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Physical review. B, Condensed matter

Cluster-variation approach to the spin-1/2XXZmodel

Dirk Jan Bukman, Guozhong An, J. M. J. van Leeuwen

Published Jun 1, 1991• 25 citations
Publisher

Abstract

We use the cluster-variation method to examine the phase diagram of the spin-1/2 XXZ model. The clusters used consist of two neighboring spins. With this method, we calculate global phase diagrams for arbitrary values of the anisotropy ${\mathit{J}}_{\mathit{z}}$/J and for arbitrary external magnetic fields. Lattices with different values of the coordination number z are considered. Analytic expressions are found for nearly all phase boundaries; the remaining phase boundary is located numerically. Despite some problems at low temperature, the results based on this approach are in quite good agreement with the results from series-expansion methods. A connection is made with the Hubbard model, which can be mapped onto the XXZ Hamiltonian in the limit of strong on-site attraction.

DOI
10.1103/physrevb.43.13352

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We use the cluster-variation method to examine the phase diagram of the spin-1/2 XXZ model. The clusters used consist of two neighboring spins. With this method, we calculate global phase diagrams for arbitrary values of the anisotropy Jz{\mathit{J}}_{\mathit{z}}Jz​/J and for arbitrary external magnetic fields. Lattices with different values of the coordination number z are considered. Analytic expressions are found for nearly all phase boundaries; the remaining phase boundary is located numerically. Despite some problems at low temperature, the results based on this approach are in quite good agreement with the results from series-expansion methods. A connection is made with the Hubbard model, which can be mapped onto the XXZ Hamiltonian in the limit of strong on-site attraction.
The phase diagram of the spin 1/2 XXZ model is examined for two different geometries, which can be changed from two- to three-dimensional. The first of these consists of an infinite stack of simple quadratic layers with a variable inter-layer coupling, where the ratio of the inter-layer and intra-layer couplings varies between zero and one. Second, the authors consider a slab geometry consisting of n such layers, with equal couplings in all directions. For both geometries they use the cluster variation method for two-spin clusters to construct the phase diagram of the model, and they examine the changes that take place as the system changes from two- to three-dimensional. For the slab geometry they also calculate the order parameter profile near T c . It turns out that Dirichlet boundary conditions for the related superfluid wavefunction are most realistic.
The one dimensional S=1 XXZ model with next-nearest-neighbor interaction ααα and Ising-type anisotropy ΔΔΔ is studied by using a numerical diagonalization technique. We discuss the ground state phase diagram of this model numerically by the twisted-boundary-condition level spectroscopy method and the phenomenological renormalization group method, and analytically by the spin wave theory. We determine the phase boundaries among the XY phase, the Haldane phase, the ferromagnetic phase and the Néel phase, and then we confirm the universality class. Moreover, we map this model onto the non-linear σσσ model and analyze the phase diagram in the ααα ≪\ll≪ -1 and ΔΔΔ ∼\sim∼ 1 region by using the renormalization group method.
We outline how the coupled cluster method of microscopic quantum many-body theory can be utilized in practice to give highly accurate results for the ground-state properties of a wide variety of highly frustrated and strongly correlated spinlattice models of interest in quantum magnetism, including their quantum phase transitions.The method itself is described, and it is shown how it may be implemented in practice to high orders in a systematically improvable hierarchy of (socalled LSUBm) approximations, by the use of computer-algebraic techniques.The method works from the outset in the thermodynamic limit of an infinite lattice at all levels of approximation, and it is shown both how the "raw" LSUBm results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, m → ∞, of the truncation index m, which denotes the only approximation made.All of this is illustrated via a specific application to a two-dimensional, frustrated, spin-half J XXZ 1 -J XXZ 2 model on a honeycomb lattice with nearestneighbor and next-nearest-neighbor interactions with exchange couplings J 1 > 0 and J 2 ≡ κJ 1 > 0, respectively, where both interactions are of the same anisotropic XXZ type.We show how the method can be used to determine the entire zerotemperature ground-state phase diagram of the model in the range 0 ≤ κ ≤ 1 of the frustration parameter and 0 ≤ Δ ≤ 1 of the spin-space anisotropy parameter.In particular, we identify a candidate quantum spin-liquid region in the phase space.
We study the ground-state phase diagram of the quantum spin-2 XXZ chain in the presence of on-site anisotropy using a matrix-product state based infinite-system density matrix renormalization group (iDMRG) algorithm. One of the interests in this system is in connecting the highly quantum-mechanical spin-1 phase diagram with the classical $S=\ensuremath{\infty}$ phase diagram. Several of the recent advances within DMRG make it possible to perform a detailed analysis of the whole phase diagram. We consider different types of on-site anisotropies, which allows us to establish the validity of the following statements: (1) the spin-2 model can be tuned into a phase, which is equivalent to the ``topologically nontrivial'' spin-1 Haldane phase, and (2) the spin-2 Haldane phase at the isotropic Heisenberg point is adiabatically connected to the ``trivial'' large-DDD phase, with a continuous change of the Hamiltonian parameters. Furthermore, we study the spin-3 XXZ chain to help explain the development of the classical phase diagram. We present details on how to use the iDMRG method to map out the phase diagram and include an extensive discussion of the numerical methods.
Quantum spins arranged on regular geometrical lattices, and interacting through a wide diversity of specified model Hamiltonians, provide a rich variety of magnetically ordered phases with many subtle properties. The zero-temperature quantum phase transitions between states of different order can be explored by the variation of internal control parameters within the interaction Hamiltonian. We illustrate how the systematic inclusion of multi-spin correlations for such lattice systems may be efficiently and effectively implemented via the coupled cluster method (CCM), which has been widely demonstrated on many previous occasions to be one of the most versatile methods of microscopic quantum many-body theory. Its power is illustrated by application to the spin-½ anisotropic Heisenberg (or XXZ) model on an infinite square lattice. The results obtained for such properties as the ground-state energy and sublattice magnetization are found to be as accurate as those obtained by any available alternative method. We show also how the CCM can now provide accurate ab initio estimates for the critical values of the parameters which mark the positions of phase boundaries between states of different quantum order, as well as for the critical indices associated with the singular behaviour near these transition points of various physical quantities.