Journal of Physics A Mathematical and General

Quantum Monte Carlo simulation of the one-dimensional spin S xxz model. I. The method

M Marcu, Andreas Wiesler

Published Sep 11, 1985• 13 citations

Abstract

The authors develop a quantum Monte Carlo procedure for the xxz spin chain. The quantum model in one dimension is mapped by a path integral method based on the Trotter formula into the infinite anisotropy limit of a classical two-dimensional spin system. The expectation values of the energy density, susceptibility and static s z -s z structure function are computed in the classical model for finite but increasing anisotropy. The computation method is a Monte Carlo simulation of the classical system. The results are systematically extrapolated to the infinite anisotropy limit.

DOI: 10.1088/0305-4470/18/13/024

Related papers

Investigation of One- and Two-Dimensional Quantum Spin Systems by Monte Carlo Simulations

Luruper Chaussee

Jan 1, 20000 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 simulation of the one-dimensional spin-S xxz model. II. High precision calculations for S=1/2

M Marcu, Jürgen Müller, F -K Schmatzer

For pt.I see ibid., vol.18, p.2479, (1985). A quantum Monte Carlo method developed in the first paper of this series is used to investigate the spin-1/2 isotropic ferromagnet and antiferromagnet. Critical indices are computed in a high precision simulation. The authors obtain alpha =-0.261+or-0.013, gamma =1.552+or-0.008 for the ferromagnet, and alpha =-1.202+or-0.009, gamma =1.132+or-0.012 for the antiferromagnet.

Journal of Physics A Mathematical and GeneralNov 11, 198524 citations

Monte Carlo studies of one-dimensional quantum Heisenberg and XY models

John J. Cullen, David Paul Landau

Physical review. B, Condensed matterJan 1, 1983118 citations

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

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

Physical review. B, Condensed matterJul 15, 199315 citations

Quantum annealing by the path-integral Monte Carlo method: The two-dimensional random Ising model

Roman Martoňák, Giuseppe E. Santoro, Erio Tosatti

Physical review. B, Condensed matterSep 13, 2002144 citations

Journal of Physics A Mathematical and General

Quantum Monte Carlo simulation of the one-dimensional spin S xxz model. I. The method

M Marcu, Andreas Wiesler

Published Sep 11, 1985• 13 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/18/13/024

Related papers

Investigation of One- and Two-Dimensional Quantum Spin Systems by Monte Carlo Simulations

Luruper Chaussee

A method for simulating quantum spin systems with nearest neighbour interaction is reviewed. In the framework of a path-integral approach based on the Trotter formula, the quantum spin system is mapped into the limit of a sequence of classical spin systems with one extra dimension. Numerical questions connected to this limit are discussed. The role of particle and winding numbers is clarified. An overview of the problems investigated by our group is given. In particular, two new results are presented. A simulation of the one-dimensional spin S isotropic Heisenberg antiferromagnet, S=1/2, 1, 3/2, 2, supports Haldane's conjecture that the mass gap is zero for half integer S and nonzero for integer S. For the two-dimensional spin 1/2 xy model we discuss evidence for a Kosterlitz-Thouless transition. In particular, we show that the helicity modulus can be computed directly from our simulaton.A method for simulating quantum spin systems with nearest neighbour interaction is reviewed. In the framework of a path-integral approach based on the Trotter formula, the quantum spin system is mapped into the limit of a sequence of classical spin systems with one extra dimension. Numerical questions connected to this limit are discussed. The role of particle and winding numbers is clarified. An overview of the problems investigated by our group is given. In particular, two new results are presented. A simulation of the one-dimensional spin S isotropic Heisenberg antiferromagnet, S=1/2, 1, 3/2, 2, supports Haldane's conjecture that the mass gap is zero for half integer S and nonzero for integer S. For the two-dimensional spin 1/2 xy model we discuss evidence for a Kosterlitz-Thouless transition. In particular, we show that the helicity modulus can be computed directly from our simulaton.

Jan 1, 20000 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 simulation of the one-dimensional spin-S xxz model. II. High precision calculations for S=1/2

M Marcu, Jürgen Müller, F -K Schmatzer

Journal of Physics A Mathematical and GeneralNov 11, 198524 citations

Monte Carlo studies of one-dimensional quantum Heisenberg and XY models

John J. Cullen, David Paul Landau

The Suzuki-Trotter transformation has been used to transform $N$ -spin $S=\frac{1}{2}$ Heisenberg and $\mathrm{XY}$ chains into two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for $\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125$ and $m\ensuremath{\le}10$. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence $\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13$. Difficulties in the method, particularly due to singular bonding, are discussed.The Suzuki-Trotter transformation has been used to transform $N$-spin $S=\frac{1}{2}$ Heisenberg and $\mathrm{XY}$ chains into two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for $\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125$ and $m\ensuremath{\le}10$. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence $\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13$. Difficulties in the method, particularly due to singular bonding, are discussed.

Physical review. B, Condensed matterJan 1, 1983118 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}}}$${\mathit{T}}_{\mathrm{\ensuremath{\rho}}}^{\mathit{K}}$-1, with \ensuremath{\sigma}=1/2.The one-dimensional Hubbard model is studied at and close to half-filling using a generalization of Handscomb's quantum Monte Carlo method. Results for spin-correlation functions and susceptibilities are presented for systems of up to 128 sites. The spin-correlation function at low temperature is well described by a recently introduced formula relating the correlation function of a finite periodic system to the corresponding T=0 correlation function of the infinite system. For the T\ensuremath{\rightarrow}0 divergence of the q=2${\mathit{k}}_{\mathit{F}}$ spin susceptibility, our results are consistent with the form \ensuremath{\chi}(2${\mathit{k}}_{\mathit{F}}$)\ensuremath{\sim}\ensuremath{\Vert}ln(T)${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\sigma}}}$${\mathit{T}}_{\mathrm{\ensuremath{\rho}}}^{\mathit{K}}$-1, with \ensuremath{\sigma}=1/2.

Physical review. B, Condensed matterJul 15, 199315 citations

Quantum annealing by the path-integral Monte Carlo method: The two-dimensional random Ising model

Roman Martoňák, Giuseppe E. Santoro, Erio Tosatti

Quantum annealing was recently found experimentally in a disordered spin- $\frac{1}{2}$ magnet to be more effective than its classical, thermal counterpart. We use the random two-dimensional Ising model as a test example and perform on it both classical and quantum (path-integral) Monte Carlo annealing. A systematic study of the dependence of the final residual energy on the annealing Monte Carlo time quantitatively demonstrates the superiority of quantum relative to classical annealing in this system. In order to determine the parameter regime for optimal efficiency of the quantum annealing procedure we explore a range of values of Trotter slice number P and temperature T. This identifies two different regimes of freezing with respect to efficiency of the algorithm, and leads to useful guidelines for the optimal choice of quantum annealing parameters.Quantum annealing was recently found experimentally in a disordered spin-$\frac{1}{2}$ magnet to be more effective than its classical, thermal counterpart. We use the random two-dimensional Ising model as a test example and perform on it both classical and quantum (path-integral) Monte Carlo annealing. A systematic study of the dependence of the final residual energy on the annealing Monte Carlo time quantitatively demonstrates the superiority of quantum relative to classical annealing in this system. In order to determine the parameter regime for optimal efficiency of the quantum annealing procedure we explore a range of values of Trotter slice number P and temperature T. This identifies two different regimes of freezing with respect to efficiency of the algorithm, and leads to useful guidelines for the optimal choice of quantum annealing parameters.

Physical review. B, Condensed matterSep 13, 2002144 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.

The Suzuki-Trotter transformation has been used to transform

N

-spin

S=\frac{1}{2}

Heisenberg and

\mathrm{XY}

chains into two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for $\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125$ and $m\ensuremath{\le}10$. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence $\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13$. Difficulties in the method, particularly due to singular bonding, are discussed.

{\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.

Quantum annealing was recently found experimentally in a disordered spin-

\frac{1}{2}

magnet to be more effective than its classical, thermal counterpart. We use the random two-dimensional Ising model as a test example and perform on it both classical and quantum (path-integral) Monte Carlo annealing. A systematic study of the dependence of the final residual energy on the annealing Monte Carlo time quantitatively demonstrates the superiority of quantum relative to classical annealing in this system. In order to determine the parameter regime for optimal efficiency of the quantum annealing procedure we explore a range of values of Trotter slice number P and temperature T. This identifies two different regimes of freezing with respect to efficiency of the algorithm, and leads to useful guidelines for the optimal choice of quantum annealing parameters.