

A. Nihat Berker, Michael Wortis

The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear ($J$) and biquadratic ($K$) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in $J$, $K$, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field ($H$) and another odd interaction ($L$) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.

Physical review. B, Solid state•Dec 1, 1976•371 citations

Real-space renormalization-group investigation of the randomly diluteq-state Potts model

A. Bakchich, A. Benyoussef, T. Biaz +1 more

The randomly bond-diluted two-dimensional nearest-neighbor q-state Potts ferromagnet and antiferromagnet on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. In the bond-diluted ferromagnet Potts model, differential recursion relations yield a phase diagram which is in quantitative agreement with all known results for q\ensuremath{\le}4. In the bond-diluted antiferromagnet Potts model, the phase diagram obtained shows that the threshold ${p}^{\mathrm{*}}$ at which the critical temperature goes to zero depends upon q.

Physical review. B, Condensed matter•Jun 1, 1988•2 citations

Real-space renormalization-group study of the Z(5) model

Victor Mizrahi, Eytan Domany

The phase diagram of the nearest-neighbor $Z(5)$ model on the square and triangular lattices is investigated using techniques of the real-space renormalization group. We use the recently introduced technique of Nienhuis, Berker, Riedel, and Schick, which yields the expected first-order transition on the Potts subspace of the $Z(5)$ model on the triangular lattice. For the general $Z(5)$ model we obtain a line of first-order transitions which crosses over into a line of second-order transitions. The same technique when applied to the square lattice is found to show this crossover behavior only in combination with a Migdal procedure.

Physical review. B, Condensed matter•Oct 1, 1981•7 citations

Application of Real-Space Renormalization Group Theory to the Antiferromagnetic Potts Model

Takehiko Oguchi, Takuma Ishikawa

The real-space renormalization group theory using the first order cumulant is applied to the q -state antiferromagnetic Potts model on 2-d (dimensional) and 3-d lattice. The special features of this paper are that the Potts states are represented by q unit vectors in q -1 dimensional space and the two-sublattice structure is preserved after the renormalization transformation. The critical exponents obtained for q =2 in 3-d are in fairly good agreement with the reliable values and those for q =3 and 4 are obtained for the first time.

Journal of the Physical Society of Japan•Jul 15, 1985•0 citations

Percolation with first- and second-neighbor bonds: Renormalization-group calculation of critical exponents

Rosane Riera, P.M.C. de Oliveira, Carlos Mauricio Chaves +1 more

We propose a real-space renormalization-group approach for the bond-percolation problem in a square lattice with first- and second-neighbor bonds. We treat the respective probabilities as independent variables. Two types of cells are constructed. In one of them we consider the lattice as two interpenetrating sublattices, first-neighbor bonds playing the role of intersublattice links. This allows the calculation of both critical exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$, without resorting to any external field. Values found for the critical indices are in good agreement with data available in the literature. The phase diagram in parameter space is also obtained in each case.

Physical review. B, Condensed matter•Oct 1, 1980•13 citations

Partial Order in Potts Models on the Generalized Decorated Square Lattice

Mingpu Qin, Jing Chen, Qiaoni Chen +5 more

We explore the Potts model on the generalized decorated square lattice, with both nearest (J 1 ) and next-nearest (J 2 ) neighbor interactions. Using the tensor renormalization-group method augmented by higher order singular value decompositions, we calculate the spontaneous magnetization of the Potts model with q = 2, 3, and 4. The results for q = 2 allow us to benchmark our numerics using the exact solution. For q = 3, we find a highly degenerate ground state with partial order on a single sublattice, but with vanishing entropy per site, and we obtain the phase diagram as a function of the ratio J 2 /J 1 . There is no finite-temperature transition for the q = 4 case when J 1 = J 2 , whereas the magnetic susceptibility diverges as the temperature goes to zero, showing that the model is critical at T = 0.

arXiv:1304.7146•Chinese Physics Letters•Jul 1, 2013•9 citations
Within a real-space renormalization-group framework, we study the qqq-state Potts model in a square lattice, assuming a positive second-neighbor coupling constant and an arbitrary sign for the first-neighbor coupling constant. The qqq evolution of the full phase diagram as well as the thermal and magnetic critical exponents are obtained. Whenever comparison with available results is possible, the agreement is either exact or very satisfactory.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
The randomly bond-diluted two-dimensional nearest-neighbor q-state Potts ferromagnet and antiferromagnet on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. In the bond-diluted ferromagnet Potts model, differential recursion relations yield a phase diagram which is in quantitative agreement with all known results for q\ensuremath{\le}4. In the bond-diluted antiferromagnet Potts model, the phase diagram obtained shows that the threshold p∗{p}^{\mathrm{*}}p∗ at which the critical temperature goes to zero depends upon q.
The phase diagram of the nearest-neighbor Z(5)Z(5)Z(5) model on the square and triangular lattices is investigated using techniques of the real-space renormalization group. We use the recently introduced technique of Nienhuis, Berker, Riedel, and Schick, which yields the expected first-order transition on the Potts subspace of the Z(5)Z(5)Z(5) model on the triangular lattice. For the general Z(5)Z(5)Z(5) model we obtain a line of first-order transitions which crosses over into a line of second-order transitions. The same technique when applied to the square lattice is found to show this crossover behavior only in combination with a Migdal procedure.
The real-space renormalization group theory using the first order cumulant is applied to the q -state antiferromagnetic Potts model on 2-d (dimensional) and 3-d lattice. The special features of this paper are that the Potts states are represented by q unit vectors in q -1 dimensional space and the two-sublattice structure is preserved after the renormalization transformation. The critical exponents obtained for q =2 in 3-d are in fairly good agreement with the reliable values and those for q =3 and 4 are obtained for the first time.
We propose a real-space renormalization-group approach for the bond-percolation problem in a square lattice with first- and second-neighbor bonds. We treat the respective probabilities as independent variables. Two types of cells are constructed. In one of them we consider the lattice as two interpenetrating sublattices, first-neighbor bonds playing the role of intersublattice links. This allows the calculation of both critical exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$, without resorting to any external field. Values found for the critical indices are in good agreement with data available in the literature. The phase diagram in parameter space is also obtained in each case.
We explore the Potts model on the generalized decorated square lattice, with both nearest (J 1 ) and next-nearest (J 2 ) neighbor interactions. Using the tensor renormalization-group method augmented by higher order singular value decompositions, we calculate the spontaneous magnetization of the Potts model with q = 2, 3, and 4. The results for q = 2 allow us to benchmark our numerics using the exact solution. For q = 3, we find a highly degenerate ground state with partial order on a single sublattice, but with vanishing entropy per site, and we obtain the phase diagram as a function of the ratio J 2 /J 1 . There is no finite-temperature transition for the q = 4 case when J 1 = J 2 , whereas the magnetic susceptibility diverges as the temperature goes to zero, showing that the model is critical at T = 0.