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Journal of the Physical Society of Japan

A Variational Approach to Understanding the Ground State of a Square-Lattice Heisenberg Antiferromagnet

Shin-ichi Nakagawa, Takehiko Hamada, Jun-ichi Kane, Yuhei Natsume

Published Apr 15, 1990• 8 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.59.1131

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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 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.
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.
'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.
We consider a resonating-valence-bond state as a trial wave function for the ground state of the antiferromagnetic Heisenberg model on the square lattice. The energy and the staggered spin-spin correlation functions are calculated on a small lattice and are compared with the results of the exact diagonalization of Oitmaa and Betts. The trial energy is only about 4.7% different from the exact ground-state energy on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The importance of the singlet pairs between the next-nearest antiferromagnetic sites is inferred. An argument against the previous interpretation of the numerical data as being evidence of long-range order is presented.
The ground state of the spin-\textonehalf{} Heisenberg model with antiferromagnetic coupling on a square lattice is analyzed in terms of nearest-neighbor resonating pair-bond states. Extrapolating from finite lattice calculations it is found that the pair-bond state provides a much closer variational estimate of the ground-state energy than the widely used Gutzwiller wave function. The effect of an orthorhombic distortion of the square lattice, such as that observed in the Cu-O planes in the high-Tc{T}_{c}Tc​ oxides, on the spin state is analyzed by exact calculation for finite lattices. From total-energy considerations it is found that the system has a slight tendency towards such a lattice distortion.
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.