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Physical review. B, Condensed matter

Exact diagonalization of finite frustrated spin-(1/2 Heisenberg models

Francisco E. Figueirido, A. Karlhede, Steven Allan Kivelson, Shivaji L. Sondhi, Martin Roček B, Daniel S. Rokhsar

Published Mar 1, 1990• 137 citations
Publisher

Abstract

We have studied the spectrum of low-spin eigenstates of a spin-(1/2 Heisenberg model on 4\ifmmode\times\else\texttimes\fi{}4 square lattices with up to third-neighbor antiferromagnetic exchange interactions. We have computed the staggered susceptibility, spin-spin correlation function, and the overlap of the exact ground state with various variational spin-density-wave ordered states as well as with the equal-amplitude, nearest-neighbor resonating-valence-bond state (which is the prototypical short-ranged RVB state). We find evidence for a complicated phase diagram with a region in which a short-ranged RVB state may be the ground state.

DOI
10.1103/physrevb.41.4619

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arXiv:cond-mat/0208371•Aug 19, 2002•0 citations
We have studied the spectrum of low-spin eigenstates of a spin-(1/2 Heisenberg model on 4\ifmmode\times\else\texttimes\fi{}4 square lattices with up to third-neighbor antiferromagnetic exchange interactions. We have computed the staggered susceptibility, spin-spin correlation function, and the overlap of the exact ground state with various variational spin-density-wave ordered states as well as with the equal-amplitude, nearest-neighbor resonating-valence-bond state (which is the prototypical short-ranged RVB state). We find evidence for a complicated phase diagram with a region in which a short-ranged RVB state may be the ground state.
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 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 phase diagram of a spin-1/2 $J\ensuremath{-}{J}^{\ensuremath{'}}\ensuremath{-}{J}_{2}$ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ model on the square lattice with two different nearest-neighbor couplings ${J,J}^{\ensuremath{'}}$ and may also be viewed as an array of coupled Heisenberg chains. The results suggest that the resonating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B 67, 024422 (2003)] for ${J}_{2}{=0.5J}^{\ensuremath{'}}\ensuremath{\ll}J$ is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both sides by a columnar dimer long range order. This columnar order could extend for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}J$ which correspond to the J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ model.
The phase diagram of a spin-1/2 J−J′−J2J-J'-J_2J−J′−J2​ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the J1−J2J_1-J_2J1​−J2​ model on the square lattice with two different nearest-neighbor couplings J,J′J,J'J,J′ and may be also viewed as an array of coupled Heisenberg chains. The results suggest that the resonnating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B {\bf 67}, 024422 (2003)] for J2=0.5J′≪JJ_2=0.5J' \ll JJ2​=0.5J′≪J is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both side by a columnar dimer long range order. This columnar order could extends for J′→JJ'\to JJ′→J which correspond to the J1−J2J_1-J_2J1​−J2​ model.
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.
Twenty-five years after the first proposal, the question whether the ground state of a frustrated spin-half system is well described by a spin-liquid Resonating Valence Bond (RVB) wave function is still controversial. A physically transparent representation of a RVB state can be obtained in fermionic representation with a standard BCS-type pairing wave function, working in the subspace with fixed number of electrons and no double occupancies. In this work, we show that, using this variational wave function with a careful parameterization of the pairing function, it is possible to obtain an extremely accurate {\em ansatz} for the ground state of the Heisenberg antiferromagnet with next-nearest neighbors interactions (J1−J2J_{1}{-}J_{2}J1​−J2​ model) in the regime of strong frustration. Indeed, in the spin-half realization of this model, it is known that the combined effect of frustration and zero-point motion interferes with the mechanism of spontaneously broken symmetry, giving rise to a non-magnetic phase of purely quantum-mechanical nature (J2/J1≃0.5J_2/J_1\simeq 0.5J2​/J1​≃0.5). This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids.