© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical review. B, Condensed matter

Quantum Monte Carlo simulation method for spin systems

Anders W. Sandvik, J. Kurkijärvi

Published Mar 1, 1991• 528 citations

Abstract

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.

DOI: 10.1103/physrevb.43.5950

Related papers

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.

Physical review. B, Condensed matter•May 1, 1991•10 citations

A generalization of Handscomb's quantum Monte Carlo scheme-application to the 1D Hubbard model

Anders W. Sandvik

A recently introduced generalization of Handscomb's quantum Monte Carlo scheme (1962) is further developed. Expressions for expectation values of various observables are studied in detail. A more efficient algorithm for importance sampling in a space of state vectors and operator strings is constructed, using the 1D Hubbard model as an illustrative example. As a test of the method, 32-site rings at band fillings 1/4, 3/8 and 1/2 are studied at a low temperature. Results for spin- and charge-density structure factors and static susceptibilities are presented.

Journal of Physics A Mathematical and General•Jul 7, 1992•274 citations

Quantum Monte Carlo study of spin correlations in the one-dimensional Hubbard model

Anders W. Sandvik, Douglas J. Scalapino, C. Singh

The one-dimensional Hubbard model is studied at and close to half-filling using a generalization of Handscomb's quantum Monte Carlo method. Results for spin-correlation functions and susceptibilities are presented for systems of up to 128 sites. The spin-correlation function at low temperature is well described by a recently introduced formula relating the correlation function of a finite periodic system to the corresponding T=0 correlation function of the infinite system. For the T\ensuremath{\rightarrow}0 divergence of the q=2${\mathit{k}}_{\mathit{F}}$ spin susceptibility, our results are consistent with the form \ensuremath{\chi}(2${\mathit{k}}_{\mathit{F}}$)\ensuremath{\sim}\ensuremath{\Vert}ln(T)${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\sigma}}}$${\mathit{T}}_{\mathrm{\ensuremath{\rho}}}^{\mathit{K}}$-1, with \ensuremath{\sigma}=1/2.

Physical review. B, Condensed matter•Jul 15, 1993•15 citations

Spontaneous staggered magnetization in antiferromagnetic Heisenberg-Ising chains

Juan Pablo Neirotti, Mário J. de Oliveira

We calculate numerically the spontaneous staggered magnetization and susceptibility of the ground state of quantum antiferromagnetic Heisenberg-Ising chains. We use a zero-temperature Monte Carlo method and consider the cases of spin 1 and 2. As a function of the anisotropy parameter, the magnetization vanishes continuously at a critical point whose critical exponents for both cases are found to be in agreement with those of the two-dimensional Ising model.

Physical review. B, Condensed matter•Sep 1, 1996•4 citations

Density-matrix quantum Monte Carlo method

Nick S. Blunt, Thomas W. Rogers, James S. Spencer +1 more

We present a quantum Monte Carlo method capable of sampling the full density matrix of a many-particle system at finite temperature. This allows arbitrary reduced density matrix elements and expectation values of complicated nonlocal observables to be evaluated easily. The method resembles full configuration interaction quantum Monte Carlo but works in the space of many-particle operators instead of the space of many-particle wave functions. One simulation provides the density matrix at all temperatures simultaneously, from $T=\ensuremath{\infty}$ to $T=0$, allowing the temperature dependence of expectation values to be studied. The direct sampling of the density matrix also allows the calculation of some previously inaccessible entanglement measures. We explain the theory underlying the method, describe the algorithm, and introduce an importance-sampling procedure to improve the stochastic efficiency. To demonstrate the potential of our approach, the energy and staggered magnetization of the isotropic antiferromagnetic Heisenberg model on small lattices, the concurrence of one-dimensional spin rings, and the Renyi ${S}_{2}$ entanglement entropy of various sublattices of the $6\ifmmode\times\else\texttimes\fi{}6$ Heisenberg model are calculated. The nature of the sign problem in the method is also investigated.

arXiv:1303.5007•Physical Review B•Jun 18, 2014•108 citations

Long-range correlations in the two-dimensional spin-(1/2 antiferromagnetic Heisenberg model: A quantum Monte Carlo study

Efstratios Manousakis, Román Salvador

Using an extended version of Handscomb's Monte Carlo method we study the spin-(1/2 quantum antiferromagnetic Heisenberg model on square lattices of various sizes. The calculated correlation length associated with antiferromagnetic order grows very rapidly as the temperature is lowered suggesting an essential singularity. Our results are consistent with those recently observed by neutron scattering done on ${\mathrm{Lo}}_{2}$${\mathrm{CuO}}_{4}$.

Physical Review Letters•Feb 29, 1988•75 citations

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.

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.

A recently introduced generalization of Handscomb's quantum Monte Carlo scheme (1962) is further developed. Expressions for expectation values of various observables are studied in detail. A more efficient algorithm for importance sampling in a space of state vectors and operator strings is constructed, using the 1D Hubbard model as an illustrative example. As a test of the method, 32-site rings at band fillings 1/4, 3/8 and 1/2 are studied at a low temperature. Results for spin- and charge-density structure factors and static susceptibilities are presented.

The one-dimensional Hubbard model is studied at and close to half-filling using a generalization of Handscomb's quantum Monte Carlo method. Results for spin-correlation functions and susceptibilities are presented for systems of up to 128 sites. The spin-correlation function at low temperature is well described by a recently introduced formula relating the correlation function of a finite periodic system to the corresponding T=0 correlation function of the infinite system. For the T\ensuremath{\rightarrow}0 divergence of the q=2

{\mathit{k}}_{\mathit{F}}

spin susceptibility, our results are consistent with the form \ensuremath{\chi}(2

{\mathit{k}}_{\mathit{F}}

)\ensuremath{\sim}\ensuremath{\Vert}ln(T)${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\sigma}}}$${\mathit{T}}_{\mathrm{\ensuremath{\rho}}}^{\mathit{K}}$-1, with \ensuremath{\sigma}=1/2.

We present a quantum Monte Carlo method capable of sampling the full density matrix of a many-particle system at finite temperature. This allows arbitrary reduced density matrix elements and expectation values of complicated nonlocal observables to be evaluated easily. The method resembles full configuration interaction quantum Monte Carlo but works in the space of many-particle operators instead of the space of many-particle wave functions. One simulation provides the density matrix at all temperatures simultaneously, from $T=\ensuremath{\infty}$ to

T=0

, allowing the temperature dependence of expectation values to be studied. The direct sampling of the density matrix also allows the calculation of some previously inaccessible entanglement measures. We explain the theory underlying the method, describe the algorithm, and introduce an importance-sampling procedure to improve the stochastic efficiency. To demonstrate the potential of our approach, the energy and staggered magnetization of the isotropic antiferromagnetic Heisenberg model on small lattices, the concurrence of one-dimensional spin rings, and the Renyi

{S}_{2}

entanglement entropy of various sublattices of the $6\ifmmode\times\else\texttimes\fi{}6$ Heisenberg model are calculated. The nature of the sign problem in the method is also investigated.

Using an extended version of Handscomb's Monte Carlo method we study the spin-(1/2 quantum antiferromagnetic Heisenberg model on square lattices of various sizes. The calculated correlation length associated with antiferromagnetic order grows very rapidly as the temperature is lowered suggesting an essential singularity. Our results are consistent with those recently observed by neutron scattering done on

{\mathrm{Lo}}_{2}

{\mathrm{CuO}}_{4}

.