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:1604.03486 · Physical review. E

Quantum correlated cluster mean-field theory applied to the transverse Ising model

Fábio M. Zimmer, M. Schmidt, Jonas Maziero

Published Jun 13, 2016• 10 citations
arXivHTMLPDF

Abstract

Mean-field theory (MFT) is one of the main available tools for analytical calculations entailed in investigations regarding many-body systems. Recently, there has been a surge of interest in ameliorating this kind of method, mainly with the aim of incorporating geometric and correlation properties of these systems. The correlated cluster MFT (CCMFT) is an improvement that succeeded quite well in doing that for classical spin systems. Nevertheless, even the CCMFT presents some deficiencies when applied to quantum systems. In this article, we address this issue by proposing the quantum CCMFT (QCCMFT), which, in contrast to its former approach, uses general quantum states in its self-consistent mean-field equations. We apply the introduced QCCMFT to the transverse Ising model in honeycomb, square, and simple cubic lattices and obtain fairly good results both for the Curie temperature of thermal phase transition and for the critical field of quantum phase transition. Actually, our results match those obtained via exact solutions, series expansions or Monte Carlo simulations.

DOI
10.1103/physreve.93.062116

Related papers

Correlated cluster mean-field theory for spin systems

Daisuke Yamamoto

A cluster mean-field method is introduced and the applications to the Ising and Heisenberg models are demonstrated. We divide the lattice sites into clusters whose size and shape are selected so that the equivalence of all sites in a cluster is preserved. Since the strength of interactions of a cluster with its surrounding clusters is strongly dependent on the spin configuration of the central cluster itself, we include this contribution in the effective fields acting on the spins. The effects of ``correlations'' between clusters can be taken into account beyond the standard mean-field level and as a result our cluster-based method gives qualitatively (and even quantitatively) correct results for the both Ising and Heisenberg models. Especially, for the Ising model on the honeycomb and square lattices, the calculated results of the critical temperature are very close (overestimated by only less than 5%) to the exact values.

arXiv:0812.4139•Physical Review B•Apr 24, 2009•69 citations

A brief account of the Ising and Ising-like models: Mean-field, effective-field and exact results

Jozef Strečka, M. Jaščur

The article provides a tutorial review on how to treat Ising models within mean-field (MF), effective-field (EF) and exact methods. MF solutions of the spin-1 Blume-Capel (BC) model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A quantum phase transition of the spin-S Ising model driven by a transverse field is explored within MF method. EF theory is elaborated within a single- and two-spin cluster approach to demonstrate an efficiency of this approximate method. The long-standing problem of this method concerned with a self-consistent determination of the free energy is addressed. EF theory is adapted for the spin-1/2 Ising model, the spin-S BC model and the transverse Ising model. The particular attention is paid to continuous and discontinuous transitions. Exact results for the spin-1/2 Ising chain, spin-1 BC chain and mixed-spin Ising chain are obtained using the transfer-matrix method, the crucial steps of which are reviewed for a spin-1/2 Ising square lattice. Critical points of the spin-1/2 Ising model on several lattices are rigorously obtained with the help of dual, star-triangle and decoration-iteration transformations. Mapping transformations are adapted to obtain exact results for the mixed-spin Ising model on planar lattices. An increase in the coordination number of the mixed-spin Ising model on decorated planar lattices gives rise to reentrant transitions, while the critical temperature of the mixed-spin Ising model on a regular honeycomb lattice is always greater than that of two semi-regular archimedean lattices. The effect of selective site dilution of the mixed-spin Ising model on a honeycomb lattice upon phase diagrams is examined. The review affords a brief account of the Ising models solved within MF, EF and exact methods along with a few comments on their future applicability.

arXiv:1511.03031•Nov 10, 2015•45 citations

The ground-state phase transition study of the transverse high-spin Ising model with different single-ion anisotropies

XU Xing-guang

The transverse high-spin Ising model with different single-ion anisotropies is studied in the ground state within the mean-field theory and the correlated effective-field theory.The expressions of magnetization and energy in the finite temperature and ground state are derived and the results are got numerically.The ground-state phase diagram is obtained within the mean-field theory,and the ground-state phase diagrams for the honeycomb lattice,the square lattice and the simple cubic lattice are gained,respectively,within the correlated effective theory.The focus of this paper is on the difference of phase transition properties within the mean-field theory and the correlated effective-field theory.

Journal of Northeast Normal University•Jan 1, 2009•0 citations

Entanglement mean-field theory and the Curie-Weiss law

Aditi Sen, Ujjwal Sen

The mean-field theory, in its different hues, forms a useful tool for investigating single-body properties, like magnetization and susceptibility, of many-body systems. We propose an "entanglement mean-field theory", which transforms a many-body system into a two-body one, while retaining footprints of the many-body parent, using which it is possible to examine its two-body properties, and predict temperature-driven as well as quantum fluctuation-driven critical phenomena, by considering two-body self-consistency equations in contrast to single-body ones in mean-field–like theories. Compared to mean-field theory, the proposed one, with little to no extra complicacy, makes better predictions for the critical points, as well as for the qualitative and quantitative behavior of single- and two-body physical quantities. In particular, the predictions of the proposed theory are in much better conformity with the Curie-Weiss law for magnetization. Also, the proposed theory predicts an order by disorder for a correlation function in the random-field transverse quantum Ising model.

arXiv:0911.4856•Europhysics Letters (EPL)•Jul 1, 2012•8 citations

AB INITIO TREATMENTS OF THE ISING MODEL IN A TRANSVERSE FIELD

R. F. Bishop, DAMIAN J. J. FARNELL, MANFRED L. RISTIG

In this article new results are presented for the zero-temperature ground-state properties of the spin-half transverse Ising model on various lattices using three different approximate techniques. These are, respectively, the coupled cluster method, the correlated basis function method, and the variational quantum Monte Carlo method. The methods, at different levels of approximation, are used to study the ground-state properties of these systems, and the results are found to be in excellent agreement both with each other and with results of exact calculations for the linear chain and results of exact cumulant series expansions for lattices of higher spatial dimension. The different techniques used are compared and contrasted in the light of these results, and the constructions of the approximate ground- state wave functions are especially discussed.

arXiv:cond-mat/0011043•International Journal of Modern Physics B•Jun 20, 2000•10 citations

Cluster Monte Carlo simulation of the transverse Ising model

Henk W. J. Blöte, Youjin Deng

We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Dec 10, 2002•282 citations
Mean-field theory (MFT) is one of the main available tools for analytical calculations entailed in investigations regarding many-body systems. Recently, there has been a surge of interest in ameliorating this kind of method, mainly with the aim of incorporating geometric and correlation properties of these systems. The correlated cluster MFT (CCMFT) is an improvement that succeeded quite well in doing that for classical spin systems. Nevertheless, even the CCMFT presents some deficiencies when applied to quantum systems. In this article, we address this issue by proposing the quantum CCMFT (QCCMFT), which, in contrast to its former approach, uses general quantum states in its self-consistent mean-field equations. We apply the introduced QCCMFT to the transverse Ising model in honeycomb, square, and simple cubic lattices and obtain fairly good results both for the Curie temperature of thermal phase transition and for the critical field of quantum phase transition. Actually, our results match those obtained via exact solutions, series expansions or Monte Carlo simulations.
A cluster mean-field method is introduced and the applications to the Ising and Heisenberg models are demonstrated. We divide the lattice sites into clusters whose size and shape are selected so that the equivalence of all sites in a cluster is preserved. Since the strength of interactions of a cluster with its surrounding clusters is strongly dependent on the spin configuration of the central cluster itself, we include this contribution in the effective fields acting on the spins. The effects of ``correlations'' between clusters can be taken into account beyond the standard mean-field level and as a result our cluster-based method gives qualitatively (and even quantitatively) correct results for the both Ising and Heisenberg models. Especially, for the Ising model on the honeycomb and square lattices, the calculated results of the critical temperature are very close (overestimated by only less than 5%) to the exact values.
The article provides a tutorial review on how to treat Ising models within mean-field (MF), effective-field (EF) and exact methods. MF solutions of the spin-1 Blume-Capel (BC) model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A quantum phase transition of the spin-S Ising model driven by a transverse field is explored within MF method. EF theory is elaborated within a single- and two-spin cluster approach to demonstrate an efficiency of this approximate method. The long-standing problem of this method concerned with a self-consistent determination of the free energy is addressed. EF theory is adapted for the spin-1/2 Ising model, the spin-S BC model and the transverse Ising model. The particular attention is paid to continuous and discontinuous transitions. Exact results for the spin-1/2 Ising chain, spin-1 BC chain and mixed-spin Ising chain are obtained using the transfer-matrix method, the crucial steps of which are reviewed for a spin-1/2 Ising square lattice. Critical points of the spin-1/2 Ising model on several lattices are rigorously obtained with the help of dual, star-triangle and decoration-iteration transformations. Mapping transformations are adapted to obtain exact results for the mixed-spin Ising model on planar lattices. An increase in the coordination number of the mixed-spin Ising model on decorated planar lattices gives rise to reentrant transitions, while the critical temperature of the mixed-spin Ising model on a regular honeycomb lattice is always greater than that of two semi-regular archimedean lattices. The effect of selective site dilution of the mixed-spin Ising model on a honeycomb lattice upon phase diagrams is examined. The review affords a brief account of the Ising models solved within MF, EF and exact methods along with a few comments on their future applicability.
The transverse high-spin Ising model with different single-ion anisotropies is studied in the ground state within the mean-field theory and the correlated effective-field theory.The expressions of magnetization and energy in the finite temperature and ground state are derived and the results are got numerically.The ground-state phase diagram is obtained within the mean-field theory,and the ground-state phase diagrams for the honeycomb lattice,the square lattice and the simple cubic lattice are gained,respectively,within the correlated effective theory.The focus of this paper is on the difference of phase transition properties within the mean-field theory and the correlated effective-field theory.
The mean-field theory, in its different hues, forms a useful tool for investigating single-body properties, like magnetization and susceptibility, of many-body systems. We propose an "entanglement mean-field theory", which transforms a many-body system into a two-body one, while retaining footprints of the many-body parent, using which it is possible to examine its two-body properties, and predict temperature-driven as well as quantum fluctuation-driven critical phenomena, by considering two-body self-consistency equations in contrast to single-body ones in mean-field–like theories. Compared to mean-field theory, the proposed one, with little to no extra complicacy, makes better predictions for the critical points, as well as for the qualitative and quantitative behavior of single- and two-body physical quantities. In particular, the predictions of the proposed theory are in much better conformity with the Curie-Weiss law for magnetization. Also, the proposed theory predicts an order by disorder for a correlation function in the random-field transverse quantum Ising model.
In this article new results are presented for the zero-temperature ground-state properties of the spin-half transverse Ising model on various lattices using three different approximate techniques. These are, respectively, the coupled cluster method, the correlated basis function method, and the variational quantum Monte Carlo method. The methods, at different levels of approximation, are used to study the ground-state properties of these systems, and the results are found to be in excellent agreement both with each other and with results of exact calculations for the linear chain and results of exact cumulant series expansions for lattices of higher spatial dimension. The different techniques used are compared and contrasted in the light of these results, and the constructions of the approximate ground- state wave functions are especially discussed.
We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.