© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical review. B, Condensed matter

Phase diagrams of two-dimensional Z(q) models

Pál Ruján, Gareth O. Williams, H. L. Frisch, Gabor Forgács

Published Feb 1, 1981• 45 citations

Abstract

The phase diagrams of general $q$-state planar spin models on a square lattice are studied for $q=4\ensuremath{-}7$. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of $Z(q)$ models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for $q=6 \mathrm{and} 7$ but are inconclusive for $q=5$. More accurate methods are needed to determine the phase structure of the $Z(5)$ model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).

DOI: 10.1103/physrevb.23.1362

Related papers

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

Mean-field-like variational and renormalisation group studies of (1+1)-dimensional spin models

Ernst Niebur, Jenő Sólyom

Two recently proposed methods, the variational cluster wavefunction approximation and the mean-field-like renormalisation group transformation, are used to describe the phase transitions in the Hamiltonian version of the two-dimensional Z(q) symmetric spin models. The convergence of the two schemes to the known results is analysed. It is shown that in their usual form neither of these models can reproduce fine details of the phase diagram, although the renormalisation group method gives in general better convergence.

Journal of Physics A Mathematical and General•Jul 11, 1987•1 citation

Propriedades críticas do modelo z(4) em sistemas bi e tridimensionais

Ezequiel Silva de Souza

A real space renormalization group method is used to investigate the criticality (phase diagrams, critical expoentes and universality classes) of Z(4) model in two and three dimensions. The values of the interaction parameters are chosen in such a way as to cover the complete phase diagrams of the model, which presents the following phases: (i) Paramagnetic (P); (ii) Ferromagnetic (F); (iii) Antiferromagnetic (AF); (iv) Intermediate Ferromagnetic (IF) and Intermediate Antiferromagnetic (IAF). In the hierarquical lattices, generated by renormalization the phase diagrams are exact. It is also possible to obtain approximated results for square and simple cubic lattices. In the bidimensional case a self-dual lattice is used and the resulting phase diagram reproduces all the exact results known for the square lattice. The Migdal-Kadanoff transformation is applied to the three dimensional case and the additional phases previously suggested by Ditzian et al, are not found

•Aug 19, 1988•0 citations

Monte Carlo study of the Z(5) model

Vera Lúcia Vieira Baltar, Gilson Carneiro, Maria Elena Pol +1 more

The phase diagram of the Z(5) model on a square lattice is studied using Monte Carlo simulation techniques. The authors confirm the existence of three phases in the ferromagnetic Z(5) model and find a critical phase for the antiferromagnetic model. The phase boundaries are determined using Monte Carlo renormalisation group methods.

Journal of Physics A Mathematical and General•Aug 1, 1985•4 citations

Phase Transition of the q -State Clock Model: Duality and Tensor Renormalization

Jing Chen, Haijun Liao, Haidong Xie +6 more

We investigate the critical behavior and the duality property of the ferromagnetic $q$-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with $q \leq 5 $ and approximate self-dual points for $q \geq 6$. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.

arXiv:1706.03455•Chinese Physics Letters•May 1, 2017•46 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

The phase diagrams of general

q

-state planar spin models on a square lattice are studied for

q=4\ensuremath{-}7

. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of

Z(q)

models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for

q=6 \mathrm{and} 7

but are inconclusive for

q=5

. More accurate methods are needed to determine the phase structure of the

Z(5)

model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).

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.

Two recently proposed methods, the variational cluster wavefunction approximation and the mean-field-like renormalisation group transformation, are used to describe the phase transitions in the Hamiltonian version of the two-dimensional Z(q) symmetric spin models. The convergence of the two schemes to the known results is analysed. It is shown that in their usual form neither of these models can reproduce fine details of the phase diagram, although the renormalisation group method gives in general better convergence.

A real space renormalization group method is used to investigate the criticality (phase diagrams, critical expoentes and universality classes) of Z(4) model in two and three dimensions. The values of the interaction parameters are chosen in such a way as to cover the complete phase diagrams of the model, which presents the following phases: (i) Paramagnetic (P); (ii) Ferromagnetic (F); (iii) Antiferromagnetic (AF); (iv) Intermediate Ferromagnetic (IF) and Intermediate Antiferromagnetic (IAF). In the hierarquical lattices, generated by renormalization the phase diagrams are exact. It is also possible to obtain approximated results for square and simple cubic lattices. In the bidimensional case a self-dual lattice is used and the resulting phase diagram reproduces all the exact results known for the square lattice. The Migdal-Kadanoff transformation is applied to the three dimensional case and the additional phases previously suggested by Ditzian et al, are not found

We investigate the critical behavior and the duality property of the ferromagnetic

q

-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with $q \leq 5 $ and approximate self-dual points for

q \geq 6

. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.

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.