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

arXiv:cond-mat/0104288 · Physical review. B, Condensed matter

Relaxational dynamics study of the classical Heisenberg spin XY model in spherical coordinate representation

Beom Jun Kim, Petter Minnhagen, Suhk Kun Oh, Jean Soo Chung

Published Jun 15, 2001• 6 citations
arXivHTMLPDF

Abstract

The two- and three-dimensional classical Heisenberg spin $\mathrm{XY}$ $(\mathrm{CHS}XY)$ models, with the spherical coordinates of spins taken as dynamic variables, are numerically investigated. We allow the polar \ensuremath{\theta} and azimuthal $\ensuremath{\varphi}$ angles to have uniform values in $[0,\ensuremath{\pi})$ and $[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}),$ respectively, and the static universality class is shown to be identical to the classical $\mathrm{XY}$ model with two-component spins, as well as the $\mathrm{CHS}\mathrm{XY}$ model with a different choice of dynamic variables, conventionally used in the literature. The relaxational dynamic simulation reveals that the dynamic critical exponent z is found to have the value $z\ensuremath{\approx}2.0$ for both two and three dimensions, in contrast to $z\ensuremath{\approx}d/2$ $(d=$ spatial dimension) found previously with spin dynamics simulation of the conventional $\mathrm{CHS}\mathrm{XY}$ model. Comparisons with the usual two-component classical $\mathrm{XY}$ model are also made.

DOI
10.1103/physrevb.64.024406

Related papers

Three-dimensional generalized xy models: A Monte Carlo study

Hassan Chamati, Silvano Romano, L. A. S. Mόl +1 more

The lattice spin models considered in the present paper consist of three-component unit vectors, associated with a D-dimensional lattice (say d), parameterized by usual spherical angles (θk,k), and interacting via a ferromagnetic potential restricted to nearest neighbours, of the form Wjk = − (sin θj sin θk)p cos (j − k), p , p ≥ 1; here is a positive quantity setting energy and temperature scales. The models were recently introduced, and rigorous comparison inequalities holding for them were investigated, and used to prove the existence of an ordering transition when D = 3 (Romano S. and Zagrebnov V. A., Phys. Lett. A, 301 (2002) 402), investigated by other approximate techniques as well (Mol L. A. S., Pereira A. R. and Moura-Melo W. A., Phys. Lett. A, 319 (2003) 114). We report here an extensive Monte Carlo study of the critical behaviour for D = 3 and p ≤ 4; our results are consistent with the same universality class as the xy model.

Europhysics Letters (EPL)•Aug 30, 2005•11 citations

Short-time critical and coarsening dynamics of the classical three-dimensional Heisenberg model

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

The dynamic critical behaviour of the three-dimensional classical Heisenberg model was studied by using short-time Monte Carlo simulations. The dynamic critical exponents z , θ and θ g , as well as the static critical exponents ν and β, were estimated by following the time evolution of the z -component of the total magnetization and its moments of higher order. The anomalous dimension of the magnetization x 0 was also calculated from a relation obtained through finite size scaling in the non-equilibrium stage. Afterwards, the local persistence exponent θ l was obtained by following the power law behaviour of the local persistence probability when the system was quenched from high temperatures to the null one ( T = 0). The time evolution of P l ( t ) was also analysed for T >0 and the expected exponential stretch was observed.

Journal of Statistical Mechanics Theory and Experiment•Oct 5, 2006•17 citations

Critical relaxational dynamics at the continuous transitions of three-dimensional spin models with Z 2 gauge symmetry

Claudio Bonati, Andrea Pelissetto C, Ettore Vicari

We characterize the dynamic universality classes of a relaxational dynamics under equilibrium conditions at the continuous transitions of three-dimensional (3D) spin systems with ${\mathbb{Z}}_{2}$ gauge symmetry. In particular, we consider the pure lattice ${\mathbb{Z}}_{2}$ gauge model and the lattice ${\mathbb{Z}}_{2}$ gauge XY model, which present various types of transitions: topological transitions without a local order parameter and transitions characterized by both gauge-invariant and non-gauge-invariant XY order parameters. We consider a standard relaxational (locally reversible) Metropolis dynamics and determine the dynamic critical exponent $z$ that characterizes the critical slowing down of the dynamics as the continuous transition is approached. At the topological ${\mathbb{Z}}_{2}$ gauge transitions we find $z=2.55(6)$. Therefore, the dynamics is significantly slower than in Ising systems---$z\ensuremath{\approx}2.02$ for the 3D Ising universality class---although 3D ${\mathbb{Z}}_{2}$ gauge systems and Ising systems have the same static critical behavior because of duality. As for the nontopological transitions in the 3D ${\mathbb{Z}}_{2}$ gauge XY model, we find that their critical dynamics belong to the same dynamic universality class as the relaxational dynamics in ungauged XY systems independently of the gauge-invariant or non-gauge-invariant nature of the order parameter at the transition.

arXiv:2501.09575•Physical review. B./Physical review. B•Mar 11, 2025•8 citations

Universal prethermal dynamics and self-similar relaxation in the two-dimensional Heisenberg model

Joaquin F. Rodriguez-Nieva, Asier Piñeiro Orioli, Jamir Marino

We characterize the universal far-from-equilibrium dynamics of the isolated two-dimensional quantum Heisenberg model. For a broad range of initial conditions, we find a long-lived universal prethermal regime characterized by self-similar behavior of spin-spin correlations. We analytically derive the spatial-temporal scaling exponents and find excellent agreement with numerics using phase space methods. The scaling exponents are insensitive to the choice of initial conditions, which include coherent and incoherent spin states as well as values of magnetization and energy in a wide range. Compared to previously studied self-similar dynamics in non-equilibrium $O(n)$ field theories and Bose gases, we find qualitatively distinct scaling behavior originating from the presence of spin modes which remain gapless at long times and which are protected by the global SU(2) symmetry. Our predictions, which suggest a new non-equilibrium universality class, are readily testable in ultra-cold atoms simulators of Heisenberg magnets.

•May 31, 2021•0 citations

CRITICAL DYNAMICS IN THE 3D SPIN-1/2 HEISENBERG MODEL: A DECOUPLED CELL MONTE CARLO STUDY

Cynthia J. Sisson

The three-dimensional spin-1/2 Heisenberg model on a simple cubic lattice is studied for ferromagnetic and antiferromagnetic interactions using the Decoupled Cell Method for quantum Monte Carlo. Results for the relaxation time τL are determined for both ferromagnetic and antiferromagnetic systems and found to be similar to those found for the classical (s → ∞) Heisenberg model. The scaling of τL with system size is used to extract the dynamical critical exponent z for the two systems. The values of z = 1.98 ± 0.12 for the ferromagnet and z = 1.94 ± 0.09 for the antiferromagnet are in good agreement with theoretical predictions and previous Monte Carlo studies of the classical Heisenberg model.

International Journal of Modern Physics C•Jun 1, 1996•1 citation

Invaded Cluster Simulations of the XY Model in Two and Three

Ilija Dukovski, Jonathan L. Machta, Lincoln Chayes

The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 20002 and 1203, respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are Kc=0.45412(2) for the three-dimensional XY model and η=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are Kc=1.120(1) and η=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.

•Jan 1, 2002•0 citations
The two- and three-dimensional classical Heisenberg spin XY\mathrm{XY}XY (CHSXY)(\mathrm{CHS}XY)(CHSXY) models, with the spherical coordinates of spins taken as dynamic variables, are numerically investigated. We allow the polar \ensuremath{\theta} and azimuthal \ensuremathφ\ensuremath{\varphi}\ensuremathφ angles to have uniform values in [0,\ensuremathπ)[0,\ensuremath{\pi})[0,\ensuremathπ) and [\ensuremath−\ensuremathπ,\ensuremathπ),[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}),[\ensuremath−\ensuremathπ,\ensuremathπ), respectively, and the static universality class is shown to be identical to the classical XY\mathrm{XY}XY model with two-component spins, as well as the CHSXY\mathrm{CHS}\mathrm{XY}CHSXY model with a different choice of dynamic variables, conventionally used in the literature. The relaxational dynamic simulation reveals that the dynamic critical exponent z is found to have the value z\ensuremath≈2.0z\ensuremath{\approx}2.0z\ensuremath≈2.0 for both two and three dimensions, in contrast to z\ensuremath≈d/2z\ensuremath{\approx}d/2z\ensuremath≈d/2 (d=(d=(d= spatial dimension) found previously with spin dynamics simulation of the conventional CHSXY\mathrm{CHS}\mathrm{XY}CHSXY model. Comparisons with the usual two-component classical XY\mathrm{XY}XY model are also made.
The lattice spin models considered in the present paper consist of three-component unit vectors, associated with a D-dimensional lattice (say d), parameterized by usual spherical angles (θk,k), and interacting via a ferromagnetic potential restricted to nearest neighbours, of the form Wjk = − (sin θj sin θk)p cos (j − k), p , p ≥ 1; here is a positive quantity setting energy and temperature scales. The models were recently introduced, and rigorous comparison inequalities holding for them were investigated, and used to prove the existence of an ordering transition when D = 3 (Romano S. and Zagrebnov V. A., Phys. Lett. A, 301 (2002) 402), investigated by other approximate techniques as well (Mol L. A. S., Pereira A. R. and Moura-Melo W. A., Phys. Lett. A, 319 (2003) 114). We report here an extensive Monte Carlo study of the critical behaviour for D = 3 and p ≤ 4; our results are consistent with the same universality class as the xy model.
The dynamic critical behaviour of the three-dimensional classical Heisenberg model was studied by using short-time Monte Carlo simulations. The dynamic critical exponents z , θ and θ g , as well as the static critical exponents ν and β, were estimated by following the time evolution of the z -component of the total magnetization and its moments of higher order. The anomalous dimension of the magnetization x 0 was also calculated from a relation obtained through finite size scaling in the non-equilibrium stage. Afterwards, the local persistence exponent θ l was obtained by following the power law behaviour of the local persistence probability when the system was quenched from high temperatures to the null one ( T = 0). The time evolution of P l ( t ) was also analysed for T >0 and the expected exponential stretch was observed.
We characterize the dynamic universality classes of a relaxational dynamics under equilibrium conditions at the continuous transitions of three-dimensional (3D) spin systems with Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry. In particular, we consider the pure lattice Z2{\mathbb{Z}}_{2}Z2​ gauge model and the lattice Z2{\mathbb{Z}}_{2}Z2​ gauge XY model, which present various types of transitions: topological transitions without a local order parameter and transitions characterized by both gauge-invariant and non-gauge-invariant XY order parameters. We consider a standard relaxational (locally reversible) Metropolis dynamics and determine the dynamic critical exponent zzz that characterizes the critical slowing down of the dynamics as the continuous transition is approached. At the topological Z2{\mathbb{Z}}_{2}Z2​ gauge transitions we find z=2.55(6)z=2.55(6)z=2.55(6). Therefore, the dynamics is significantly slower than in Ising systems---$z\ensuremath{\approx}2.02$ for the 3D Ising universality class---although 3D Z2{\mathbb{Z}}_{2}Z2​ gauge systems and Ising systems have the same static critical behavior because of duality. As for the nontopological transitions in the 3D Z2{\mathbb{Z}}_{2}Z2​ gauge XY model, we find that their critical dynamics belong to the same dynamic universality class as the relaxational dynamics in ungauged XY systems independently of the gauge-invariant or non-gauge-invariant nature of the order parameter at the transition.
We characterize the universal far-from-equilibrium dynamics of the isolated two-dimensional quantum Heisenberg model. For a broad range of initial conditions, we find a long-lived universal prethermal regime characterized by self-similar behavior of spin-spin correlations. We analytically derive the spatial-temporal scaling exponents and find excellent agreement with numerics using phase space methods. The scaling exponents are insensitive to the choice of initial conditions, which include coherent and incoherent spin states as well as values of magnetization and energy in a wide range. Compared to previously studied self-similar dynamics in non-equilibrium O(n)O(n)O(n) field theories and Bose gases, we find qualitatively distinct scaling behavior originating from the presence of spin modes which remain gapless at long times and which are protected by the global SU(2) symmetry. Our predictions, which suggest a new non-equilibrium universality class, are readily testable in ultra-cold atoms simulators of Heisenberg magnets.
The three-dimensional spin-1/2 Heisenberg model on a simple cubic lattice is studied for ferromagnetic and antiferromagnetic interactions using the Decoupled Cell Method for quantum Monte Carlo. Results for the relaxation time τL are determined for both ferromagnetic and antiferromagnetic systems and found to be similar to those found for the classical (s → ∞) Heisenberg model. The scaling of τL with system size is used to extract the dynamical critical exponent z for the two systems. The values of z = 1.98 ± 0.12 for the ferromagnet and z = 1.94 ± 0.09 for the antiferromagnet are in good agreement with theoretical predictions and previous Monte Carlo studies of the classical Heisenberg model.
The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 20002 and 1203, respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are Kc=0.45412(2) for the three-dimensional XY model and η=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are Kc=1.120(1) and η=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.