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

Coarse grained free energies with gradient corrections from Monte Carlo simulations in Fourier space

A. Tröster

Published Jul 9, 2007• 24 citations
Publisher

Abstract

By performing Wang-Landau simulations in Fourier space, we study coarse grained free energies of generalized lattice spin models. The temperature dependence of the lowest order Landau-Ginzburg free energy expansion coefficients of the ${s}^{4}$ metamodel for phase transitions of the Ising universality class for a given coarse-graining length is calculated for the first time and compared to the standard assumptions made in the literature. In particular, the assumption that the so-called gradient coefficient is constant with temperature is in excellent agreement with the simulations. Our results also indicate that the coarse grained potentials follow finite size scaling. Finally, a coarse grained free energy with a temperature-dependent coarse-graining length equal to the system's correlation length is computed. The resulting potentials are compared to their mean-field counterparts.

DOI
10.1103/physrevb.76.012402

Related papers

Topologically correct phase boundaries and transition temperatures for Ising Hamiltonians via self-consistent coarse-grained cluster-lattice models

Teck Leong Tan, Duane D. Johnson

We derive a cluster mean-field theory for an Ising Hamiltonian using a cluster-lattice Fourier transform with a cluster of size ${N}_{\mathrm{c}}$ and a coarse-grained (CG) lattice into cells of size ${N}_{\mathrm{cell}}$. We explore forms with ${N}_{\mathrm{cell}}\ensuremath{\geqslant}{N}_{\mathrm{c}}$, including a non-CG (NCG) version with ${N}_{\mathrm{cell}}\ensuremath{\rightarrow}\ensuremath{\infty}$. For ${N}_{\text{c}}={N}_{\mathrm{cell}}$, the set of static, self-consistent equations relating cluster and CG lattice correlations is analogous to that in dynamical cluster approximation and cellular dynamical mean-field theory used in correlated electron physics. A variational ${N}_{\mathrm{c}}$-site cluster grand potential based on ${N}_{\mathrm{c}}={N}_{\mathrm{cell}}$ CG lattice maintains thermodynamic consistency and improves predictions, recovering Monte Carlo and series expansion results upon finite-size scaling; notably, the ${N}_{\mathrm{c}}=1$ CG results already predict well the first- and second-order phase boundary topology and transition temperatures for frustrated lattices. The NCG version is significantly faster computationally than the CG case and more accurate at fixed ${N}_{\mathrm{c}}$ for ferromagnetism, which is potentially useful for cluster expansion and quantum cluster applications.

Physical Review B•Apr 27, 2011•12 citations

Direct Calculation of Absolute Free Energy for Lattice Systems by Monte Carlo Sampling of Finite-Size Dependence

K. K. Mon

It is shown that the absolute free energy of lattice spin systems can be calculated directly by a novel application of Monte Carlo sampling of finite-size dependence. Results are obtained for the two- and three-dimensional Ising models at ${T}_{c}$ and are consistent with recent finite-size scaling-theory predictions. The free energy is in excellent agreement with Onsager's exact solution for two dimensions and with series expansions for three dimensions. The ${U}_{0}$ is measured for the first time. Results are consistent with the proposed universality.

Physical Review Letters•Jun 24, 1985•43 citations

Monte Carlo simulation of the fcc antiferromagnetic Ising model

T. Polgreen

Monte Carlo methods are employed to measure the internal energy, sublattice magnetization, and spin-spin correlation function of the fcc antiferromagnetic Ising model as a function of temperature. The internal energy of both the ordered and disordered phases is fitted by appropriate series expansions, and the free energy is obtained analytically from the series. The ordering transition is seen to be of first order with a transition temperature of 1.736\ifmmode\pm\else\textpm\fi{}0.001 in units of the nearest-neighbor coupling $J$. These results are compared with earlier approximations of the model, in particular, the low-temperature series expansion and the Kikuchi tetrahedron approximation, and other Monte Carlo results. The spin-spin correlation function was measured in the disordered phase up to eight lattice spacings in the [100] direction. The correlation length at the transition is found to be $\ensuremath{\sim}2.5a$. The behavior of the correlation length is approximately mean-field-like.

Physical review. B, Condensed matter•Feb 1, 1984•60 citations

Critical Properties from Monte Carlo Coarse Graining and Renormalization

Kurt Binder

The distribution function ${P}_{L}(s)$ of the local order parameter $s$ in finite blocks of size ${L}^{d}$ is studied for Ising models for dimensionalities $d=2, 3, \mathrm{and} 4$ by Monte Carlo methods. A real-space renormalization group based on phenomenological scaling yields fairly accurate results for rather small $L$ (e.g., the standard exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$ for $d=3$ are found as $\frac{2\ensuremath{\beta}}{\ensuremath{\nu}}=1.03\ifmmode\pm\else\textpm\fi{}0.01$, $\frac{1}{\ensuremath{\nu}}=1.60\ifmmode\pm\else\textpm\fi{}0.05$). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities $n>1$.

Physical Review Letters•Aug 31, 1981•801 citations

Coarse-grained belief propagation for simulation of interacting quantum systems at all temperatures

Ersen Bilgin, David Poulin

We continue our numerical study of quantum belief propagation initiated in [Phys. Rev. A 77, 052318 (2008)]. We demonstrate how the method can be expressed in terms of an effective thermal potential that materializes when the system presents quantum correlations, but is insensitive to classical correlations. The thermal potential provides an efficient means to assess the precision of belief propagation on graphs with no loops. We illustrate these concepts using the one-dimensional quantum Ising model and compare our results with exact solutions. We also use the method to study the transverse field quantum Ising spin glass for which we obtain a phase diagram that is largely in agreement with the one obtained in [Phys. Rev. B 78, 134424 (2008).] using a different approach. Finally, we introduce the coarse-grained belief propagation (CGBP) algorithm to improve belief propagation at low temperatures. This method combines the reliability of belief propagation at high temperatures with the ability of entanglement renormalization to efficiently describe low-energy subspaces of quantum systems with local interactions. With CGBP, thermodynamic properties of quantum systems can be calculated with a high degree of accuracy at all temperatures.

arXiv:0910.2299•Physical Review B•Feb 9, 2010•19 citations

Coarse graining, Monte Carlo renormalisation, percolation threshold and critical temperature in the Ising model

Dietrich Stauffer

Finite-size scaling suggests that the super-spins corresponding to large cells in a Monte Carlo renormalisation scheme form at T=T c and infinite network of connected down spins. The finite-size scaling assumption, M b =f((T-T c )b 1 nu /) for the renormalised block spin magnetisation is tested by computer simulation for lattice sizes 128 3 and 256 3 .

Journal of Physics A Mathematical and General•Dec 1, 1984•8 citations
By performing Wang-Landau simulations in Fourier space, we study coarse grained free energies of generalized lattice spin models. The temperature dependence of the lowest order Landau-Ginzburg free energy expansion coefficients of the s4{s}^{4}s4 metamodel for phase transitions of the Ising universality class for a given coarse-graining length is calculated for the first time and compared to the standard assumptions made in the literature. In particular, the assumption that the so-called gradient coefficient is constant with temperature is in excellent agreement with the simulations. Our results also indicate that the coarse grained potentials follow finite size scaling. Finally, a coarse grained free energy with a temperature-dependent coarse-graining length equal to the system's correlation length is computed. The resulting potentials are compared to their mean-field counterparts.
We derive a cluster mean-field theory for an Ising Hamiltonian using a cluster-lattice Fourier transform with a cluster of size Nc{N}_{\mathrm{c}}Nc​ and a coarse-grained (CG) lattice into cells of size Ncell{N}_{\mathrm{cell}}Ncell​. We explore forms with ${N}_{\mathrm{cell}}\ensuremath{\geqslant}{N}_{\mathrm{c}}$, including a non-CG (NCG) version with ${N}_{\mathrm{cell}}\ensuremath{\rightarrow}\ensuremath{\infty}$. For Nc=Ncell{N}_{\text{c}}={N}_{\mathrm{cell}}Nc​=Ncell​, the set of static, self-consistent equations relating cluster and CG lattice correlations is analogous to that in dynamical cluster approximation and cellular dynamical mean-field theory used in correlated electron physics. A variational Nc{N}_{\mathrm{c}}Nc​-site cluster grand potential based on Nc=Ncell{N}_{\mathrm{c}}={N}_{\mathrm{cell}}Nc​=Ncell​ CG lattice maintains thermodynamic consistency and improves predictions, recovering Monte Carlo and series expansion results upon finite-size scaling; notably, the Nc=1{N}_{\mathrm{c}}=1Nc​=1 CG results already predict well the first- and second-order phase boundary topology and transition temperatures for frustrated lattices. The NCG version is significantly faster computationally than the CG case and more accurate at fixed Nc{N}_{\mathrm{c}}Nc​ for ferromagnetism, which is potentially useful for cluster expansion and quantum cluster applications.
It is shown that the absolute free energy of lattice spin systems can be calculated directly by a novel application of Monte Carlo sampling of finite-size dependence. Results are obtained for the two- and three-dimensional Ising models at Tc{T}_{c}Tc​ and are consistent with recent finite-size scaling-theory predictions. The free energy is in excellent agreement with Onsager's exact solution for two dimensions and with series expansions for three dimensions. The U0{U}_{0}U0​ is measured for the first time. Results are consistent with the proposed universality.
Monte Carlo methods are employed to measure the internal energy, sublattice magnetization, and spin-spin correlation function of the fcc antiferromagnetic Ising model as a function of temperature. The internal energy of both the ordered and disordered phases is fitted by appropriate series expansions, and the free energy is obtained analytically from the series. The ordering transition is seen to be of first order with a transition temperature of 1.736\ifmmode\pm\else\textpm\fi{}0.001 in units of the nearest-neighbor coupling JJJ. These results are compared with earlier approximations of the model, in particular, the low-temperature series expansion and the Kikuchi tetrahedron approximation, and other Monte Carlo results. The spin-spin correlation function was measured in the disordered phase up to eight lattice spacings in the [100] direction. The correlation length at the transition is found to be $\ensuremath{\sim}2.5a$. The behavior of the correlation length is approximately mean-field-like.
The distribution function PL(s){P}_{L}(s)PL​(s) of the local order parameter sss in finite blocks of size Ld{L}^{d}Ld is studied for Ising models for dimensionalities d=2,3,and4d=2, 3, \mathrm{and} 4d=2,3,and4 by Monte Carlo methods. A real-space renormalization group based on phenomenological scaling yields fairly accurate results for rather small LLL (e.g., the standard exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$ for d=3d=3d=3 are found as $\frac{2\ensuremath{\beta}}{\ensuremath{\nu}}=1.03\ifmmode\pm\else\textpm\fi{}0.01$, $\frac{1}{\ensuremath{\nu}}=1.60\ifmmode\pm\else\textpm\fi{}0.05$). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities n>1n>1n>1.
We continue our numerical study of quantum belief propagation initiated in [Phys. Rev. A 77, 052318 (2008)]. We demonstrate how the method can be expressed in terms of an effective thermal potential that materializes when the system presents quantum correlations, but is insensitive to classical correlations. The thermal potential provides an efficient means to assess the precision of belief propagation on graphs with no loops. We illustrate these concepts using the one-dimensional quantum Ising model and compare our results with exact solutions. We also use the method to study the transverse field quantum Ising spin glass for which we obtain a phase diagram that is largely in agreement with the one obtained in [Phys. Rev. B 78, 134424 (2008).] using a different approach. Finally, we introduce the coarse-grained belief propagation (CGBP) algorithm to improve belief propagation at low temperatures. This method combines the reliability of belief propagation at high temperatures with the ability of entanglement renormalization to efficiently describe low-energy subspaces of quantum systems with local interactions. With CGBP, thermodynamic properties of quantum systems can be calculated with a high degree of accuracy at all temperatures.