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:1211.3471 · Physical Review E

Path-integral Monte Carlo method for the local Z2 Berry phase

Yuichi Motoyama, Synge Todo

Published Feb 6, 2013• 17 citations
arXivHTMLPDF

Abstract

We present a loop cluster algorithm Monte Carlo method for calculating the local Z(2) Berry phase of the quantum spin models. The Berry connection, which is given as the inner product of two ground states with different local twist angles, is expressed as a Monte Carlo average on the worldlines with fixed spin configurations at the imaginary-time boundaries. The "complex weight problem" caused by the local twist is solved by adopting the meron cluster algorithm. We present the results of simulation on the antiferromagnetic Heisenberg model on an out-of-phase bond-alternating ladder to demonstrate that our method successfully detects the change in the valence bond pattern at the quantum phase transition point. We also propose that the gauge-fixed local Berry connection can be an effective tool to estimate precisely the quantum critical point.

DOI
10.1103/physreve.87.021301

Related papers

Path-integral Monte Carlo for the Gauge-fixed Berry Connection and the Local Z_2 Berry Phase

Yuichi Motoyama, Synge Todo

We present a general formula of the gauge-fixed Berry connection which can be evaluated by path- integral Monte Carlo method. We also propose that the gauge-fixed local Berry connection can be an- other effective tool to estimate precisely the quantum critical point. For a demonstration, we calculate the gauge-fixed Berry connection and the local Z2 Berry phase of the antiferromagnetic Heisenberg model on a staggered bond-alternating ladder, and estimated quantum critical point is consistent with other methods.

arXiv:1310.0205•Oct 1, 2013•0 citations

ZN Berry phase and symmetry-protected topological phases of the SU( N ) antiferromagnetic Heisenberg chain

Yuichi Motoyama, Synge Todo

The local ${Z}_{N}$ quantized Berry phase for the $\mathrm{SU}(N)$ antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local ${Z}_{2}$ Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional $\mathrm{SU}(N)$ spin systems. We extend the path-integral quantum Monte Carlo method for the ${Z}_{2}$ Berry phase in order to calculate the ${Z}_{N}$ Berry phase numerically. We demonstrate our method by calculating the ${Z}_{4}$ Berry phase for the bond-alternating $\mathrm{SU}(4)$ antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.

arXiv:1508.00960•Physical review. B./Physical review. B•Nov 20, 2018•10 citations

The loop algorithm

Hans Gerd Evertz

A review of the loop algorithm , its generalizations, and its relation to some other Monte Carlo techniques is given. The loop algorithm is a quantum Monte Carlo procedure that employs non-local changes of worldline configurations, determined by local stochastic decisions. It is based on a formulation of quantum models of any dimension in an extended ensemble of worldlines and graphs, and is related to Swendsen-Wang algorithms. It can be represented directly on an operator level, both with a continuous imaginary time path integral and with the stochastic series expansion. It overcomes many of the difficulties of traditional worldline simulations. Autocorrelations are reduced by orders of magnitude. Grand-canonical ensembles, off-diagonal operators, and variance reduced estimators are accessible. In some cases, infinite systems can be simulated. For a restricted class of models, the fermion sign problem can be overcome. Transverse magnetic fields are handled efficiently, in contrast to strong diagonal fields. The method has been applied successfully to a variety of models for spin and charge degrees of freedom, including Heisenberg and XYZ spin models, hard-core bosons, Hubbard and t - J -models. Owing to the improved efficiency, precise calculations of asymptotic behaviour and of quantum critical exponents have been possible.

arXiv:cond-mat/9707221•Advances In Physics•Jan 1, 2003•295 citations

Randomness-driven Quantum Phase Transition in Bond-alternating Haldane Chain

Takayuki Arakawa, Synge Todo, Hajime Takayama

The effect of bond randomness on the spin-gapped ground state of the spin-1 bond-alternating antiferromagnetic Heisenberg chain is discussed. By using the loop cluster quantum Monte Carlo method, we investigate the stability of topological order in terms of the recently proposed twist order parameter [M. Nakamura and S. Todo: Phys. Rev. Lett. 89 (2002) 077204]. It is observed that the dimer phases as well as the Haldane phase of the spin-1 Heisenberg chain are robust against a weak randomness, though the valence-bond-solid-like topological order in the latter phase is destroyed by introducing a disorder stronger than the critical value.

arXiv:cond-mat/0410755•Journal of the Physical Society of Japan•Apr 1, 2005•6 citations

Monte Carlo approach to phase transitions in quantum systems

Naoki Kawashima, Yasuyuki Kato

We propose algorithms of the path-ingetral-based quantum Monte Carlo simulation, which is otherwise prohibitively slow. While the basic idea is the loop-cluster update, there are some important 'tricks' that are vital to make the simulation practical. In the present paper, we show two such techniques and their successful applications to the two-dimensional SU( N ) Heisenberg model and the three-dimensional Bose Hubbard model. In the former, we obtain a new type of the valence-bond-solid state for the two-boson representation, while in the latter we equillibrate a system of which the size is comparable to a typical experiment of optical lattices.

Journal of Physics Conference Series•Jan 7, 2009•2 citations

Meron-cluster simulation of the quantum antiferromagnetic Heisenberg model in a magnetic field in one- and two-dimensions

Palma, A. Riveros

Motivated by the numerical simulation of systems which display quantum phase transitions, we present a novel application of the meron-cluster algorithm to simulate the quantum antiferromagnetic Heisenberg model coupled to an external uniform magnetic field both in one and in two dimensions.In the infinite volume limit and at zero temperature we found numerical evidence that supports a quantum phase transition very close to the critical values B c = 2 and B c = 4 for the system in one and two dimensions, respectively.For the one dimensional system, we have compared the numerical data obtained with analytical predictions for the magnetization density as a function of the external field obtained by scaling-behaviour analysis and Bethe Ansatz techniques.Since there is no analytical solution for the two dimensional case, we have compared our results with the magnetization density obtained by scaling relations for small lattice sizes and with the approximated thermodynamical limit at zero temperature guessed by scaling relations.Moreover, we have compared the numerical data with other numerical simulations performed by using different algorithms in one and two dimensions, like the directed loop method.The numerical data obtained are in perfect agreement with all these previous results, which confirms that the meron-algorithm is reliable for quantum Monte Carlo simulations and applicable both in one and two dimensions.Finally, we have computed the integrated autocorrelation time to measure the efficiency of the meron algorithm in one dimension.

arXiv:1312.4915•Condensed Matter Physics•May 25, 2015•8 citations
We present a loop cluster algorithm Monte Carlo method for calculating the local Z(2) Berry phase of the quantum spin models. The Berry connection, which is given as the inner product of two ground states with different local twist angles, is expressed as a Monte Carlo average on the worldlines with fixed spin configurations at the imaginary-time boundaries. The "complex weight problem" caused by the local twist is solved by adopting the meron cluster algorithm. We present the results of simulation on the antiferromagnetic Heisenberg model on an out-of-phase bond-alternating ladder to demonstrate that our method successfully detects the change in the valence bond pattern at the quantum phase transition point. We also propose that the gauge-fixed local Berry connection can be an effective tool to estimate precisely the quantum critical point.
We present a general formula of the gauge-fixed Berry connection which can be evaluated by path- integral Monte Carlo method. We also propose that the gauge-fixed local Berry connection can be an- other effective tool to estimate precisely the quantum critical point. For a demonstration, we calculate the gauge-fixed Berry connection and the local Z2 Berry phase of the antiferromagnetic Heisenberg model on a staggered bond-alternating ladder, and estimated quantum critical point is consistent with other methods.
The local ZN{Z}_{N}ZN​ quantized Berry phase for the SU(N)\mathrm{SU}(N)SU(N) antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local Z2{Z}_{2}Z2​ Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional SU(N)\mathrm{SU}(N)SU(N) spin systems. We extend the path-integral quantum Monte Carlo method for the Z2{Z}_{2}Z2​ Berry phase in order to calculate the ZN{Z}_{N}ZN​ Berry phase numerically. We demonstrate our method by calculating the Z4{Z}_{4}Z4​ Berry phase for the bond-alternating SU(4)\mathrm{SU}(4)SU(4) antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.
A review of the loop algorithm , its generalizations, and its relation to some other Monte Carlo techniques is given. The loop algorithm is a quantum Monte Carlo procedure that employs non-local changes of worldline configurations, determined by local stochastic decisions. It is based on a formulation of quantum models of any dimension in an extended ensemble of worldlines and graphs, and is related to Swendsen-Wang algorithms. It can be represented directly on an operator level, both with a continuous imaginary time path integral and with the stochastic series expansion. It overcomes many of the difficulties of traditional worldline simulations. Autocorrelations are reduced by orders of magnitude. Grand-canonical ensembles, off-diagonal operators, and variance reduced estimators are accessible. In some cases, infinite systems can be simulated. For a restricted class of models, the fermion sign problem can be overcome. Transverse magnetic fields are handled efficiently, in contrast to strong diagonal fields. The method has been applied successfully to a variety of models for spin and charge degrees of freedom, including Heisenberg and XYZ spin models, hard-core bosons, Hubbard and t - J -models. Owing to the improved efficiency, precise calculations of asymptotic behaviour and of quantum critical exponents have been possible.
The effect of bond randomness on the spin-gapped ground state of the spin-1 bond-alternating antiferromagnetic Heisenberg chain is discussed. By using the loop cluster quantum Monte Carlo method, we investigate the stability of topological order in terms of the recently proposed twist order parameter [M. Nakamura and S. Todo: Phys. Rev. Lett. 89 (2002) 077204]. It is observed that the dimer phases as well as the Haldane phase of the spin-1 Heisenberg chain are robust against a weak randomness, though the valence-bond-solid-like topological order in the latter phase is destroyed by introducing a disorder stronger than the critical value.
We propose algorithms of the path-ingetral-based quantum Monte Carlo simulation, which is otherwise prohibitively slow. While the basic idea is the loop-cluster update, there are some important 'tricks' that are vital to make the simulation practical. In the present paper, we show two such techniques and their successful applications to the two-dimensional SU( N ) Heisenberg model and the three-dimensional Bose Hubbard model. In the former, we obtain a new type of the valence-bond-solid state for the two-boson representation, while in the latter we equillibrate a system of which the size is comparable to a typical experiment of optical lattices.
Motivated by the numerical simulation of systems which display quantum phase transitions, we present a novel application of the meron-cluster algorithm to simulate the quantum antiferromagnetic Heisenberg model coupled to an external uniform magnetic field both in one and in two dimensions.In the infinite volume limit and at zero temperature we found numerical evidence that supports a quantum phase transition very close to the critical values B c = 2 and B c = 4 for the system in one and two dimensions, respectively.For the one dimensional system, we have compared the numerical data obtained with analytical predictions for the magnetization density as a function of the external field obtained by scaling-behaviour analysis and Bethe Ansatz techniques.Since there is no analytical solution for the two dimensional case, we have compared our results with the magnetization density obtained by scaling relations for small lattice sizes and with the approximated thermodynamical limit at zero temperature guessed by scaling relations.Moreover, we have compared the numerical data with other numerical simulations performed by using different algorithms in one and two dimensions, like the directed loop method.The numerical data obtained are in perfect agreement with all these previous results, which confirms that the meron-algorithm is reliable for quantum Monte Carlo simulations and applicable both in one and two dimensions.Finally, we have computed the integrated autocorrelation time to measure the efficiency of the meron algorithm in one dimension.