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International Journal of Modern Physics B

TWO-DIMENSIONAL ANTIFERROMAGNETIC HEISENBERG MODEL: EXACT AND RVB STATES

Didier Poilblanc

Published Oct 1, 1988• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217979288000810

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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 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 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.
'Resonating-valence-bond' (RVB) variational wavefunctions are numerically investigated in order to describe the ground state and the singlet low-lying excited states of the spin-1/2, J 1 -J 2 Heisenberg antiferromagnet with 1/8<or approximately=J 2 /J 1 <or approximately=1 on triangular lattices of N=4p sites compatible with the classical Neel orders. Comparison with previous exact diagonalization results indicates that the singlet eigenstates associated with collinear Neel order, among which the ground state is found, can be accurately described with appropriate superpositions of two-sublattice RVB trial states, while the other singlet low-lying eigenstates associated with four-sublattice Neel order can be well approximated from the four-sublattice RVB trial states.
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.
The theory of the resonating valence bond (RVB) for the antiferromagnetic Heisenberg model of spin 1/2 proposed by Anderson is generalized by taking account of singlet bonds connecting distant spins, and is solved exactly for the finite lattice. It is shown that the RVB wave function is S tot (total spin quantum number)=0, and that replacing a part of singlet bonds by triplet bonds, eigenfunctions with S tot =1, 2, … can be constructed. These are the new representation of eigenfunctions of the Heisenberg Hamiltonian.