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arXiv:cond-mat/0412464 · Physical Review B

Theoretical studies of the phase transition in the anisotropic two-dimensional square spin lattice

Mohamad Al Hajj, Nathalie Guihéry, Jean‐Paul Malrieu, Peter Wind

Published Sep 21, 2004• 14 citations
arXivHTMLPDF

Abstract

The phase transition occurring in a square two-dimensional spin lattice governed by an anisotropic Heisenberg Hamiltonian has been studied according to two recently proposed methods. The first one, the Dressed Cluster Method, provides excellent evaluations of the cohesive energy. The discontinuity of its derivative around the critical (isotropic) value of the anisotropy parameter confirms the first-order character of the phase transition. Nevertheless, the method introduces two distinct reference functions (either N\'eel or $XY$) which may, in principle, force the discontinuity. The Real-Space Renormalization Group with Effective Interactions does not reach the same numerical accuracy, but it does not introduce a reference function and the phase transition appears qualitatively, due to the existence of two domains with specific fixed points. The method confirms the dependence of the spin gap on the anisotropy parameter occurring in the Heisenberg-Ising domain.

DOI
10.1103/physrevb.70.094415

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The phase transition occurring in a square two-dimensional spin lattice governed by an anisotropic Heisenberg Hamiltonian has been studied according to two recently proposed methods. The first one, the Dressed Cluster Method, provides excellent evaluations of the cohesive energy. The discontinuity of its derivative around the critical (isotropic) value of the anisotropy parameter confirms the first-order character of the phase transition. Nevertheless, the method introduces two distinct reference functions (either N'eel or XYXYXY) which may, in principle, force the discontinuity. The Real-Space Renormalization Group with Effective Interactions does not reach the same numerical accuracy, but it does not introduce a reference function and the phase transition appears qualitatively, due to the existence of two domains with specific fixed points. The method confirms the dependence of the spin gap on the anisotropy parameter occurring in the Heisenberg-Ising domain.
The coupled cluster method (CCM) is applied to the spin-one anisotropic Heisenberg antiferromagnet (HAF) on the square lattice at zero temperature using a new high-order CCM ground-state formalism for general quantum spin number (s≥1/2s \ge 1/2s≥1/2). The results presented constitute the first such application of this new formalism, and they are shown to be among the most accurate results for the ground-state energy and sublattice magnetisation of this model as yet determined. We ``track'' the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter, ΔΔΔ, from the trivial Ising limit (Δ→∞Δ\to \inftyΔ→∞) down to a critical value ΔcΔ_cΔc​, at which point this solution terminates. This behaviour is associated with a phase transition in the system, and hence a primary result of these high-order CCM calculations is that they provide an accurate and unbiased (i.e., {\it ab initio}) estimation of the position of the quantum phase transition point as a function of the anisotropy parameter. Our result is, namely, that this point occurs at (or slightly below) the isotropic Heisenberg point at Δ=1Δ=1Δ=1, for this model.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
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.
In this work we apply two different real-space renormalization-group (RSRG) approaches to the anisotropic antiferromagnetic spin-1/2 Heisenberg model on the square lattice. Our calculations allow for an approximate evaluation of the T vs \ensuremath{\Delta} phase diagram: the results suggest the existence of a critical value of $\ensuremath{\Delta}>0,$ at which the critical temperature goes to zero, and the presence of reentrant behavior on the critical line between the ordered and disordered phases. This whole critical line is found to belong to the same universality class as the Ising model. Our results are in accordance with previous RSRG approaches but not with numerical simulations and spin-wave calculations.
We investigate entanglement and quantum phase transition in a two-dimensional Heisenberg anisotropic spin-1/2XY1/2\mathit{XY}1/2XY model, using the quantum renormalization-group method (QRG) on a square lattice of $N\ifmmode\times\else\texttimes\fi{}N$ sites. The entanglement through geometric average of concurrences is calculated after each step of the QRG. We show that the concurrence achieves a nonzero value at the critical point more rapidly as compared to the one-dimensional case. The relationship between the entanglement and the quantum phase transition is studied. The evolution of entanglement develops two saturated values corresponding to two different phases. We compute the derivative of the concurrence, which is found to be discontinuous at the critical point $\ensuremath{\gamma}=0$, and indicates a second-order phase transition in the spin system. Further, the scaling behavior of the system is investigated by computing the derivative of the concurrence in terms of the system size.
A real space renormalization group method is applied to study the spin-1/2 two-dimensional anisotropic Heisenberg antiferromagnets. We carry out the calculation on a 5×5 cluster by using a variational approach. The ground-state energy per site is estimated as e0=−0.6845±0.0005 which is in good agreement with other numerical estimates. We also calculate the different ground state energies when anisotropy changes. By comparing with 3×3 cluster calculation the resulting threshold for long range order drops and the curve of the renormalization group parameter vs. anisotropy shows no hint of discontinuity over a wide range of anisotropic parameters. Hence, our calculation suggests that long-range order exists for the ground-state of the spin-1/2 Heisenberg antiferromagnets in two dimensions.