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Progress of Theoretical Physics

Spin-Wave Wave Function for Quantum Spin Models

Franjo Franjic, Sandro Sorella

Published Mar 1, 1997• 36 citations
Publisher

Abstract

We present a new approach to determine an accurate variational wave function for general quantum spin models, completely defined by a consistency requirement with the simple and well-known linear spin-wave expansion. With this wave function, it is also possible to obtain the correct behavior of the long distance correlation functions for the 1D S = 1/2 antiferromagnet. In 2D the proposed spin-wave wave function represents an excellent approximation to the exact ground state of the S = 1/2 XY model. We obtained accurate values for the correlation functions and discuss their physical relevance.

DOI
10.1143/ptp.97.399

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Physical review. B, Condensed matter•Mar 1, 1990•238 citations
We present a new approach to determine an accurate variational wave function for general quantum spin models, completely defined by a consistency requirement with the simple and well-known linear spin-wave expansion. With this wave function, it is also possible to obtain the correct behavior of the long distance correlation functions for the 1D S = 1/2 antiferromagnet. In 2D the proposed spin-wave wave function represents an excellent approximation to the exact ground state of the S = 1/2 XY model. We obtained accurate values for the correlation functions and discuss their physical relevance.
We present here a new approach to determine an accurate variational wavefunction for general quantum antiferromagnets, completely defined by the requirement to reproduce the simple and well known spin-wave expansion. By this wavefunction, it is possible to obtain the correct behavior of the long distance correlation functions for the one dimensional S=1/2S=1/2S=1/2 antiferromagnet, i.e. {\em a system without long range order}. The qualitative difference between the integer and half integer case is also easily understood with this variational approach. Finally we present numerical results for the 2D XY model, showing that the present wavefunction has an overlap $> 0.99 $ to the exact ground state of the S=1/2S=1/2S=1/2 model for finite system up to 6×66 \times 66×6 clusters.
We present a new method for a systematic spin-wave expansion for the quantum fluctuations of a generic spin Hamiltonian in a finite lattice, where the inverse spin magnitude 1/ S is a well-defined expansion parameter. The first two leading contributions of the spin-spin correlation function are evaluated for the J 1 - J 2 Heisenberg model. Very good agreement between our finite-size predictions and the exact diagonalization and Monte Carlo results is found for J 2 / J 1 < 0.2 and S = 1/2, thus confirming the existence of antiferromagnetic long-range order in this J region. For J 2 / J 1 > 0.3 the expansion is poorly converging, suggesting a possible breakdown of the spin-wave approximation. Here our calculation seems consistent with a possible spin liquid ground state.
The ground-state properties of the spin-(1/2 Heisenberg antiferromagnet on a square lattice are studied by using a simple variational wave function that interpolates continuously between the N\'eel state and short-range resonating-valence-bond states. Exact calculations of the variational energy for small systems show that the state with the lowest energy has long-range antiferromagnetic order. The staggered magnetization in this state is approximately 70% of its maximum possible value. The variational estimate of the ground-state energy is substantially lower than the value obtained for the nearest-neighbor resonating-valence-bond wave function.
We present a systematic formalism for the variational evaluation of ground-state properties of Hubbard-type models in finite dimensions d. The formalism starts from generalized Gutzwiller correlated wave functions, which are then studied in a systematic (1/d) expansion around the limit of high dimensions (d=\ensuremath{\infty}). The limit of d\ensuremath{\rightarrow}\ensuremath{\infty} has recently been introduced by Metzner and Vollhardt (MV) for itinerant lattice fermion systems. The approach, presented in this paper, is particularly efficient since results in d=\ensuremath{\infty} are obtained without having to calculate a single graph. Our results confirm the finding of MV that counting approximations in the spirit of the Gutzwiller approximation become exact in d=\ensuremath{\infty} for translationally invariant wave functions. This type of approximation is no longer exact for more complicated (e.g., antiferromagnetic) wave functions. In addition, we completely reproduce the results of the Kotliar-Ruckenstein path-integral approach to the Hubbard model. Performing a (1/d) expansion for the Gutzwiller wave function, we show that the lowest orders in (1/d) are sufficient to reproduce all numerical findings in d=2,3 quantitatively. We therefore conclude that the limit of d=\ensuremath{\infty} is a very fruitful starting point for the study of finite-dimensional systems. On the basis of our study we propose new variational wave functions for the numerical investigation of antiferromagnetism in the Hubbard, the t-J, and the spin-1/2 Heisenberg model. For the first two models we calculate in d=\ensuremath{\infty} only, for the Heisenberg model we also derive corrections up to order (1/d). To this order we obtain complete agreement with linear spin-wave theory. Since our trial state is based on a fermionic description of the Heisenberg model, we interpret this analytically determined wave function as Fermi sea of spin-1/2 quasiparticles (``spinons'').
In this article, the correlated basis-function (CBF) method is applied for the first time to the quantum spin-half XY model on the linear chain, the square lattice, and the simple cubic lattice. In this treatment of the quantum spin-half XY model a Jastrow ansatz is utilised to approximate the ground-state wave function. Results for the ground-state energy and the sublattice magnetisation are presented, and evidence that the CBF detects the quantum phase transition point in this model is also presented. The CBF results are compared to previous coupled cluster method (CCM) results for the spin-half XY model, and the two formalisms are then compared and contrasted. In this paper we consider the T = 0 properties of the quantum spin system known as the spin-half XY model, described by the Hamiltonian
We have studied via a Green-function Monte Carlo (GFMC) method the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions. We use a well-known transformation to map the spin problem onto a system of hard-core bosons that allows us to exploit interesting analogies between magnetism and superfluidity. The GFMC method is a zero-temperature stochastic method that projects out the component of the true ground state in a given variational wave function. This method is complementary to previously used finite-temperature Monte Carlo methods and is well suited to studying the ground state and low-lying excited states. Starting with even a simple wave function, e.g., the classical N\'eel state, the GFMC method can obtain the short-range correlations very accurately, and we find the ground-state energy per site E0{\mathit{E}}_{0}E0​/J=-0.6692(2). We show that it is important to include the zero-point motion of the elementary excitations in the ground state and by a spin-wave analysis find that it produces long-range correlations in the wave function. Upon inclusion of such long-range correlations, we obtain a staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) and the structure factor scrS(q)\ensuremath{\sim}q at long wavelengths. Using the Feynman-Bijl relation, from the slope we deduce the renormalization of the spin-wave velocity by quantum fluctuations to be Zc{\mathit{Z}}_{\mathit{c}}Zc​==c/cs{\mathit{c}}_{\mathit{s}}cs​=1.14(5).