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Modern Physics Letters B

GRASSMANNIAN APPROACH TO THE RESONATING-VALENCE-BOND STATE OF THE 2D HEISENBERG ANTIFERROMAGNET

Yonathan Shapir, Thomas Blum

Published Aug 1, 1989• 2 citations
Publisher

Abstract

A systematic representation of the resonating-valence-bond state of the 2D spin-1/2 Heisenberg antiferromagnet in terms of anticommuting Grassmann variables is derived. This representation is extended to the dilute case and the holon statistics are considered. Independent phase fluctuations on each valence-bond yield a U(1) gauge theory with low-lying gapless excitations in accordance with other approaches.

DOI
10.1142/s0217984989001424

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A systematic representation of the resonating-valence-bond state of the 2D spin-1/2 Heisenberg antiferromagnet in terms of anticommuting Grassmann variables is derived. This representation is extended to the dilute case and the holon statistics are considered. Independent phase fluctuations on each valence-bond yield a U(1) gauge theory with low-lying gapless excitations in accordance with other approaches.
We consider a resonating-valence-bond state as a variational estimate for the ground state of a Heisenberg antiferromagnet. We derive an expression for the normalization of this wave function, which is a superposition of a very large number of singlet pair states, which we then use to show that the singularities, correlations, and low-lying excitations of this quantum system can be calculated from the partition function of a related dimerlike classical statistical mechanics problem. Possible calculational schemes are discussed.
'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 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.
The low-energy spin excitations in the flux phase of the (2+1)-dimensional large-nnn SU(n)\mathrm{SU}(n)SU(n)Heisenberg antiferromagnet are relativistic fermions coupled to a U(1) gauge field. These excitations are studied in the physical n=2n=2n=2 case with the use of a 1n\frac{1}{n}n1​ expansion. A Chern-Simons term is included to remove gauge configurations forbidden in the n=2n=2n=2 theory.