arXiv:cond-mat/9911047 · Physical Review Letters

Cluster Algorithms for General- S Quantum Spin Systems

Synge Todo, Kiyoshi Katō

Published Jul 6, 2001• 270 citations

Abstract

We present a general strategy to extend quantum cluster algorithms for $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ spin systems, such as the loop algorithm, to those with an arbitrary size of spins. The partition function of a high- $S$ spin system is generally represented by the path integral of a $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ model with special boundary conditions in the imaginary-time direction. We introduce additional graphs for the boundary part and give the labeling probability explicitly, which completes the algorithm together with an existing $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ algorithm. As a demonstration, we simulate the integer-spin antiferromagnetic Heisenberg chains. The magnitude of the first excitation gap is estimated to be 0.41048(6), 0.08917(4), and 0.01002(3) for $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$, 2, and 3, respectively.

DOI: 10.1103/physrevlett.87.047203

Related papers

Loop Algorithms for Monte Carlo Simulations of Quantum Spin Systems

Naoki Kawashima, James E. Gubernatis

Physical Review LettersAug 29, 199459 citations

Continuous-Time Loop Algorithm for General-SSpin Systems

Synge Todo, K. Katō

A loop cluster algorithm, which can be applied to quantum spin systems with arbitrary size of spins, is proposed. We also present a generalized Swendsen-Wang algorithm for general-S classical Ising models.

Progress of Theoretical Physics SupplementJan 1, 20009 citations

Extended cluster algorithm in quantum simulations

Hiromi Otsuka

Physical review. B, Condensed matterJun 20, 20011 citation

Efficient Algorithms for Weakly-Interacting Quantum Spin Systems

Ryan L. Mann, Gabriel Waite

We establish efficient algorithms for weakly-interacting quantum spin systems at arbitrary temperature. In particular, we obtain a fully polynomial-time approximation scheme for the partition function and an efficient approximate sampling scheme for the thermal distribution over a classical spin space. Our approach is based on the cluster expansion method and a standard reduction from approximate sampling to approximate counting.

quant-pharXiv:2601.21140Physical Review AJan 30, 20260 citations

Exact S matrices for integrable quantum spin chains

Luca Mezincescu, Rafael I. Nepomechie

arXiv:hep-th/9501109Jan 24, 19952 citations

Exact numerical results on finite one- and two-dimensional Heisenberg systems

C. S. Jain, Kanwar Krishan, Chanchal K. Majumdar +1 more

Physical review. B, Solid stateDec 1, 197511 citations

arXiv:cond-mat/9911047 · Physical Review Letters

Cluster Algorithms for General- S Quantum Spin Systems

Synge Todo, Kiyoshi Katō

Published Jul 6, 2001• 270 citations

arXiv HTML PDF

Abstract

We present a general strategy to extend quantum cluster algorithms for

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

spin systems, such as the loop algorithm, to those with an arbitrary size of spins. The partition function of a high-

S

spin system is generally represented by the path integral of a

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

model with special boundary conditions in the imaginary-time direction. We introduce additional graphs for the boundary part and give the labeling probability explicitly, which completes the algorithm together with an existing

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

algorithm. As a demonstration, we simulate the integer-spin antiferromagnetic Heisenberg chains. The magnitude of the first excitation gap is estimated to be 0.41048(6), 0.08917(4), and 0.01002(3) for

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1

, 2, and 3, respectively.

DOI: 10.1103/physrevlett.87.047203

Related papers

Loop Algorithms for Monte Carlo Simulations of Quantum Spin Systems

Naoki Kawashima, James E. Gubernatis

We present a new algorithm for world line Monte Carlo simulations of quantum spin systems. The algorithm consists of a stochastic pairing and a stochastic flipping procedure, and the formula for the pairing probability is explicitly given for Heisenberg models. As an illustration of the algorithm, we calculated autocorrelation times for simulations of the $S=1$ antiferromagnetic Heisenberg chain. The rapid increase in autocorrelation times which is seen in the conventional world line method as the imaginary-time spacing decreases is eliminated by the present algorithm.We present a new algorithm for world line Monte Carlo simulations of quantum spin systems. The algorithm consists of a stochastic pairing and a stochastic flipping procedure, and the formula for the pairing probability is explicitly given for Heisenberg models. As an illustration of the algorithm, we calculated autocorrelation times for simulations of the $S=1$ antiferromagnetic Heisenberg chain. The rapid increase in autocorrelation times which is seen in the conventional world line method as the imaginary-time spacing decreases is eliminated by the present algorithm.

Physical Review LettersAug 29, 199459 citations

Continuous-Time Loop Algorithm for General-SSpin Systems

Synge Todo, K. Katō

Progress of Theoretical Physics SupplementJan 1, 20009 citations

Extended cluster algorithm in quantum simulations

Hiromi Otsuka

We propose an extension of the quantum cluster algorithm by a combined use of the Fortuin-Kasteleyn mapping and the Hubbard-Stratonovich transformation. To describe the idea, we consider the $S=\frac{1}{2}$ $\mathrm{XXZ}$ chain model and express the partition function as the sum in an extended configuration space of spins, graphs, and fields. Then it is clarified that this algorithm possesses a computationally tractable continuous-time limit and maintains virtues of the quantum cluster algorithm. Numerical simulations are performed and its applicability is demonstrated. Further, we discuss potential gains in our algorithm.We propose an extension of the quantum cluster algorithm by a combined use of the Fortuin-Kasteleyn mapping and the Hubbard-Stratonovich transformation. To describe the idea, we consider the $S=\frac{1}{2}$ $\mathrm{XXZ}$ chain model and express the partition function as the sum in an extended configuration space of spins, graphs, and fields. Then it is clarified that this algorithm possesses a computationally tractable continuous-time limit and maintains virtues of the quantum cluster algorithm. Numerical simulations are performed and its applicability is demonstrated. Further, we discuss potential gains in our algorithm.

Physical review. B, Condensed matterJun 20, 20011 citation

Efficient Algorithms for Weakly-Interacting Quantum Spin Systems

Ryan L. Mann, Gabriel Waite

quant-pharXiv:2601.21140Physical Review AJan 30, 20260 citations

Exact S matrices for integrable quantum spin chains

Luca Mezincescu, Rafael I. Nepomechie

We begin with a review of the antiferromagnetic spin 1/2 Heisenberg chain. In particular, we show that the model has particle-like excitations with spin 1/2, and we compute the exact bulk S matrix. We then review our recent work which generalizes these results. We first consider an integrable alternating spin 1/2 - spin 1 chain. In addition to having excitations with spin 1/2, this model also has excitations with spin 0. We compute the bulk S matrix, which has some unusual features. We then consider the open antiferromagnetic spin 1/2 Heisenberg chain with boundary magnetic fields. We give a direct calculation of the boundary S matrix. (Talk presented at the conference on Statistical Mechanics and Quantum Field Theory at USC, 16 - 21 May 1994)We begin with a review of the antiferromagnetic spin 1/2 Heisenberg chain. In particular, we show that the model has particle-like excitations with spin 1/2, and we compute the exact bulk S matrix. We then review our recent work which generalizes these results. We first consider an integrable alternating spin 1/2 - spin 1 chain. In addition to having excitations with spin 1/2, this model also has excitations with spin 0. We compute the bulk S matrix, which has some unusual features. We then consider the open antiferromagnetic spin 1/2 Heisenberg chain with boundary magnetic fields. We give a direct calculation of the boundary S matrix. (Talk presented at the conference on Statistical Mechanics and Quantum Field Theory at USC, 16 - 21 May 1994)

arXiv:hep-th/9501109Jan 24, 19952 citations

Exact numerical results on finite one- and two-dimensional Heisenberg systems

C. S. Jain, Kanwar Krishan, Chanchal K. Majumdar +1 more

Results of exact numerical work on the spectra and thermodynamic properties are presented for finite clusters in one and two dimensions, for the isotropic Heisenberg Hamiltonian and the Heisenberg Hamiltonian with longitudinal anisotropy. Eigenvalues of the isotropic Heisenberg loops (that is, those with periodic boundary conditions) of $N\ensuremath{\le}12, 8, 6, 5$, 5 particles, each of spin \textonehalf{}, 1, $\frac{3}{2}$ , 2 and $\frac{5}{2}$ , respectively, are found. It is shown that the spin- $S$ linear chain ($N\ensuremath{\rightarrow}\ensuremath{\infty}$) will have ground-state energy per spin $\frac{{\ensuremath{\epsilon}}_{\ensuremath{\infty}}^{s}}{2J}=\ensuremath{-}1.404\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{-}2.85\ifmmode\pm\else\textpm\fi{}0.03, \ensuremath{-}4.7\ifmmode\pm\else\textpm\fi{}0.4 \mathrm{and} \ensuremath{-}7.1\ifmmode\pm\else\textpm\fi{}0.5$, for $S=1, \frac{3}{2}, 2, \mathrm{and} \frac{5}{2}$ , respectively. For antiferromagnetic interaction, the lowest excited states of the loops with even number of particles have spin 1 and the familiar double sine spectrum; while for the spin-\textonehalf{} case the eigenvalues are very different from the result of the spin-wave theory, they approach this limit as the spin increases, and for spin $\frac{5}{2}$ , the infinite-spin results of the spin-wave theory should be quite accurate. For ferromagnetic loops, there exist spin-wave bound states for higher spins just like those familiar in the spin-\textonehalf{} linear chain. Effective magnetic moments for these loops are tabulated. We also present the moments for the $N=8$ spin-\textonehalf{} chain (with open ends), and find the difference to be small. Results on the 3\ifmmode\times\else\texttimes\fi{}4 two-dimensional Heisenberg antiferromagnet are presented and compared with experimental data. Finally, we show some examples of the violation of the nondegeneracy rule, similar to those found by Heilmann and Lieb; levels belonging to different representations of a symmetry group remain permanently degenerate when a parameter is varied.Results of exact numerical work on the spectra and thermodynamic properties are presented for finite clusters in one and two dimensions, for the isotropic Heisenberg Hamiltonian and the Heisenberg Hamiltonian with longitudinal anisotropy. Eigenvalues of the isotropic Heisenberg loops (that is, those with periodic boundary conditions) of $N\ensuremath{\le}12, 8, 6, 5$, 5 particles, each of spin \textonehalf{}, 1, $\frac{3}{2}$, 2 and $\frac{5}{2}$, respectively, are found. It is shown that the spin-$S$ linear chain ($N\ensuremath{\rightarrow}\ensuremath{\infty}$) will have ground-state energy per spin $\frac{{\ensuremath{\epsilon}}_{\ensuremath{\infty}}^{s}}{2J}=\ensuremath{-}1.404\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{-}2.85\ifmmode\pm\else\textpm\fi{}0.03, \ensuremath{-}4.7\ifmmode\pm\else\textpm\fi{}0.4 \mathrm{and} \ensuremath{-}7.1\ifmmode\pm\else\textpm\fi{}0.5$, for $S=1, \frac{3}{2}, 2, \mathrm{and} \frac{5}{2}$, respectively. For antiferromagnetic interaction, the lowest excited states of the loops with even number of particles have spin 1 and the familiar double sine spectrum; while for the spin-\textonehalf{} case the eigenvalues are very different from the result of the spin-wave theory, they approach this limit as the spin increases, and for spin $\frac{5}{2}$, the infinite-spin results of the spin-wave theory should be quite accurate. For ferromagnetic loops, there exist spin-wave bound states for higher spins just like those familiar in the spin-\textonehalf{} linear chain. Effective magnetic moments for these loops are tabulated. We also present the moments for the $N=8$ spin-\textonehalf{} chain (with open ends), and find the difference to be small. Results on the 3\ifmmode\times\else\texttimes\fi{}4 two-dimensional Heisenberg antiferromagnet are presented and compared with experimental data. Finally, we show some examples of the violation of the nondegeneracy rule, similar to those found by Heilmann and Lieb; levels belonging to different representations of a symmetry group remain permanently degenerate when a parameter is varied.

Physical review. B, Solid stateDec 1, 197511 citations

We present a general strategy to extend quantum cluster algorithms for

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

spin systems, such as the loop algorithm, to those with an arbitrary size of spins. The partition function of a high-

S

spin system is generally represented by the path integral of a

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1

, 2, and 3, respectively.

S=1

antiferromagnetic Heisenberg chain. The rapid increase in autocorrelation times which is seen in the conventional world line method as the imaginary-time spacing decreases is eliminated by the present algorithm.

We propose an extension of the quantum cluster algorithm by a combined use of the Fortuin-Kasteleyn mapping and the Hubbard-Stratonovich transformation. To describe the idea, we consider the

S=\frac{1}{2}

\mathrm{XXZ}

chain model and express the partition function as the sum in an extended configuration space of spins, graphs, and fields. Then it is clarified that this algorithm possesses a computationally tractable continuous-time limit and maintains virtues of the quantum cluster algorithm. Numerical simulations are performed and its applicability is demonstrated. Further, we discuss potential gains in our algorithm.

\frac{3}{2}

, 2 and

\frac{5}{2}

, respectively, are found. It is shown that the spin-

S

linear chain ($N\ensuremath{\rightarrow}\ensuremath{\infty}$) will have ground-state energy per spin $\frac{{\ensuremath{\epsilon}}_{\ensuremath{\infty}}^{s}}{2J}=\ensuremath{-}1.404\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{-}2.85\ifmmode\pm\else\textpm\fi{}0.03, \ensuremath{-}4.7\ifmmode\pm\else\textpm\fi{}0.4 \mathrm{and} \ensuremath{-}7.1\ifmmode\pm\else\textpm\fi{}0.5$, for

S=1, \frac{3}{2}, 2, \mathrm{and} \frac{5}{2}

, respectively. For antiferromagnetic interaction, the lowest excited states of the loops with even number of particles have spin 1 and the familiar double sine spectrum; while for the spin-\textonehalf{} case the eigenvalues are very different from the result of the spin-wave theory, they approach this limit as the spin increases, and for spin

\frac{5}{2}

, the infinite-spin results of the spin-wave theory should be quite accurate. For ferromagnetic loops, there exist spin-wave bound states for higher spins just like those familiar in the spin-\textonehalf{} linear chain. Effective magnetic moments for these loops are tabulated. We also present the moments for the

N=8

spin-\textonehalf{} chain (with open ends), and find the difference to be small. Results on the 3\ifmmode\times\else\texttimes\fi{}4 two-dimensional Heisenberg antiferromagnet are presented and compared with experimental data. Finally, we show some examples of the violation of the nondegeneracy rule, similar to those found by Heilmann and Lieb; levels belonging to different representations of a symmetry group remain permanently degenerate when a parameter is varied.