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

Dal nevostochnyi Matematicheskii Zhurnal

Finite-size scaling in ferromagnetic spin systems on the pyrochlore lattice

Konstantin S Soldatov, M.A. Padalko, Vladislav Sergeevich Strongin, Dmitrii Yu Kapitan, Egor Vasiliev, Alexey Rybin, Vitalii Yu. Kapitan, Konstantin Valentinovich Nefedev

Published Nov 25, 2020• 2 citations
Publisher

Abstract

In this paper we present the results of the high-performance computations for the Ising model, the XY-model and the classical Heisenberg model for the pyrochlore lattice. We used Wolff and Swendsen-Wang cluster algorithms with GPU parallelization for the calculations. We obtained critical exponents and critical temperatures using finite-size scaling approach.

DOI
10.47910/femj202026

Related papers

A new MCRG (Monte Carlo Renormalization Group) calculation of the critical behavior of the 3d ising model - preliminary results

CLIVE F. BAILLIE, Kenneth N. Barish, Rajan Gupta +1 more

We have undertaken a large-scale computer simulation of the 3d Ising model using the Monte Carlo Renormalization Group (MCRG) method. Our calculation improves on past calculations in three ways. Firstly, larger lattices are used -- we have run on 128{sup 3} as well as 64{sup 3}. Secondly, many more spin operators -- 53 even and 46 odd -- are measured. Lastly, we have incorporated more efficient ways of spin update -- Wolff's single-cluster variant of the Swendsen and Wang algorithm. In addition to calculating the critical temperature the thermal and magnetic exponents, and improving the estimate of the correction-to-scaling exponent, we are aiming to verify the presence of redundant operators. We do not yet have sufficient statistics to give final results so here we describe our calculation and present preliminary results which demonstrate that we have control over both finite size effects and systematic errors arising from truncation. 16 refs., 7 figs.

•Oct 5, 1989•0 citations

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

Phase diagram of the XXZ pyrochlore model from pseudo-Majorana functional renormalization group

Yannik Schaden, M. G. Gonzalez, Johannes Reuther

We calculate the magnetic phase diagram of the spin- 1 / 2 nearest neighbor XXZ pyrochlore model using the pseudo-Majorana functional renormalization group in the temperature flow formalism. Our phase diagram as a function of temperature and coupling ratio, allowing both longitudinal and transverse couplings to be ferromagnetic and antiferromagnetic, reveals a large nonmagnetic regime at low temperatures, which includes the quantum spin ice phase near the antiferromagnetic Ising model, as well as the antiferromagnetic Heisenberg and XY models. We are able to detect magnetic phase transitions via critical finite size scaling down to temperatures two orders of magnitude smaller than the spin interactions, demonstrating the remarkably good performance of our method upon approaching the ground state. Specifically, the low-temperature transition from the zero-flux quantum spin ice phase into the transverse ferromagnetic phase shows very good agreement with previous quantum Monte Carlo results. Comparing our findings with classical results, we identify a quantum order-by-disorder effect near the antiferromagnetic XY model. In magnetically disordered regimes, we find characteristic patterns of broadened pinch points in the spin structure factor and investigate their evolution when approaching magnetically ordered phases. We also compute linear responses to lattice symmetry breaking perturbations and identify a possible lattice nematic ground state of the antiferromagnetic XY model.

Physical review. B./Physical review. B•Apr 24, 2025•2 citations

Static critical behavior of three-dimensional classical Heisenberg models: A high-resolution Monte Carlo study

Kun Chen, Alan M. Ferrenberg, David Paul Landau

Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).

Physical review. B, Condensed matter•Aug 1, 1993•331 citations

Transfer Matrix and Finite-Size Scaling for the Ising Model on Two- and Three-Dimensional Lattices

Takehiko Oguchi, Y‐h. Taguchi

The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.

Progress of Theoretical Physics Supplement•Jan 1, 1986•8 citations

Critical exponents of the classical three-dimensional Heisenberg model: A single-cluster Monte Carlo study

Christian Holm, Wolfhard Janke E

We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to ${100}^{3}$ spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.

arXiv:hep-lat/9301002•Physical review. B, Condensed matter•Jul 1, 1993•232 citations
In this paper we present the results of the high-performance computations for the Ising model, the XY-model and the classical Heisenberg model for the pyrochlore lattice. We used Wolff and Swendsen-Wang cluster algorithms with GPU parallelization for the calculations. We obtained critical exponents and critical temperatures using finite-size scaling approach.
We have undertaken a large-scale computer simulation of the 3d Ising model using the Monte Carlo Renormalization Group (MCRG) method. Our calculation improves on past calculations in three ways. Firstly, larger lattices are used -- we have run on 128{sup 3} as well as 64{sup 3}. Secondly, many more spin operators -- 53 even and 46 odd -- are measured. Lastly, we have incorporated more efficient ways of spin update -- Wolff's single-cluster variant of the Swendsen and Wang algorithm. In addition to calculating the critical temperature the thermal and magnetic exponents, and improving the estimate of the correction-to-scaling exponent, we are aiming to verify the presence of redundant operators. We do not yet have sufficient statistics to give final results so here we describe our calculation and present preliminary results which demonstrate that we have control over both finite size effects and systematic errors arising from truncation. 16 refs., 7 figs.
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.
We calculate the magnetic phase diagram of the spin- 1 / 2 nearest neighbor XXZ pyrochlore model using the pseudo-Majorana functional renormalization group in the temperature flow formalism. Our phase diagram as a function of temperature and coupling ratio, allowing both longitudinal and transverse couplings to be ferromagnetic and antiferromagnetic, reveals a large nonmagnetic regime at low temperatures, which includes the quantum spin ice phase near the antiferromagnetic Ising model, as well as the antiferromagnetic Heisenberg and XY models. We are able to detect magnetic phase transitions via critical finite size scaling down to temperatures two orders of magnitude smaller than the spin interactions, demonstrating the remarkably good performance of our method upon approaching the ground state. Specifically, the low-temperature transition from the zero-flux quantum spin ice phase into the transverse ferromagnetic phase shows very good agreement with previous quantum Monte Carlo results. Comparing our findings with classical results, we identify a quantum order-by-disorder effect near the antiferromagnetic XY model. In magnetically disordered regimes, we find characteristic patterns of broadened pinch points in the spin structure factor and investigate their evolution when approaching magnetically ordered phases. We also compute linear responses to lattice symmetry breaking perturbations and identify a possible lattice nematic ground state of the antiferromagnetic XY model.
Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).
The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.
We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to 1003{100}^{3}1003 spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.