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Journal of Physics A Mathematical and General

Exact ground states for a class of antiferromagnetic Heisenberg models with short-range interactions

Daniel Klein

Published Feb 1, 1982• 100 citations
Publisher

Abstract

A special class of Heisenberg spin Hamiltonians with short-range antiferromagnetically signed interactions is described. It is demonstrated via a simple group-theoretic argument that the ground states to many of these Hamiltonians are Kekule states, involving singlet spin-pairing between nearest-neighbour pairs of sites. A number of examples are given and previous related work is indicated. In addition to standard applications to exchange-coupled atomic ions, the relevance of this work for the simple valence-bond model of conjugated pi -electron networks is mentioned.

DOI
10.1088/0305-4470/15/2/032

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A special class of Heisenberg spin Hamiltonians with short-range antiferromagnetically signed interactions is described. It is demonstrated via a simple group-theoretic argument that the ground states to many of these Hamiltonians are Kekule states, involving singlet spin-pairing between nearest-neighbour pairs of sites. A number of examples are given and previous related work is indicated. In addition to standard applications to exchange-coupled atomic ions, the relevance of this work for the simple valence-bond model of conjugated pi -electron networks is mentioned.
We give an explicit proof that the completely dimerized (’Kekulè or nearest-neighbor resonant valence bond) state is the degenerate ground state of a linear chain S =1/2 Heisenberg antiferromagnet, with competing couplings extending up to 2n nearest neighbors, if the relation J1 = 2n, J2 = 2n − 1,...,J2n = 1, holds between the various exchange integrals. The previously known result for J2 =1/2 J1 is then just a special case corresponding to n=1. We study in detail n=2 (fourth-nearest neighbors) and show that this state is the ground state of the Hamiltonian when J1 − 2J2 + J3 = 0, J3 = 2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0 ≤ J4 ≤1/4 J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising models is made.
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.
We study the spin-1/2 antiferromagnetic Heisenberg Hamiltonians with nearest-neighbour pair interaction on some graphs with an odd number of vertices, called triangulated trees. The model has valence bond ground states in which there is a localized spinon (1/2 spin) on some site. The space of ground states for a tree of L triangles is (L+1) dimensional. In the limit of infinite volume we find an infinity of pure ground states. To any (infinite) branch of the tree there belongs a spin-Peierls ground state without spinon. There is an infinite family of equivalent pure ground states differing only locally, in the position of the spinon. In any finite-volume ground state the pair correlations decay exponentially. For the chain of triangles we show the existence of a gap in the energy spectrum above the ground state
We study a class of infinite-range spin-(1/2 quantum Heisenberg antiferromagnets. We show that, in addition to ground states with macroscopic net spin in one or several sublattices (``N\'eel states''), there are valence-bond (VB) states that are local spin singlets. We show that there is a class of perturbations that stabilizes the VB state. We characterize the excitation spectrum and find an order parameter for the VB states. Finally, we extend our method to find perturbations that stabilize microscopic as well as macroscopic chiral states.
We present a new class of variational wavefunctions that can be used to investigate quantum antiferromagnetic systems. The states are a natural generalization of the nearest-neighbour dimer phases which have been thoroughly studied in the literature. We describe a one-dimensional class of total-spin-singlet wavefunctions, which include as special cases the two nearest-neighbour dimer phases and the exact solution to the total-spin-two projector Hamiltonian. These different solutions are 'smoothly' interpolated by a range of singlets with short-range correlations. Although the states have only short-range order, we can obtain over 98% of the ground-state energy of the nearest-neighbour Heisenberg model. We apply our basis to the J 1 -J 2 Heisenberg model and the single hole in the t 1 -t 2 infinite-U Hubbard model, successfully predicting the behaviour observed in finite-size scaling calculations.