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

Quantum Monte Carlo study of the spin-1/2 Heisenberg model

Y.-C. Chen, H. H. Chen, Felix Lee

Published May 1, 1991• 10 citations
Publisher

Abstract

Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/${\mathit{k}}_{\mathit{B}}$T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature ${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}}$/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.

DOI
10.1103/physrevb.43.11082

Related papers

Critical behavior of the S=1/2 Heisenberg ferromagnet: A Handscomb quantum Monte Carlo study

Adauto J. F. de Souza, U.M.S. Costa, M. L. Lyra

We investigate the critical relaxational dynamics of the $S=1/2$ Heisenberg ferromagnet on a simple cubic lattice within the Handscomb prescription, on which a diagrammatic series expansion of the partition function is computed by means of a Monte Carlo procedure. Using a phenomenological renormalization-group analysis of graph quantities related to the spin susceptibility and order parameter, we obtain precise estimates of the critical exponents relations $\ensuremath{\gamma}/\ensuremath{\nu}=1.98\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.512\ifmmode\pm\else\textpm\fi{}0.002$ and of the Curie temperature ${k}_{B}{T}_{c}/J=1.6778\ifmmode\pm\else\textpm\fi{}0.0002.$ The critical correlation time ${\ensuremath{\tau}}_{\mathrm{int}}$ of both energy and susceptibility is also computed. We found the number of Monte Carlo steps needed to generate uncorrelated diagram configurations scales with the system's volume. We estimate the efficiency of the Handscomb method, comparing its ability in dealing with the critical slowing down to that of other quantum and classical Monte Carlo prescriptions.

arXiv:cond-mat/0004176•Physical review. B, Condensed matter•Oct 1, 2000•13 citations

Quantum Monte Carlo simulation method for spin systems

Anders W. Sandvik, J. Kurkijärvi

A quantum Monte Carlo simulation scheme for spin systems is presented. The method is a generalization of Handscomb's method but applicable to any length of the spin, i.e., when the spin traces cannot be evaluated analytically. The Monte Carlo sampling is extended to the space of spin vectors in addition to the usual operator-index sequences. An important technical point is that the index sequences are augmented with the aid of unit operators to a constant, self-consistently determined length. The scheme is applied to the one-dimensional antiferromagnetic spin-S Heisenberg model. Results at low temperatures are reported for S=1 and S=3/2 and system sizes up to N=64. The computed magnetic structure factor in the S=1 chain is in agreement with earlier ground-state calculations. For S=3/2 we find the exponent \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=0.49\ifmmode\pm\else\textpm\fi{}0.04 for the divergence of the antiferromagnetic structure factor. Further, the susceptibility as a function of the wave number is computed. For S=1 the staggered susceptibility \ensuremath{\chi}(\ensuremath{\pi}) at T=0 is found to take the value 20.0\ifmmode\pm\else\textpm\fi{}1.5 in units such that \ensuremath{\chi}(q)\ensuremath{\rightarrow}${\mathit{T}}^{\mathrm{\ensuremath{-}}1}$ at high temperatures (with the temperature scale defined by ${\mathit{k}}_{\mathit{B}}$=1). For S=3/2 we obtain the exponent \ensuremath{\gamma}=1.45\ifmmode\pm\else\textpm\fi{}0.05 for the divergence of the staggered susceptibility.

Physical review. B, Condensed matter•Mar 1, 1991•528 citations

Quantum Monte Carlo study of a Heisenberg spin system in two dimensions with long-range interactions

Oleg N. Vassiliev, Michael G Cottam, I. V. Rojdestvenski

The thermodynamic properties and critical behavior of a Heisenberg ferromagnet in two dimensions are studied using the Handscomb quantum Monte Carlo method. The spins (with quantum number S=12) are arranged on a square lattice and have a long-range interaction with a power-law decrease, J(r)∝r−p, with distance r. The parameter p is taken to be in the range 2<p⩽6. The system has an ordering transition at a nonzero temperature Tc when 2<p<4. The dependence of Tc on p and the critical behavior of the susceptibility are investigated. Properties of spin correlation functions in different regimes (of p and T) are discussed. The results are compared with previous approximate analytic quantum theories.

Journal of Applied Physics•Jun 1, 2001•5 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

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

Reweighting histograms in quantum Monte Carlo studies of the Heisenberg model

Adauto J. F. de Souza, M. L. Lyra

In this communication, the histogram-reweighting technique is extended to quantum Monte Carlo studies of spin systems. The proposed method is applied to obtain equilibrium critical parameters of the quantum $S=1/2$ Heisenberg ferromagnet on simple cubic lattices. Our estimates are ${k}_{B}{T}_{c}/J=1.67764(2),$ ${U}^{*}=0.4504(1),$ $\ensuremath{\gamma}=1.397(20),$ $\ensuremath{\beta}=0.360(6),$ and $\ensuremath{\nu}=0.705(10)$, which are in perfect agreement with the best field-theory and high-temperature-series results. The method can readily be applied to general quantum-spin models, standing as an important tool for the study of quantum phase transitions.

Physical review. B, Condensed matter•Mar 1, 2002•12 citations
Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/kB{\mathit{k}}_{\mathit{B}}kB​T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature {\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}}/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.
We investigate the critical relaxational dynamics of the S=1/2S=1/2S=1/2 Heisenberg ferromagnet on a simple cubic lattice within the Handscomb prescription, on which a diagrammatic series expansion of the partition function is computed by means of a Monte Carlo procedure. Using a phenomenological renormalization-group analysis of graph quantities related to the spin susceptibility and order parameter, we obtain precise estimates of the critical exponents relations $\ensuremath{\gamma}/\ensuremath{\nu}=1.98\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.512\ifmmode\pm\else\textpm\fi{}0.002$ and of the Curie temperature ${k}_{B}{T}_{c}/J=1.6778\ifmmode\pm\else\textpm\fi{}0.0002.$ The critical correlation time ${\ensuremath{\tau}}_{\mathrm{int}}$ of both energy and susceptibility is also computed. We found the number of Monte Carlo steps needed to generate uncorrelated diagram configurations scales with the system's volume. We estimate the efficiency of the Handscomb method, comparing its ability in dealing with the critical slowing down to that of other quantum and classical Monte Carlo prescriptions.
A quantum Monte Carlo simulation scheme for spin systems is presented. The method is a generalization of Handscomb's method but applicable to any length of the spin, i.e., when the spin traces cannot be evaluated analytically. The Monte Carlo sampling is extended to the space of spin vectors in addition to the usual operator-index sequences. An important technical point is that the index sequences are augmented with the aid of unit operators to a constant, self-consistently determined length. The scheme is applied to the one-dimensional antiferromagnetic spin-S Heisenberg model. Results at low temperatures are reported for S=1 and S=3/2 and system sizes up to N=64. The computed magnetic structure factor in the S=1 chain is in agreement with earlier ground-state calculations. For S=3/2 we find the exponent \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=0.49\ifmmode\pm\else\textpm\fi{}0.04 for the divergence of the antiferromagnetic structure factor. Further, the susceptibility as a function of the wave number is computed. For S=1 the staggered susceptibility \ensuremath{\chi}(\ensuremath{\pi}) at T=0 is found to take the value 20.0\ifmmode\pm\else\textpm\fi{}1.5 in units such that \ensuremath{\chi}(q)\ensuremath{\rightarrow}${\mathit{T}}^{\mathrm{\ensuremath{-}}1}$ at high temperatures (with the temperature scale defined by kB{\mathit{k}}_{\mathit{B}}kB​=1). For S=3/2 we obtain the exponent \ensuremath{\gamma}=1.45\ifmmode\pm\else\textpm\fi{}0.05 for the divergence of the staggered susceptibility.
The thermodynamic properties and critical behavior of a Heisenberg ferromagnet in two dimensions are studied using the Handscomb quantum Monte Carlo method. The spins (with quantum number S=12) are arranged on a square lattice and have a long-range interaction with a power-law decrease, J(r)∝r−p, with distance r. The parameter p is taken to be in the range 2<p⩽6. The system has an ordering transition at a nonzero temperature Tc when 2<p<4. The dependence of Tc on p and the critical behavior of the susceptibility are investigated. Properties of spin correlation functions in different regimes (of p and T) are discussed. The results are compared with previous approximate analytic quantum theories.
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 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.
In this communication, the histogram-reweighting technique is extended to quantum Monte Carlo studies of spin systems. The proposed method is applied to obtain equilibrium critical parameters of the quantum S=1/2S=1/2S=1/2 Heisenberg ferromagnet on simple cubic lattices. Our estimates are kBTc/J=1.67764(2),{k}_{B}{T}_{c}/J=1.67764(2),kB​Tc​/J=1.67764(2), U∗=0.4504(1),{U}^{*}=0.4504(1),U∗=0.4504(1), $\ensuremath{\gamma}=1.397(20),$ $\ensuremath{\beta}=0.360(6),$ and $\ensuremath{\nu}=0.705(10)$, which are in perfect agreement with the best field-theory and high-temperature-series results. The method can readily be applied to general quantum-spin models, standing as an important tool for the study of quantum phase transitions.