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

Critical properties of a general class of Z(5)-symmetric spin models

B. Bonnier, K. Rouidi

Published Nov 1, 1990• 9 citations
Publisher

Abstract

We study the isotropic, ferromagnetic two-dimensional spin systems with Z(5) symmetry, which, in the general case, depend upon two couplings, a temperature and a parameter \ensuremath{\Omega}, used to interpolate continuously between the clock (\ensuremath{\Omega}=0) and the Potts (\ensuremath{\Omega}=1) models, passing through the integrable Fateev-Zamolodchikov case (\ensuremath{\Omega}=${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$\ensuremath{\approxeq}0.618). Two methods, the high-temperature expansion and the exact \ensuremath{\beta} function on finite lattices, are applied within the Hamiltonian formulation to study critical properties. The transitions found are first order (${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$\ensuremath{\le}1), second order of the Ising type (\ensuremath{\Omega}\ensuremath{\simeq}${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$), or of the Kosterlitz-Thouless type (0\ensuremath{\le}\ensuremath{\Omega}${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$) with a varying exponent.

DOI
10.1103/physrevb.42.8157

Related papers

Critical bifurcation point of theopenZ(5)-symmetric spin model

B. Bonnier

The critical behavior of the general isotropic, ferromagnetic two-dimensional spin system with openZ(5) symmetry is studied with use of high-temperature expansions of its mass gap. On the basis of these expansions we propose a simple analytic representation of the mass gap which naturally reproduces all the different phase transitions exhibited by this model (first order and second order of the Ising and of the Kosterlitz-Thouless types). In addition, the bifurcation point where the soft phases originate is clearly identified with the Fateev-Zamolodchikov value.

Physical review. B, Condensed matter•Jul 1, 1991•4 citations

Critical properties of the Z(p)-symmetric spin models from an exact finite-lattice approach

B. Bonnier, M. Hontebeyrie, C. Meyers

The Hamiltonian formulation of the two-dimensional spin systems with Z(p) symmetry, ranging from the Ising to the XY-plane rotor model, is used together with the Lanczos scheme to compute exactly the mass gap of finite systems. Critical properties are then deduced from a Callan-Symanzik \ensuremath{\beta} function. We find in particular a Kosterlitz-Thouless transition for p\ensuremath{\ge}5, with a p-dependent exponent which coincides with its renormalization-group estimate only in the p=\ensuremath{\infty} limit.

Physical review. B, Condensed matter•Mar 1, 1989•16 citations

Nonequilibrium scaling explorations on a two-dimensional Z (5)-symmetric model

Roberto da Silva, Henrique A. Fernandes, J. R. Drugowich de Felício

We have investigated the dynamic critical behavior of the two-dimensional Z(5)-symmetric spin model by using short-time Monte Carlo (MC) simulations. We have obtained estimates of some critical points in its rich phase diagram and included, among the usual critical lines the study of first-order (weak) transition by looking into the order-disorder phase transition. In addition, we also investigated the soft-disorder phase transition by considering empiric methods. A study of the behavior of β/νz along the self-dual critical line has been performed and special attention has been devoted to the critical bifurcation point, or Fateev-Zamolodchikov (FZ) point. First, by using a refinement method and taking into account simulations out of equilibrium, we were able to localize parameters of this point. In a second part of our study, we turned our attention to the behavior of the model at the early stage of its time evolution in order to find the dynamic critical exponent z as well as the static critical exponents β and ν of the FZ point on square lattices. The values of the static critical exponents and parameters are in good agreement with the exact results, and the dynamic critical exponent z≈2.28 very close to the four-state Potts model (z≈2.29).

arXiv:1409.3915•Physical Review E•Oct 1, 2014•16 citations

On the Phase Transition and Critical Behaviour in the Two-Dimensional Anisotropic Heisenberg Model

R. Shanker, R. A. Singh

The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.

Journal of the Physical Society of Japan•Mar 15, 1976•2 citations

The critical behaviour of self-dual Z(N) spin systems: finite-size scaling and conformal invariance

Francisco Castilho Alcaraz

This paper is concerned with the critical properties of a family of self-dual two-dimensional Z(N) models whose bulk free energy is exactly known at the self-dual point. The author's analysis is performed by studying the finite-size behaviour of the corresponding one-dimensional quantum Hamiltonians which also possess an exact solution at their self-dual point. By exploring finite-size scaling ideas and the conformal invariance of the critical infinite system the author calculates, for N up to 8, the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra. The results strongly suggest that the Z(N) quantum field theory of Zamolodchikov and Fateev is the underlying field theory associated with these statistical mechanical systems. The author also tests for the Z(5) case, the conjecture that these models correspond to the bifurcation points in the phase diagram of the General Z(N) spin model, where a massless phase originates.

Journal of Physics A Mathematical and General•Jun 21, 1987•48 citations

Clock model interpolation and symmetry breaking in O(2) models

Leon Hostetler, Jin Zhang, Ryo Sakai +3 more

Motivated by recent attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we define an extended-O(2) model by adding a $\ensuremath{\gamma}\mathrm{cos}(q\ensuremath{\varphi})$ term to the ordinary O(2) model with angular values restricted to a $2\ensuremath{\pi}$ interval. In the $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the model becomes an extended $q$-state clock model that reduces to the ordinary $q$-state clock model when $q$ is an integer and otherwise is a continuation of the clock model for noninteger $q$. By shifting the $2\ensuremath{\pi}$ integration interval, the number of angles selected can change discontinuously and two cases need to be considered. What we call case 1 has one more angle than what we call case 2. We investigate this class of clock models in two space-time dimensions using Monte Carlo and tensor renormalization group methods. Both the specific heat and the magnetic susceptibility show a double-peak structure for fractional $q$. In case 1, the small-$\ensuremath{\beta}$ peak is associated with a crossover, and the large-$\ensuremath{\beta}$ peak is associated with an Ising critical point, while both peaks are crossovers in case 2. When $q$ is close to an integer by an amount $\mathrm{\ensuremath{\Delta}}q$ and the system is close to the small-$\ensuremath{\beta}$ Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as $\ensuremath{\sim}1/(\mathrm{\ensuremath{\Delta}}q{)}^{1\ensuremath{-}1/{\ensuremath{\delta}}^{\ensuremath{'}}}$ with ${\ensuremath{\delta}}^{\ensuremath{'}}$ estimates consistent with the magnetic critical exponent $\ensuremath{\delta}=15$. The crossover peak and the Ising critical point move to Berezinskii-Kosterlitz-Thouless transition points with the same power-law scaling. A phase diagram for this model in the $(\ensuremath{\beta},q)$ plane is sketched. These results are possibly relevant for configurable Rydberg-atom arrays where the interpolations among phases with discrete symmetries can be achieved by varying continuously the distances among atoms and the detuning frequency.

arXiv:2105.10450•Physical review. D/Physical review. D.•Sep 14, 2021•16 citations
We study the isotropic, ferromagnetic two-dimensional spin systems with Z(5) symmetry, which, in the general case, depend upon two couplings, a temperature and a parameter \ensuremath{\Omega}, used to interpolate continuously between the clock (\ensuremath{\Omega}=0) and the Potts (\ensuremath{\Omega}=1) models, passing through the integrable Fateev-Zamolodchikov case (\ensuremath{\Omega}=\ensuremathΩc{\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}\ensuremathΩc​\ensuremath{\approxeq}0.618). Two methods, the high-temperature expansion and the exact \ensuremath{\beta} function on finite lattices, are applied within the Hamiltonian formulation to study critical properties. The transitions found are first order (\ensuremathΩc{\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}\ensuremathΩc​\ensuremath{\le}1), second order of the Ising type (\ensuremath{\Omega}\ensuremath{\simeq}\ensuremathΩc{\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}\ensuremathΩc​), or of the Kosterlitz-Thouless type (0\ensuremath{\le}\ensuremath{\Omega}\ensuremathΩc{\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}\ensuremathΩc​) with a varying exponent.
The critical behavior of the general isotropic, ferromagnetic two-dimensional spin system with openZ(5) symmetry is studied with use of high-temperature expansions of its mass gap. On the basis of these expansions we propose a simple analytic representation of the mass gap which naturally reproduces all the different phase transitions exhibited by this model (first order and second order of the Ising and of the Kosterlitz-Thouless types). In addition, the bifurcation point where the soft phases originate is clearly identified with the Fateev-Zamolodchikov value.
The Hamiltonian formulation of the two-dimensional spin systems with Z(p) symmetry, ranging from the Ising to the XY-plane rotor model, is used together with the Lanczos scheme to compute exactly the mass gap of finite systems. Critical properties are then deduced from a Callan-Symanzik \ensuremath{\beta} function. We find in particular a Kosterlitz-Thouless transition for p\ensuremath{\ge}5, with a p-dependent exponent which coincides with its renormalization-group estimate only in the p=\ensuremath{\infty} limit.
We have investigated the dynamic critical behavior of the two-dimensional Z(5)-symmetric spin model by using short-time Monte Carlo (MC) simulations. We have obtained estimates of some critical points in its rich phase diagram and included, among the usual critical lines the study of first-order (weak) transition by looking into the order-disorder phase transition. In addition, we also investigated the soft-disorder phase transition by considering empiric methods. A study of the behavior of β/νz along the self-dual critical line has been performed and special attention has been devoted to the critical bifurcation point, or Fateev-Zamolodchikov (FZ) point. First, by using a refinement method and taking into account simulations out of equilibrium, we were able to localize parameters of this point. In a second part of our study, we turned our attention to the behavior of the model at the early stage of its time evolution in order to find the dynamic critical exponent z as well as the static critical exponents β and ν of the FZ point on square lattices. The values of the static critical exponents and parameters are in good agreement with the exact results, and the dynamic critical exponent z≈2.28 very close to the four-state Potts model (z≈2.29).
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
This paper is concerned with the critical properties of a family of self-dual two-dimensional Z(N) models whose bulk free energy is exactly known at the self-dual point. The author's analysis is performed by studying the finite-size behaviour of the corresponding one-dimensional quantum Hamiltonians which also possess an exact solution at their self-dual point. By exploring finite-size scaling ideas and the conformal invariance of the critical infinite system the author calculates, for N up to 8, the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra. The results strongly suggest that the Z(N) quantum field theory of Zamolodchikov and Fateev is the underlying field theory associated with these statistical mechanical systems. The author also tests for the Z(5) case, the conjecture that these models correspond to the bifurcation points in the phase diagram of the General Z(N) spin model, where a massless phase originates.
Motivated by recent attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we define an extended-O(2) model by adding a $\ensuremath{\gamma}\mathrm{cos}(q\ensuremath{\varphi})$ term to the ordinary O(2) model with angular values restricted to a $2\ensuremath{\pi}$ interval. In the $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the model becomes an extended qqq-state clock model that reduces to the ordinary qqq-state clock model when qqq is an integer and otherwise is a continuation of the clock model for noninteger qqq. By shifting the $2\ensuremath{\pi}$ integration interval, the number of angles selected can change discontinuously and two cases need to be considered. What we call case 1 has one more angle than what we call case 2. We investigate this class of clock models in two space-time dimensions using Monte Carlo and tensor renormalization group methods. Both the specific heat and the magnetic susceptibility show a double-peak structure for fractional qqq. In case 1, the small-$\ensuremath{\beta}$ peak is associated with a crossover, and the large-$\ensuremath{\beta}$ peak is associated with an Ising critical point, while both peaks are crossovers in case 2. When qqq is close to an integer by an amount $\mathrm{\ensuremath{\Delta}}q$ and the system is close to the small-$\ensuremath{\beta}$ Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as $\ensuremath{\sim}1/(\mathrm{\ensuremath{\Delta}}q{)}^{1\ensuremath{-}1/{\ensuremath{\delta}}^{\ensuremath{'}}}$ with ${\ensuremath{\delta}}^{\ensuremath{'}}$ estimates consistent with the magnetic critical exponent $\ensuremath{\delta}=15$. The crossover peak and the Ising critical point move to Berezinskii-Kosterlitz-Thouless transition points with the same power-law scaling. A phase diagram for this model in the $(\ensuremath{\beta},q)$ plane is sketched. These results are possibly relevant for configurable Rydberg-atom arrays where the interpolations among phases with discrete symmetries can be achieved by varying continuously the distances among atoms and the detuning frequency.