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:1012.0823 · Journal of Statistical Mechanics Theory and Experiment

Off-equilibrium relaxational dynamics with an improved Ising Hamiltonian

Mario Collura

Published Dec 23, 2010• 11 citations
arXivHTMLPDF

Abstract

We study the off-equilibrium relaxational dynamics at criticality in the three-dimensional Blume–Capel model whose static critical behaviour belongs to the 3D Ising universality class. Using an 'improved' Hamiltonian (the leading corrections to scaling have vanishing amplitude) we perform Monte Carlo simulations of the relaxational dynamics after a quench from to T c . Analysing the off-equilibrium dynamics at T c we obtain an estimate of the dynamical critical exponent z = 2.020(8) that is perfectly consistent with the field theory predictions.

DOI
10.1088/1742-5468/2010/12/p12036

Related papers

Dynamic critical exponent z of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model

Martin Hasenbusch B

We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.

arXiv:1908.01702•Physical review. E•Feb 24, 2020•39 citations

Relaxational dynamics in 3D randomly diluted Ising models

Martin Hasenbusch B, Andrea Pelissetto C, Ettore Vicari

We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the ± J Ising model along the paramagnetic–ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size dependence of suitably defined autocorrelation times at the critical point provides the estimate z = 2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from to T c . In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behaviour gives an estimate of z that is perfectly consistent with the equilibrium estimate z = 2.35(2).

arXiv:0709.4179•Journal of Statistical Mechanics Theory and Experiment•Nov 23, 2007•23 citations

Critical dynamics of three-dimensional Z N gauge models and the inverted X Y universality class

Claudio Bonati, H. Panagopoulos, Ettore Vicari

We investigate the critical relaxational dynamics of the three-dimensional (3D) lattice Z_{N} gauge models with N=6 and N=8, whose equilibrium critical behavior at their topological transitions belongs to the inverted XY (IXY) universality class [this is also the universality class of the continuous transitions of the 3D lattice U(1) gauge Higgs models with a one-component complex scalar field], which is connected to the standard XY universality class by a nonlocal duality relation of the partition functions. Specifically, we consider the purely relaxational dynamics realized by a locally reversible Metropolis dynamics, as commonly used in Monte Carlo simulations. To determine the corresponding dynamic exponent z, we focus on the out-of-equilibrium critical relaxational flows arising from instantaneous quenches to the critical point, which are analyzed within an out-of-equilibrium finite-size scaling framework. We obtain the estimate z=2.59(3). A numerical analysis of the equilibrium critical dynamics give consistent, but less accurate, results. This dynamic exponent is expected to characterize the critical slowing down of the purely relaxational dynamics of all topological transitions that belong to the 3D IXY universality class. We note that this result implies that the critical relaxational dynamics of the 3D IXY universality class is slower than that of the standard 3D XY universality class, whose relaxational dynamic exponent z≈2.02 is significantly smaller, although they share the same length-scale critical exponent ν≈0.6717.

Physical review. E•Jul 30, 2025•2 citations

Two- and three-point functions at criticality: Monte Carlo simulations of the improved three-dimensional Blume-Capel model

Martin Hasenbusch B

We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to $L=1600$. As a check, also simulations of the spin-$\frac{1}{2}$ Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of $N$-point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where $L$ is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading ${Z}_{2}$-even scalar $\ensuremath{\epsilon}$.

arXiv:1711.10946•Physical review. E•Jan 16, 2018•14 citations

Equilibrium and out-of-equilibrium critical dynamics of the three-dimensional Heisenberg model with random cubic anisotropy

Antonio Astillero, Juan Jesus Ruiz-Lorenzo

We study the critical dynamics of the three-dimensional Heisenberg model with random cubic anisotropy in the out-of-equilibrium and equilibrium regimes. Analytical approaches based on field theory predict that the universality class of this model is that of the three-dimensional site-diluted Ising model. We have been able to estimate the dynamic critical exponent by working in the equilibrium regime and by computing the integrated autocorrelation times obtaining z=2.50(5) (without taking into account scaling corrections) and z=2.29(11) (by fixing the scaling corrections to that predicted by field theory). In the out-of-equilibrium regime we have focused on the study of the dynamic correlation length, allowing us to compute the dynamic critical exponent obtaining z=2.38(2), which is compatible with the equilibrium ones. Finally, both estimates are also compatible with the most accurate prediction z=2.35(2), from numerical simulations of the 3D site-diluted Ising model, in agreement with the predictions based on field theory.

arXiv:2410.14275•Physical review. E•Jun 5, 2025•1 citation

Finite size scaling study of lattice models in the three-dimensional Ising universality class

Martin Hasenbusch B

We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter $D$ on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter $D$, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for $D=0.641$ and 0.655 on lattices of a linear size up to $L=360$ we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.

arXiv:1004.4486•Physical Review B•Nov 23, 2010•333 citations
We study the off-equilibrium relaxational dynamics at criticality in the three-dimensional Blume–Capel model whose static critical behaviour belongs to the 3D Ising universality class. Using an 'improved' Hamiltonian (the leading corrections to scaling have vanishing amplitude) we perform Monte Carlo simulations of the relaxational dynamics after a quench from to T c . Analysing the off-equilibrium dynamics at T c we obtain an estimate of the dynamical critical exponent z = 2.020(8) that is perfectly consistent with the field theory predictions.
We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.
We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the ± J Ising model along the paramagnetic–ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size dependence of suitably defined autocorrelation times at the critical point provides the estimate z = 2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from to T c . In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behaviour gives an estimate of z that is perfectly consistent with the equilibrium estimate z = 2.35(2).
We investigate the critical relaxational dynamics of the three-dimensional (3D) lattice Z_{N} gauge models with N=6 and N=8, whose equilibrium critical behavior at their topological transitions belongs to the inverted XY (IXY) universality class [this is also the universality class of the continuous transitions of the 3D lattice U(1) gauge Higgs models with a one-component complex scalar field], which is connected to the standard XY universality class by a nonlocal duality relation of the partition functions. Specifically, we consider the purely relaxational dynamics realized by a locally reversible Metropolis dynamics, as commonly used in Monte Carlo simulations. To determine the corresponding dynamic exponent z, we focus on the out-of-equilibrium critical relaxational flows arising from instantaneous quenches to the critical point, which are analyzed within an out-of-equilibrium finite-size scaling framework. We obtain the estimate z=2.59(3). A numerical analysis of the equilibrium critical dynamics give consistent, but less accurate, results. This dynamic exponent is expected to characterize the critical slowing down of the purely relaxational dynamics of all topological transitions that belong to the 3D IXY universality class. We note that this result implies that the critical relaxational dynamics of the 3D IXY universality class is slower than that of the standard 3D XY universality class, whose relaxational dynamic exponent z≈2.02 is significantly smaller, although they share the same length-scale critical exponent ν≈0.6717.
We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to L=1600L=1600L=1600. As a check, also simulations of the spin-12\frac{1}{2}21​ Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of NNN-point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where LLL is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading Z2{Z}_{2}Z2​-even scalar $\ensuremath{\epsilon}$.
We study the critical dynamics of the three-dimensional Heisenberg model with random cubic anisotropy in the out-of-equilibrium and equilibrium regimes. Analytical approaches based on field theory predict that the universality class of this model is that of the three-dimensional site-diluted Ising model. We have been able to estimate the dynamic critical exponent by working in the equilibrium regime and by computing the integrated autocorrelation times obtaining z=2.50(5) (without taking into account scaling corrections) and z=2.29(11) (by fixing the scaling corrections to that predicted by field theory). In the out-of-equilibrium regime we have focused on the study of the dynamic correlation length, allowing us to compute the dynamic critical exponent obtaining z=2.38(2), which is compatible with the equilibrium ones. Finally, both estimates are also compatible with the most accurate prediction z=2.35(2), from numerical simulations of the 3D site-diluted Ising model, in agreement with the predictions based on field theory.
We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter DDD on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter DDD, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for D=0.641D=0.641D=0.641 and 0.655 on lattices of a linear size up to L=360L=360L=360 we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.