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Physical review. B, Condensed matter

Investigation of different classes of variational functions for the triangular andkagome´ spin-1/2 Heisenberg antiferromagnets

Philippe Sindzingre, P. Lecheminant, C. Lhuillier

Published Aug 1, 1994• 55 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.50.3108

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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 kagomeaa−−−latticeconstructedfromthe\ensuremath√3\ifmmode×\else\texttimes\fi\ensuremath√3andq=0Neeˊlstates.Numericalcalculationsshowthata\ensuremath√3\ifmmode×\else\texttimes\fi\ensuremath√3wavefunctiongivesforthetriangularlatticeavariationalenergyandspin−spincorrelationsinverygoodagreementwithdiagonalizationresultson12−and36−sitesystems.Ratherlowvariationalenergiesarealsoobtainedwithtrialfunctionsofthe\ensuremath√3\ifmmode×\else\texttimes\fi\ensuremath√3andq=0typeforthekagomeaa--- 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 kagomeaa−−−latticeconstructedfromthe\ensuremath√3\ifmmode×\else\texttimes\fi\ensuremath√3andq=0Neeˊlstates.Numericalcalculationsshowthata\ensuremath√3\ifmmode×\else\texttimes\fi\ensuremath√3wavefunctiongivesforthetriangularlatticeavariationalenergyandspin−spincorrelationsinverygoodagreementwithdiagonalizationresultson12−and36−sitesystems.Ratherlowvariationalenergiesarealsoobtainedwithtrialfunctionsofthe\ensuremath√3\ifmmode×\else\texttimes\fi\ensuremath√3andq=0typeforthekagomeaa--- 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 class of singlet resonanting-valence-bond wave functions on a square lattice, with the bond-length distribution as a variational parameter. This class contains the two limiting cases of the dimer wave function and of the N\'eel state. We present numerical calculations of the energy and the spin-spin correlation functions up to very large lattice sizes (180\ifmmode\times\else\texttimes\fi{}180) both for disordered states with exponentially decaying correlation functions and for ordered states. The energy of a disordered state can be within 0.1% of our best ordered state ($\frac{\ensuremath{-}0.3344J}{\mathrm{bond}}$).
'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-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 study the spin-1∕21∕21∕2 Heisenberg model in a lattice that interpolates between the triangular and the kagome lattices. The exchange interaction along the bonds of the kagome lattice is JJJ, and the one along the bonds connecting kagome and nonkagome sites is ${J}^{\ensuremath{'}}$, so that ${J}^{\ensuremath{'}}=J$ corresponds to the triangular limit and ${J}^{\ensuremath{'}}=0$ to the kagome one. We use variational and exact diagonalization techniques. We analyze the behavior of the order parameter for the antiferromagnetic phase of the triangular lattice, the spin gap, and the structure of the spin excitations as functions of ${J}^{\ensuremath{'}}∕J$. Our results indicate that the antiferromagnetic order is not affected by the reduction of ${J}^{\ensuremath{'}}$ down to ${J}^{\ensuremath{'}}∕J\ensuremath{\simeq}0.2$. Below this value, antiferromagnetic correlations grow weaker, a description of the ground state in terms of a N\'eel phase renormalized by quantum fluctuations becomes inadequate, and the finite-size spectra develop features that are not compatible with antiferromagnetic ordering. However, this phase does not appear to be connected to the kagome phase as well, as the low-energy spectra do not evolve with continuity for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$ to the kagome limit. In particular, for any nonzero value of ${J}^{\ensuremath{'}}$, the latter interaction sets the energy scale for the low-lying spin excitations, and a gapless triplet spectrum, destabilizing the kagome phase, is expected.
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.