Journal of Physics Condensed Matter

Monte Carlo computation of the free energy in quantum two-dimensional Heisenberg ferromagnets using the expanded-ensemble method

Tatyana V. Kuznetsova, P. N. Vorontsov‐Velyaminov

Published Feb 8, 1993• 11 citations

Abstract

The expanded-ensemble method, initially formulated for classical systems, is combined with the Handscomb quantum Monte Carlo approach to develop a new method for the free-energy MC calculation of quantum systems. It was applied to a quantum 2D Heisenberg ferromagnetic system and proved effective. Comparisons were performed between the Takahashi spin-wave approximation for free energy and the authors' numerical results.

DOI: 10.1088/0953-8984/5/6/008

Related papers

A New Monte Carlo Approach to Quantum Spin Systems. II

Shinji Kadowaki, Akira Ueda

Progress of Theoretical PhysicsSep 1, 19892 citations

Quantum-Statistical Monte Carlo Method for Heisenberg Spins

J. W. Lyklema

An exact Monte Carlo method for calculating thermodynamic properties of quantum spin systems is described. Results for one-dimensional ferromagnetic and antiferromagnetic systems and for three-dimensional ferromagnetic systems show that the method can be used to study quantum spin systems as extensively as classical spin systems are studied with conventional Monte Carlo methods.

Physical Review LettersJul 12, 198267 citations

Quantum Monte Carlo simulation method for spin systems

Anders W. Sandvik, J. Kurkijärvi

Physical review. B, Condensed matterMar 1, 1991528 citations

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

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

Physical review. B, Condensed matterMay 1, 199110 citations

Finite- and long-range spin-wave free-energy asymptotics for the two-dimensional quantum Heisenberg ferromagnet

Andreas Klippel

arXiv:2610.10736Oct 7, 20260 citations

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

arXiv:cond-mat/0004176Physical review. B, Condensed matterOct 1, 200013 citations

Journal of Physics Condensed Matter

Monte Carlo computation of the free energy in quantum two-dimensional Heisenberg ferromagnets using the expanded-ensemble method

Tatyana V. Kuznetsova, P. N. Vorontsov‐Velyaminov

Published Feb 8, 1993• 11 citations

Publisher

Abstract

DOI: 10.1088/0953-8984/5/6/008

Related papers

A New Monte Carlo Approach to Quantum Spin Systems. II

Shinji Kadowaki, Akira Ueda

A simple Monte Carlo method1) which was shown to be applicable to finite spin-systems of rather large spin size ragardless of dimensionality of the system is applied to the spin-1/2 isotropic Heisenberg ferromagnetic system on the square lattice of spin size 10 × 10, 20 × 20 and 30 × 30. Rather accurate values are obtained for thermodynamic quantities for temperature down to k B T / J =0.01. The accuracy and efficiency of the MC method are discussed in detail in comparison with the other works.A simple Monte Carlo method1) which was shown to be applicable to finite spin-systems of rather large spin size ragardless of dimensionality of the system is applied to the spin-1/2 isotropic Heisenberg ferromagnetic system on the square lattice of spin size 10 × 10, 20 × 20 and 30 × 30. Rather accurate values are obtained for thermodynamic quantities for temperature down to k B T / J =0.01. The accuracy and efficiency of the MC method are discussed in detail in comparison with the other works.

Progress of Theoretical PhysicsSep 1, 19892 citations

Quantum-Statistical Monte Carlo Method for Heisenberg Spins

J. W. Lyklema

Physical Review LettersJul 12, 198267 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.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 matterMar 1, 1991528 citations

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

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

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.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.

Physical review. B, Condensed matterMay 1, 199110 citations

Finite- and long-range spin-wave free-energy asymptotics for the two-dimensional quantum Heisenberg ferromagnet

Andreas Klippel

For the two-dimensional quantum Heisenberg ferromagnet with asymptotically radial power-law exchange, we prove the leading low-temperature free energy at every fixed spin, uniformly on the weak-field scale. We obtain fractional and logarithmically corrected laws associated with the dispersions predicted in [16, 17]. Quantum Monte Carlo simulations support the associated anomalous magnon dispersions for spin one-half [20]. For kernels with finite second moment, we also extend the nearest-neighbour theorem [14], generalizing Takahashi's quadratic law [21] and its weak-field dependence [11].For the two-dimensional quantum Heisenberg ferromagnet with asymptotically radial power-law exchange, we prove the leading low-temperature free energy at every fixed spin, uniformly on the weak-field scale. We obtain fractional and logarithmically corrected laws associated with the dispersions predicted in [16, 17]. Quantum Monte Carlo simulations support the associated anomalous magnon dispersions for spin one-half [20]. For kernels with finite second moment, we also extend the nearest-neighbour theorem [14], generalizing Takahashi's quadratic law [21] and its weak-field dependence [11].

arXiv:2610.10736Oct 7, 20260 citations

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.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/0004176Physical review. B, Condensed matterOct 1, 200013 citations

{\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.

{\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.

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.