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

arXiv:0809.2969 · Physical Review Letters

Efficient Monte Carlo Algorithm in Quasi-One-Dimensional Ising Spin Systems

Tota Nakamura

Published Nov 20, 2008• 20 citations
arXivHTMLPDF

Abstract

We develop an efficient Monte Carlo algorithm, which accelerates slow Monte Carlo dynamics in quasi-one-dimensional Ising spin systems. The loop algorithm of the quantum Monte Carlo method is applied to the classical spin models with highly anisotropic exchange interactions. Both correlation time and real CPU time are reduced drastically. The algorithm is demonstrated in the layered triangular-lattice antiferromagnetic Ising model. We have obtained the relation between the transition temperature and the exchange interaction parameters, which modifies the result of the chain-mean-field theory.

DOI
10.1103/physrevlett.101.210602

Related papers

Auxiliary-field Monte Carlo method for frustrated spin systems

Ning Cai, Yuan 源 Gao 高, Wei Li +1 more

Abstract We extend a semiclassical numerical method, bosonic auxiliary-field Monte Carlo, to quantum spin systems. This method breaks the lattice into clusters, solves each cluster precisely and couples them with classical auxiliary fields through classical Monte Carlo simulation. We test the method with antiferromagnetic spin models in one-dimensional chains, square lattices and triangular lattices, and obtain reasonable results at finite temperatures. This algorithm builds a bridge between classical Monte Carlo method and quantum methods. The algorithm can be improved with either progress in classical Monte Carlo sampling or the development of quantum solvers, and can also be further applied to systems with different lattices or interactions.

Chinese Physics B•Jan 1, 2025•0 citations

Investigation of Correlation Functions in Low-Dimensional Quantum Antiferromagnets by Quantum Monte Carlo Methods

Carsten H. Aits

In this thesis the quantum Monte Carlo loop algorithm is applied to study the properties of correlation functions in low-dimensional quantum antiferromagnets. Two different types of lattice models are investigated. The first system is the quantum critical XXZ chain. The second class of models emerges when coupling the square lattice Heisenberg model to phonons. In case of the quantum critical XXZ chain, the loop algorithm is used to compute the two-point spin and dimer correlation functions at low temperatures. Starting from the well-known asymptotic expressions from conformal field theory for the decay of the correlations, the crossover from the finite-temperature to the ground state behavior is investigated. In addtion, the Monte Carlo data are used to extract numerical estimates for the correlation amplitudes. The deviations from asymptotic scaling at low finite temperatures and intermediate distances are examined. The results are also used to study the algebraic and logarithmic corrections in the ground state. The method is further applied to compute the low-temperature spin correlations of the square lattice Heisenberg model coupled to dispersionless Einstein phonons. Also, statically dimerized Heisenberg models in two dimensions are investigated. Here the quantum Monte Carlo technique is used to determine the specific pattern of dimerized bonds which has the largest gain in the magnetic energy. A direct comparison of ground state energies is only possible for a small number of dimerized systems. For this reason an alternative approach is developed, which allows to study a significantly larger class of models. Starting from a Landau expansion of the free energy, this amounts to a computation of specific structure factors of the dimer correlations of the uniform Heisenberg model. As a consequence, a Landau theory for the two-dimensional counterpart of the spin-Peierls transition is established. This adiabatic approach also applies to the Heisenberg model coupled to dispersive phonons. In this case the data for the dimer correlations are used to study the impact of the spin-phonon coupling on the phonon dispersion relations. In particular, the softening of phonon modes is investigated, which is relevant for experimental studies on substances with a quasi two-dimensional magnetic structure.

•Jan 1, 2005•0 citations

Critical behavior of the classical spin-1 Ising model: a combined low-temperature series expansion and Metropolis Monte Carlo analysis

Amir Taheridehkordi, Roberto Zivieri

In this paper, we theoretically study the critical properties of the classical spin-1 Ising model using two approaches: 1) the analytical low-temperature series expansion and 2) the numerical Metropolis Monte Carlo technique. Within this analysis, we discuss the critical behavior of one-, two- and three-dimensional systems modeled by the first-neighbor spin-1 Ising model for different types of exchange interactions. The comparison of the results obtained according the Metropolis Monte Carlo simulations allows us to highlight the limits of the widely used mean-field theory approach. We also show, via a simple transformation, that for the special case where the bilinear and bicubic terms are set equal to zero in the Hamiltonian the partition function of the spin-1 Ising model can be reduced to that of the spin-1/2 Ising model with temperature dependent external field and temperature independent exchange interaction times an exponential factor depending on the other terms of the Hamiltonian and confirm this result numerically by using the Metropolis Monte Carlo simulation. Finally, we investigate the dependence of the critical temperature on the strength of long-range interactions included in the Ising Hamiltonian comparing it with that of the first-neighbor spin-1/2 Ising model.

arXiv:2007.08593•Jul 16, 2020•0 citations

Deterministic Loop Stochastic Series Expansion Algorithm for Quantum Spin Models in Magnetic Fields

Liuyun Dao, Yan-Cheng Wang, Hui Shao

The stochastic series expansion (SSE) algorithm is one of the most powerful quantum Monte Carlo methods and has been extensively applied to the study of quantum many body systems. Its efficiency is particularly enhanced with a deterministic loop update scheme in the study of the S=1/2 quantum spin systems that preserve SU(2) spin rotational symmetry. Once the symmetry is broken, such as by an external field, a directed loop method is typically required, resulting in a significant reduction in efficiency. Inspired by the SSE approach developed for the quantum Ising model, we introduce a deterministic loop SSE method that is particularly suited for antiferromagnetic systems under a staggered magnetic field. This method enables separate investigations of longitudinal and transverse modes in magnetically ordered phases arising from spontaneous symmetry breaking. We benchmark the performance of our algorithm against the standard directed loop approach applied to the antiferromagnetic Heisenberg chain and demonstrate that our method substantially reduces CPU time per Monte Carlo step, thereby can outperform the directed loop algorithm in efficiency.

arXiv:2604.04635•Apr 6, 2026•0 citations

Quasi-adiabatic quantum Monte Carlo algorithm for quantum evolution in imaginary time

Chengwei Liu, Anatoli Polkovnikov, Anders W. Sandvik

We propose a quantum Monte Carlo (QMC) algorithm for nonequilibrium dynamics in a system with a parameter varying as a function of imaginary time. The method is based on successive applications of an evolving Hamiltonian to an initial state and delivers results for a whole range of the tuning parameter in a single run, allowing for access to both static and dynamic properties of the system. This approach reduces to the standard Schr\"odinger dynamics in imaginary time for quasi-adiabatic evolutions, i.e., including the leading nonadiabatic correction to the adiabatic limit. We here demonstrate this quasi-adiabatic QMC (QAQMC) method for linear ramps of the transverse-field Ising model across its quantum-critical point in one and two dimensions. The critical behavior can be described by generalized dynamic scaling. For the two-dimensional square-lattice system, we use the method to obtain a high-precision estimate of the quantum-critical point ${(h/J)}_{c}=3.04458(7)$, where $h$ is the transverse magnetic field and $J$ is the nearest-neighbor Ising coupling. The QAQMC method can also be used to extract the Berry curvature and the metric tensor.

arXiv:1212.4815•Physical Review B•May 23, 2013•32 citations

Quantum cluster algorithm for frustrated Ising models in a transverse field

Sounak Biswas, Geet Rakala, Kedar Damle

Working within the stochastic series expansion framework, we introduce and characterize a plaquette-based quantum cluster algorithm for quantum Monte Carlo simulations of transverse field Ising models with frustrated Ising exchange interactions. As a demonstration of the capabilities of this algorithm, we show that a relatively small ferromagnetic next-nearest-neighbor coupling drives the transverse field Ising antiferromagnet on the triangular lattice from an antiferromagnetic three-sublattice ordered state at low temperature to a ferrimagnetic three-sublattice ordered state.

arXiv:1512.00931•Physical review. B./Physical review. B•Jun 1, 2016•14 citations
We develop an efficient Monte Carlo algorithm, which accelerates slow Monte Carlo dynamics in quasi-one-dimensional Ising spin systems. The loop algorithm of the quantum Monte Carlo method is applied to the classical spin models with highly anisotropic exchange interactions. Both correlation time and real CPU time are reduced drastically. The algorithm is demonstrated in the layered triangular-lattice antiferromagnetic Ising model. We have obtained the relation between the transition temperature and the exchange interaction parameters, which modifies the result of the chain-mean-field theory.
Abstract We extend a semiclassical numerical method, bosonic auxiliary-field Monte Carlo, to quantum spin systems. This method breaks the lattice into clusters, solves each cluster precisely and couples them with classical auxiliary fields through classical Monte Carlo simulation. We test the method with antiferromagnetic spin models in one-dimensional chains, square lattices and triangular lattices, and obtain reasonable results at finite temperatures. This algorithm builds a bridge between classical Monte Carlo method and quantum methods. The algorithm can be improved with either progress in classical Monte Carlo sampling or the development of quantum solvers, and can also be further applied to systems with different lattices or interactions.
In this thesis the quantum Monte Carlo loop algorithm is applied to study the properties of correlation functions in low-dimensional quantum antiferromagnets. Two different types of lattice models are investigated. The first system is the quantum critical XXZ chain. The second class of models emerges when coupling the square lattice Heisenberg model to phonons. In case of the quantum critical XXZ chain, the loop algorithm is used to compute the two-point spin and dimer correlation functions at low temperatures. Starting from the well-known asymptotic expressions from conformal field theory for the decay of the correlations, the crossover from the finite-temperature to the ground state behavior is investigated. In addtion, the Monte Carlo data are used to extract numerical estimates for the correlation amplitudes. The deviations from asymptotic scaling at low finite temperatures and intermediate distances are examined. The results are also used to study the algebraic and logarithmic corrections in the ground state. The method is further applied to compute the low-temperature spin correlations of the square lattice Heisenberg model coupled to dispersionless Einstein phonons. Also, statically dimerized Heisenberg models in two dimensions are investigated. Here the quantum Monte Carlo technique is used to determine the specific pattern of dimerized bonds which has the largest gain in the magnetic energy. A direct comparison of ground state energies is only possible for a small number of dimerized systems. For this reason an alternative approach is developed, which allows to study a significantly larger class of models. Starting from a Landau expansion of the free energy, this amounts to a computation of specific structure factors of the dimer correlations of the uniform Heisenberg model. As a consequence, a Landau theory for the two-dimensional counterpart of the spin-Peierls transition is established. This adiabatic approach also applies to the Heisenberg model coupled to dispersive phonons. In this case the data for the dimer correlations are used to study the impact of the spin-phonon coupling on the phonon dispersion relations. In particular, the softening of phonon modes is investigated, which is relevant for experimental studies on substances with a quasi two-dimensional magnetic structure.
In this paper, we theoretically study the critical properties of the classical spin-1 Ising model using two approaches: 1) the analytical low-temperature series expansion and 2) the numerical Metropolis Monte Carlo technique. Within this analysis, we discuss the critical behavior of one-, two- and three-dimensional systems modeled by the first-neighbor spin-1 Ising model for different types of exchange interactions. The comparison of the results obtained according the Metropolis Monte Carlo simulations allows us to highlight the limits of the widely used mean-field theory approach. We also show, via a simple transformation, that for the special case where the bilinear and bicubic terms are set equal to zero in the Hamiltonian the partition function of the spin-1 Ising model can be reduced to that of the spin-1/2 Ising model with temperature dependent external field and temperature independent exchange interaction times an exponential factor depending on the other terms of the Hamiltonian and confirm this result numerically by using the Metropolis Monte Carlo simulation. Finally, we investigate the dependence of the critical temperature on the strength of long-range interactions included in the Ising Hamiltonian comparing it with that of the first-neighbor spin-1/2 Ising model.
The stochastic series expansion (SSE) algorithm is one of the most powerful quantum Monte Carlo methods and has been extensively applied to the study of quantum many body systems. Its efficiency is particularly enhanced with a deterministic loop update scheme in the study of the S=1/2 quantum spin systems that preserve SU(2) spin rotational symmetry. Once the symmetry is broken, such as by an external field, a directed loop method is typically required, resulting in a significant reduction in efficiency. Inspired by the SSE approach developed for the quantum Ising model, we introduce a deterministic loop SSE method that is particularly suited for antiferromagnetic systems under a staggered magnetic field. This method enables separate investigations of longitudinal and transverse modes in magnetically ordered phases arising from spontaneous symmetry breaking. We benchmark the performance of our algorithm against the standard directed loop approach applied to the antiferromagnetic Heisenberg chain and demonstrate that our method substantially reduces CPU time per Monte Carlo step, thereby can outperform the directed loop algorithm in efficiency.
We propose a quantum Monte Carlo (QMC) algorithm for nonequilibrium dynamics in a system with a parameter varying as a function of imaginary time. The method is based on successive applications of an evolving Hamiltonian to an initial state and delivers results for a whole range of the tuning parameter in a single run, allowing for access to both static and dynamic properties of the system. This approach reduces to the standard Schr\"odinger dynamics in imaginary time for quasi-adiabatic evolutions, i.e., including the leading nonadiabatic correction to the adiabatic limit. We here demonstrate this quasi-adiabatic QMC (QAQMC) method for linear ramps of the transverse-field Ising model across its quantum-critical point in one and two dimensions. The critical behavior can be described by generalized dynamic scaling. For the two-dimensional square-lattice system, we use the method to obtain a high-precision estimate of the quantum-critical point (h/J)c=3.04458(7){(h/J)}_{c}=3.04458(7)(h/J)c​=3.04458(7), where hhh is the transverse magnetic field and JJJ is the nearest-neighbor Ising coupling. The QAQMC method can also be used to extract the Berry curvature and the metric tensor.
Working within the stochastic series expansion framework, we introduce and characterize a plaquette-based quantum cluster algorithm for quantum Monte Carlo simulations of transverse field Ising models with frustrated Ising exchange interactions. As a demonstration of the capabilities of this algorithm, we show that a relatively small ferromagnetic next-nearest-neighbor coupling drives the transverse field Ising antiferromagnet on the triangular lattice from an antiferromagnetic three-sublattice ordered state at low temperature to a ferrimagnetic three-sublattice ordered state.