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Journal of Physics Condensed Matter

Variational RVB wavefunctions for the spin-1/2 J1-J2Heisenberg antiferromagnet on the triangular lattice

Philippe Sindzingre, B. Bernu, P. Lecheminant

Published Nov 13, 1995• 4 citations
Publisher

Abstract

'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.

DOI
10.1088/0953-8984/7/46/010

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'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 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 investigate a class of resonating-valence-bond wave functions on the triangular lattice which interpolate between the \ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 N\'eel state and a dimer state according to the range of the bonds and the similar two classes of resonating-valence-bond wave functions on the kagome$aa--- lattice constructed from the \ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 and q=0 Ne\'el states. Numerical calculations show that a \ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 wave function gives for the triangular lattice a variational energy and spin-spin correlations in very good agreement with diagonalization results on 12- and 36-site systems. Rather low variational energies are also obtained with trial functions of the \ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 and q=0 type for the kagome$aa--- lattice but spin-spin correlations beyond first neighbors are not in good agreement with diagonalization results on 12- and 36-site systems. For the 12-site system, the spin-spin correlations of the best q=0 wave function most resemble those of the first excited state. The q=0 and perhaps the \ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 wave functions may describe excited states close to the ground state in the case of larger systems.
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 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 spin-1/2 antiferromagnetic Heisenberg and X Y models are investigated on the triangular lattice by numerically diagonalizing finite size systems (up to 21 spins). The ground state energies of both models are found to be close to other theoretical predictions except those of Marland and Betts who performed similar analysis to ours. We have numerically constructed a resonating valence bond state of Anderson. It has a relatively large projection on the ground state of the finite Heisenberg model, although the absolute value of the projection decreases linearly with the system size. The trial function proposed by Miyashita for the X Y model does not have an appreciable projection on the numerically exact state. We present evidence for the absence of a sublattice long range order in the Heisenberg model and the existence of the same quantity in the X Y model.