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Physical review. B, Condensed matter

Resonating-valence-bond states with antiferromagnetic long-range order

Michael Ma

Published Oct 1, 1988• 13 citations
Publisher

Abstract

We derive the singlet versions of the N\'eel states for certain n-sublattice systems, including the two-dimensional square lattice (previously considered by Carlson and Sethna) and the triangular lattice. These states are written explicitly in resonating-valence-bond form through the help of the Young-tableau formalism. Since these states are exact ground states for infinite-range models, they can be used as a test for the fermion and Schwinger boson mean-field theories (MFT). We show that the fermion MFT is incompatible with the ground state, while the boson MFT gives the exact ground state. Implications of the results with respect to the ability of the MFT's to describe antiferromagnetic long-range order are discussed.

DOI
10.1103/physrevb.38.6813

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We derive the singlet versions of the N'eel states for certain n-sublattice systems, including the two-dimensional square lattice (previously considered by Carlson and Sethna) and the triangular lattice. These states are written explicitly in resonating-valence-bond form through the help of the Young-tableau formalism. Since these states are exact ground states for infinite-range models, they can be used as a test for the fermion and Schwinger boson mean-field theories (MFT). We show that the fermion MFT is incompatible with the ground state, while the boson MFT gives the exact ground state. Implications of the results with respect to the ability of the MFT's to describe antiferromagnetic long-range order are discussed.
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.
A two-dimensional square-lattice Hubbard model is investigated. A slave-fermion formulation, in which the electron operator is expressed by a combination of bose operators for the spin freedom, and fermi operators for the charge freedom, is used. The phase diagram for the resonating valence bond (RVB) state is obtained. The RVB state shows long-range antiferromagnetic order at T =0, the period of which deviates from that of the commensurate Néel order as the system deviates from the half-filled case.
We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type, the spin-lattice coupling-induced VBS state can occur when 2S2S2S is not an integer multiple of the coordination number zzz. S=2S=2S=2 spins on the triangular lattice with z=6z=6z=6 is discussed as an example. Within the Schwinger boson mean-field theory the ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ modulation of the valence bond amplitudes. Excitation spectrum for the modulated valence bond state is worked out within the single-mode approximation. The spectrum offers a new collective mode, distinct from the spin wave excitations of the magnetically ordered ground state, and observable in neutron scattering.
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.