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Physical review. B, Condensed matter

Quantum phase transition in square- and triangular-lattice spin-½ antiferromagnets

Chen Zeng, I. Staples, R. F. Bishop

Published Apr 1, 1996• 8 citations
Publisher

Abstract

We use the coupled-cluster method to study ground-state properties of anisotropic $S=\frac{1}{2}$ antiferromagnets on square and triangular lattices, with the inclusion of arbitrarily long-ranged two-spin correlations. We detect the singularities of various quantities associated with the quantum phase transitions and also compute their critical exponents. The two-spin correlation coefficients for the triangular lattice are found to exhibit an interesting oscillatory behavior in their signs, knowledge of which could assist the implementation of quantum Monte Carlo simulations.

DOI
10.1103/physrevb.53.9168

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We use the coupled-cluster method to study ground-state properties of anisotropic S=12S=\frac{1}{2}S=21​ antiferromagnets on square and triangular lattices, with the inclusion of arbitrarily long-ranged two-spin correlations. We detect the singularities of various quantities associated with the quantum phase transitions and also compute their critical exponents. The two-spin correlation coefficients for the triangular lattice are found to exhibit an interesting oscillatory behavior in their signs, knowledge of which could assist the implementation of quantum Monte Carlo simulations.
We extend our own previous applications of the microscopic coupled-cluster method (CCM) to quantum antiferromagnets. In particular, we carry out a systematic calculation involving high-order multispin correlations for the spin-1/2 anisotropic Heisenberg models on the one-dimensional chain and the two-dimensional square lattice. Their ground-state properties are obtained as functions of the anisotropy parameter. Our CCM analysis not only produces accurate results for such physical quantities as the ground-state energy which are comparable to the best results from other techniques, but it also enables us to study the quantum phase transitions of the spin models in a systematic and unbiased manner. © 1996 John Wiley & Sons, Inc.
We study the spin-1/2 and spin-1 [Formula: see text] Heisenberg antiferromagnets (HAFs) on an infinite, anisotropic, two-dimensional triangular lattice, using the coupled cluster method. With respect to an underlying square-lattice geometry the model contains antiferromagnetic (J1 > 0) bonds between nearest neighbours and competing [Formula: see text] bonds between next-nearest-neighbours across only one of the diagonals of each square plaquette, the same diagonal in each square. In a topologically equivalent triangular-lattice geometry the model has two sorts of nearest-neighbour bonds, with [Formula: see text] bonds along parallel chains and with J1 bonds providing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at one extreme (κ = 0) and a set of decoupled chains at the other (κ → ∞), with the isotropic HAF on the triangular lattice in between at κ = 1. For the spin-1/2 [Formula: see text] model, we find a weakly first-order (or possibly second-order) quantum phase transition from a Néel-ordered state to a helical state at a first critical point at κc1 = 0.80 ± 0.01, and a second critical point at κc2 = 1.8 ± 0.4 where a first-order transition occurs between the helical state and a collinear stripe-ordered state. For the corresponding spin-1 model we find an analogous transition of the second-order type at κc1 = 0.62 ± 0.01 between states with Néel and helical ordering, but we find no evidence of a further transition in this case to a stripe-ordered phase.
We study the spin-1/2 and spin-1 J1-J2 Heisenberg antiferromagnets (HAFs) on an in–nite, anisotropic, two-dimensional triangular lattice, using the coupled cluster method. With respect to an underlying square–lattice geometry the model contains antiferromag- netic (J1 > 0) bonds between nearest neighbours and competing (J02 > 0) bonds between next-nearest-neighbours across only one of the diagonals of each square plaquette, the same diagonal in each square. In a topologically equivalent triangular-lattice geometry the model has two sorts of nearest-neighbour bonds, with bonds along parallel chains and with J1 bonds providing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at one extreme and a set of de- coupled chains at the other, with the isotropic HAF on the triangular lattice in between at k = 1. For the spin-1/2 J1-J2 2 model, we a weakly order (or possibly second-order) quantum phase transition from a eel-ordered state to a helical state at critical point at and a second critical point at where a transition occurs between the helical state and a collinear stripe- ordered state. For the corresponding spin-1 model we an analogous transition of the second-order type at between states with and helical ordering, but we no evidence of a further transition in thiSocase to a stripe-ordered phase.
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.
The coupled cluster method (CCM) is a well-known method of quantum many-body theory, and in this article we present an application of the CCM to the spin-half ${J}_{1}{\ensuremath{-}J}_{2}$ quantum spin model with nearest- and next-nearest-neighbor interactions on the linear chain and the square lattice. We present results for ground-state expectation values of such quantities as the energy and the sublattice magnetization. The presence of critical points in the solution of the CCM equations, which are associated with phase transitions in the real system, is investigated. Completely distinct from the investigation of the critical points, we also make a link between the expansion coefficients of the ground-state wave function in terms of an Ising basis and the CCM ket-state correlation coefficients. We are thus able to present evidence of the breakdown, at a given value of J2/J1,{J}_{2}{/J}_{1},J2​/J1​, of the Marshall-Peierls sign rule which is known to be satisfied at the pure Heisenberg point (J2=0){(J}_{2}=0)(J2​=0) on any bipartite lattice. For the square lattice, our best estimates of the points at which the sign rule breaks down and at which the phase transition from the antiferromagnetic phase to the frustrated phase occurs are, respectively, given by ${J}_{2}{/J}_{1}\ensuremath{\approx}0.26$ and ${J}_{2}{/J}_{1}\ensuremath{\approx}0.61.$
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.