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Physical review. B, Condensed matter

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

Edward B. Brown

Published Jul 1, 1987• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.36.266

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Physical Review•Jan 18, 1965•23 citations
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.
The system described by the Hamiltonian shown is investigated on the basis of exact high-temperature series expansions and the Green function technique. Expansions for the zero-field ( B = 0) free energy and susceptibility (valid for arbitrary lattice structure) are developed to fifth order in reciprocal temperature for the S = ½ system with nearest-neighbour forces. The critical behaviour is discussed in two and three dimensions using the random phase approximation and Padé approximants. In particular the variation of the critical temperature with η is accurately determined for common two- and three-dimensional lattices. It is conjectured that the susceptibility index γ remains constant for 0 ⩽ η < 1 but changes discontinuously at η = 1.
A real-space effective-field renormalization-group method (ERFG) recently derived for computing critical properties of Ising spins is extended to treat the quantum spin-1/2 anisotropic Heisenberg model. The formalism is based on a generalized but approximate Callen–Suzuki spin relation and utilizes a convenient differential operator expansion technique. The method is illustrated in several lattice structures by employing its simplest approximation version in which clusters with one (N′=1) and two (N=2) spins are used. The results are compared with those obtained from the standard mean-field (MFRG) and Migdal–Kadanoff (MKRG) renormalization-group treatments and it is shown that this technique leads to rather accurate results. It is shown that, in contrast with the MFRG and MKRG predictions, the EFRG, besides correctly distinguishing the geometries of different lattice structures, also provides a vanishing critical temperature for all two-dimensional lattices in the isotropic Heisenberg limit. For the simple cubic lattice, the dependence of the transition temperature Tc with the exchange anisotropy parameter Δ [i.e., Tc(Δ)], and the resulting value for the critical thermal crossover exponent φ [i.e., Tc≂Tc(0)+AΔ1/φ ] are in quite good agreement with results available in the literature in which more sophisticated treatments are used.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
In the usual theory of the two-time Green's function, decoupling techniques are introduced in order to obtain a self-closed equation for the lowest order Green's function. These decouplings give neither the correct temperature dependence (the absence of a T3{T}^{3}T3 term) of magnetization nor the correct correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$, which should be zero by definition. In this paper the equation of motion for the higher order Green's function is solved in the approximation which will lead to correct results up to terms of order T3{T}^{3}T3 in these quantities. Employing this higher order Green's function, it is found that the correction to the magnetization consists of a term of order T4{T}^{4}T4 and a divergent term and that the correlation function consists of a term of order T3{T}^{3}T3 and a divergent term. A method extracting a significant part from this divergent term is found by analyzing a system of two interacting \textonehalf{} spins. By this method, it is shown that the T3{T}^{3}T3 term in the magnetization is completely eliminated and that the correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$ vanishes identically.