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Physical review. B, Condensed matter

Exact states and variational states of a periodic two-dimensional cluster of the Heisenberg model

Didier Poilblanc

Published Jan 1, 1989• 25 citations
Publisher

Abstract

The exact many-body states of a 10-site square cluster with periodic boundary conditions of an S=(1/2 antiferromagnetic Heisenberg lattice model are investigated. A method that takes full advantage of the symmetries of the system (the cluster space-group symmetry and the total spin symmetry) greatly reduces the size of the Hamiltonian matrix and gives an analytic solution. A general form of a resonating-valence-bond wave function, written in a fermion representation, is examined. It is shown that the optimum variational wave function has complex singlet bond distribution amplitudes of s-wave + imaginary d-wave symmetry. It has a very large overlap with the ground state (0.995), belongs to the same space-group representation, and its energy functional has a minimum very close to the ground-state energy. On the other hand, if the amplitudes are restricted to real numbers, the local minimum has slightly higher energy and is nondegenerate but has an infinite manifold of equivalent representations in terms of projected BCS states related by a local SU(2) gauge transformation.

DOI
10.1103/physrevb.39.140

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International Journal of Modern Physics B•Oct 1, 1988•2 citations
The exact many-body states of a 10-site square cluster with periodic boundary conditions of an S=(1/2 antiferromagnetic Heisenberg lattice model are investigated. A method that takes full advantage of the symmetries of the system (the cluster space-group symmetry and the total spin symmetry) greatly reduces the size of the Hamiltonian matrix and gives an analytic solution. A general form of a resonating-valence-bond wave function, written in a fermion representation, is examined. It is shown that the optimum variational wave function has complex singlet bond distribution amplitudes of s-wave + imaginary d-wave symmetry. It has a very large overlap with the ground state (0.995), belongs to the same space-group representation, and its energy functional has a minimum very close to the ground-state energy. On the other hand, if the amplitudes are restricted to real numbers, the local minimum has slightly higher energy and is nondegenerate but has an infinite manifold of equivalent representations in terms of projected BCS states related by a local SU(2) gauge transformation.
The exact many-body states of a 10-site square cluster with periodic boundary conditions of a S = 1/2 antiferromagnetic Heisenberg lattice model are investigated. A general form of a Resonating Valence Bond wavefunction (RVB), written in fermion representation is examined. It is shown that the optimum variational wavefunction has complex singlet bond distribution amplitudes of s+id symmetry . It has a very large overlap with the groundstate, belongs to the same space group representation and its energy functional has a minimum very close to the ground-state energy. On the other hand, if the amplitudes are restricted to real numbers, the local minimum has slightly higher energy and d-wave symmetry.
The trial function including a few variational parameters introduced by the consideration of geometrical characters is proposed for the purpose of discussing the exact ground-state wave function of spin-1/2 Heisenberg antiferromagnet on a square lattice. In this wave function, the essential contribution of singlet pairings with larger separations between spin sites rather than the 1st nearest neighbor is taken into account. The obtained solution for spin number N ≤26 on the basis of this extended RVB (resonating-valence-bond) picture comes quite close to the exact solution, though we use only a few geometrical parameters for the systems constituted by many degrees of freedom.
A resonating-valence-bond (RVB) wave function including the nearest- and next-nearest-neighbor singlets is proposed for the S=1/2 Heisenberg antiferromagnetic chain with competing interactions. A variational upper bound for the ground-state energy is provided. Spin-spin correlation functions are calculated analytically and compared with those obtained by finite-size exact diagonalizations. Within this RVB ansatz, the spatial periodicity of the ground-state wave function, which is defined as the phase modulation of spin-spin correlation functions, is found to undergo a discontinuous transformation from two-fold to four-fold as the relative strength of the competing interactions is varied. We further discuss this transformation in terms of the Marshall-Peierls sign rule.
The ground states of the antiferromagnetic Heisenberg model on the triangular and square lattices are discussed with use of the combinatorial method on the basis of Anderson's RVB (resonating-valence-bond) variational wave-function. Some properties of RVB states are studied systematically for finite system-size N up to 20 and are compared to the exact values. For the triangular lattice the estimated energy value E RVB =-2.08 is closer to the exact value than that of the spin-wave theory. The value E RVB =-2.36 is estimated for the square lattice. For the square lattice our RVB state is closer to the exact ground state than the Néel state is in the sense of a wave-function. The dimer problem has a close relation to RVB states and is used to count exactly the number of singlet-pair configurations.
We study a variational wave function for the ground state of the two-dimensional S=1∕2S=1∕2S=1∕2 Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes h(x,y)h(x,y)h(x,y) for valence bonds connecting spins separated by (x,y)(x,y)(x,y) lattice spacings. In contrast to previous studies, in which a functional form for h(x,y)h(x,y)h(x,y) was assumed, we here optimize all the amplitudes for lattices with up to $32\ifmmode\times\else\texttimes\fi{}32$ spins. We use two different schemes for optimizing the amplitudes; a Newton conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only $\ensuremath{\approx}0.06%$ from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling $\ensuremath{\approx}2%$ below the exact ones at long distances (corresponding to an $\ensuremath{\approx}1%$ underestimation of the sublattice magnetization). The amplitudes h(r)h(r)h(r) for valence bonds of long length rrr decay as ${r}^{\ensuremath{-}3}$. We also discuss some results for small frustrated lattices.
Following the discovery of high Tc superconductivity in the copper oxides, there has been a great deal of interest in the RVB wave function proposed by Anderson [1]. As a warm-up exercise we have considered a valence-bond wave function for the one dimensional spin-1/2 Heisenberg chain. The main virtue of our work is to propose a new variational singlet wavefunction which is almost analytically tractable by a transfer-matrix technique. We have obtained the ground state energy for finite as well as infinite chains, in good agreement with exact results. Correlation functions, excited states, and the effects of other interactions (e.g., spin-Peierls) are also accessible within this scheme [2]. Since the ground state of the chain is known to be a singlet (Lieb & Mattis [3]), we write the appropriate wave function as a superposition of valence-bond singlets, [Formula: see text], where | k > is a spin configuration obtained by pairing all spins into singlet pairs, in a way which is common in valence-bond calculations of large molecules. As in that case, each configuration, | k >, can be represented by a Rümer diagram, with directed bonds connecting each pair of spins on the chain. The c k 's are variational co-efficients, the form of which is determined as follows: Each singlet configuration (Rümer diagram) is divided into "zones", a "zone" corresponding to the region between two consecutive sites. Each zone is indexed by its distance from the end of the chain and by the number of bonds crossing it. Our procedure assigns a variational parameter, x ij , to the j th zone, when crossed by i bonds. The resulting wavefunction for an N-site chain is written as [Formula: see text] where m ij(k) equals 1 when zone j is crossed by i bonds and zero otherwise. To make the calculation tractable we reduce the number of variational parameters by disallowing configurations with bonds connecting any two sites separated by more than 2M lattice points. (For simplicity, we have limited ourselves to M=3, but the scheme can be used for any M). With the simple ansatz, matrix elements can be calculated by a transfer-matrix method. To understand the transfer-matrix method note that since only local zone parameters appear in the description of each state | k >, matrix elements and overlaps, [Formula: see text] and < k | k '>, are completely specified by a small number of "local states" associated with each zone. Within a given zone a local state is defined by (i) the number of bonds crossing the zone and (ii), by whether the bonds originate from the initial (| k >) or final (| k '>) state. It is then easy to see that "local states" of consecutive zones are connected by a 15 × 15 transfer matrix (for the case M=3). Furthermore, the overlap matrix element can be written as a product of transfer-matrices associated with each zone of the chain. When calculating matrix elements of the Hamiltonian, an additional matrix, U , must be defined, to represent the particular zone involving the two spins connected by the Heisenberg interaction. The relevant details as well as the comparison with exact results will be given elsewhere. We are planning to ultimately apply this method to the two dimensional case, and hope to include the effects of holes.