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

Physical review. B, Condensed matter

Mean-field Green’s-function theory of the two-sublattice Ising model in a transverse field

Edward B. Brown

Published Jun 1, 1991• 2 citations
Publisher

Abstract

Double-time Green's functions are used within a mean-field treatment of the two-sublattice Ising model in a transverse field. In contrast to previous treatments, a correct version of the spectral theorem is used to obtain a correct mean-field treatment for all spin. In particular, the dielectric constant is obtained for S=1. The results are markedly different from those of earlier treatments and are readily extended to four-sublattice models.

DOI
10.1103/physrevb.43.13664

Related papers

Green’s-function theory of the anisotropic Heisenberg model in a transverse field

Edward B. Brown

The S=(1/2 anisotropic Heisenberg model in a transverse field (AHMTF) is studied by means of double-time Green's functions. A new approximation scheme is employed which is based upon the assumption of the statistical independence of the ordering operator. The basic approximations are required to satisfy all relevant operator and correlation identities. Emphasis is placed upon obtaining accurate approximations for those Green's functions which are related to the ordering susceptibility. The scheme is relatively simple to use and applicable to a wide variety of systems. Standard Green's-function techniques are shown to provide results for the critical transverse field for Ising-like systems which violate S=(1/2 identities. They are also shown to be not applicable to the ordered phase of XY-like systems. In contrast, the new approximation provides reasonable results for all temperatures and couplings for the S=(1/2 AHMTF.

Physical review. B, Condensed matter•Jul 1, 1987•14 citations

Green's-function theory of the transverse Ising model

Philip E. Bloomfield, Edward B. Brown

Recent attempts to develop double-time Green's-function theories for the transverse Ising model (TIM) have been subject to certain difficulties; the appearance of zero-frequency poles in the commutator Green's function (CGF) led to the avoidance of CGF and an incomplete treatment based only upon anticommutator Green's functions (AGF), and ambiguities in the determination of some thermal averages. Upon decoupling the equations-of-motion hierarchy, we develop a consistent Green's-function theory of the TIM. The AGF and CGF are shown to provide consistent results, the problem of zero-frequency poles in CGF being treated appropriately. We resolve the ambiguity problem and obtain reasonable values for the Weiss critical field and the ground-state magnetization.

Physical review. B, Condensed matter•Aug 1, 1980•12 citations

Comment on ‘‘Undetermined-constant method in the boson Green’s-function theory’’

Edward B. Brown

The apparent fundamental inconsistency in the behavior of double-time Green's functions and the correlations related to them by the spectral theorem claimed by Zhang and Qin [Phys. Rev. A 36, 915 (1987)] is shown to arise from the use of an incorrect version of the spectral theorem. The ``method of undetermined constants'' constructed by Zhang and Qin to remove this apparent inconsistency is not an intrinsic characteristic of the boson Green's-function method. Rather, it is unnecessary and leads to incorrect results in both systems considered by Zhang and Qin, i.e., the mean-field transverse Ising model and the symmetrically decoupled zero-field Heisenberg model. Use of the correct spectral theorem is shown to remove the inconsistency. Correct treatments based on the correct spectral theorem of the models considered by Zhang and Qin are also presented.

Physical Review A•Apr 1, 1991•1 citation

Second Order Treatment of a Pseudo-Spin System. II: Application to the Transverse-Field Ising Model

Carla Buzano, Mario Rasetti, M. Vadacchino

The transverse-field Ising model is solved within the scheme of the generalized mean-field method earlier proposed by the authors. The uncompatibility between the self-consistency condition and the thermal equilibrium is corrected through the introduction of a correlated mean-field, realized by a virtual-field parameter. Numerical results are discussed in detail, from which the specific heat and the transverse-field/critical-temperature phase boundary diagram are derived.

Progress of Theoretical Physics•Sep 1, 1982•3 citations

On the Theory of the Ising Model with a Transverse Field

G. Kamieniarz

Abstract The zero‐temperature behaviour of the transverse Ising model is investigated in the paramagnetic region. Two versions of the Green's function technique are used and the results are in excellent agreement with the high density and series expansions results in both, the weak and strong interaction limit.

physica status solidi (b)•Jun 1, 1978•1 citation

Dynamics of the two-sublattice Ising model in a transverse field studied using a coupled-operators continued-fraction expansion

C E SaMotta, Antonio S. T. Pires

The dynamics of the longitudinal spin fluctuations of the two-sublattice three-dimensional Ising model in a transverse field is studied using a continued-fraction expansion for coupled operators.

Journal of Physics C Solid State Physics•Nov 20, 1984•1 citation
Double-time Green's functions are used within a mean-field treatment of the two-sublattice Ising model in a transverse field. In contrast to previous treatments, a correct version of the spectral theorem is used to obtain a correct mean-field treatment for all spin. In particular, the dielectric constant is obtained for S=1. The results are markedly different from those of earlier treatments and are readily extended to four-sublattice models.
The S=(1/2 anisotropic Heisenberg model in a transverse field (AHMTF) is studied by means of double-time Green's functions. A new approximation scheme is employed which is based upon the assumption of the statistical independence of the ordering operator. The basic approximations are required to satisfy all relevant operator and correlation identities. Emphasis is placed upon obtaining accurate approximations for those Green's functions which are related to the ordering susceptibility. The scheme is relatively simple to use and applicable to a wide variety of systems. Standard Green's-function techniques are shown to provide results for the critical transverse field for Ising-like systems which violate S=(1/2 identities. They are also shown to be not applicable to the ordered phase of XY-like systems. In contrast, the new approximation provides reasonable results for all temperatures and couplings for the S=(1/2 AHMTF.
Recent attempts to develop double-time Green's-function theories for the transverse Ising model (TIM) have been subject to certain difficulties; the appearance of zero-frequency poles in the commutator Green's function (CGF) led to the avoidance of CGF and an incomplete treatment based only upon anticommutator Green's functions (AGF), and ambiguities in the determination of some thermal averages. Upon decoupling the equations-of-motion hierarchy, we develop a consistent Green's-function theory of the TIM. The AGF and CGF are shown to provide consistent results, the problem of zero-frequency poles in CGF being treated appropriately. We resolve the ambiguity problem and obtain reasonable values for the Weiss critical field and the ground-state magnetization.
The apparent fundamental inconsistency in the behavior of double-time Green's functions and the correlations related to them by the spectral theorem claimed by Zhang and Qin [Phys. Rev. A 36, 915 (1987)] is shown to arise from the use of an incorrect version of the spectral theorem. The ``method of undetermined constants'' constructed by Zhang and Qin to remove this apparent inconsistency is not an intrinsic characteristic of the boson Green's-function method. Rather, it is unnecessary and leads to incorrect results in both systems considered by Zhang and Qin, i.e., the mean-field transverse Ising model and the symmetrically decoupled zero-field Heisenberg model. Use of the correct spectral theorem is shown to remove the inconsistency. Correct treatments based on the correct spectral theorem of the models considered by Zhang and Qin are also presented.