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Physical review. B, Condensed matter

Ground-state spin pairing of general-spin linear-chain Heisenberg models

S. A. Alexander, Daniel Klein

Published Jan 1, 1985• 0 citations
Publisher

Abstract

Simple products of singlet-coupled spin pairs are identified as exact eigenstates for a certain class of antiferromagnetically signed linear-chain Heisenberg models with both bond-length alternation and next-nearest-neighbor interactions. Where such an eigenstate becomes a ground state, a cusp is noted to arise in the ground-state energy. Differences between the spin-1/2 and higher-spin cases appear.

DOI
10.1103/physrevb.31.574

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Simple products of singlet-coupled spin pairs are identified as exact eigenstates for a certain class of antiferromagnetically signed linear-chain Heisenberg models with both bond-length alternation and next-nearest-neighbor interactions. Where such an eigenstate becomes a ground state, a cusp is noted to arise in the ground-state energy. Differences between the spin-1/2 and higher-spin cases appear.
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 consider a class of geometrically frustrated Heisenberg spin systems which admit exact ground states. The systems consist of suitably coupled antiferromagnetic spin trimers with integer spin quantum numbers s and their ground state Φ will be the product state of the local singlet ground states of the trimers. We provide linear equations for the inter-trimer coupling constants which are equivalent to Φ being an eigenstate of the corresponding Heisenberg Hamiltonian and sufficient conditions for Φ being a ground state. The classical case s → ∞ can be completely analyzed. For the quantum case we consider a couple of examples, where the critical values of the inter-trimer couplings are numerically determined. These examples include chains of corner sharing tetrahedra as well as certain spin tubes. Φ is proven to be gapped in the case of trimer chains. This follows from a more general theorem on quantum chains with product ground states.
Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction J2{\mathit{J}}_{2}J2​=1/2J1{\mathit{J}}_{1}J1​, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations J1{\mathit{J}}_{1}J1​-2J2{\mathit{J}}_{2}J2​+J3{\mathit{J}}_{3}J3​=0 and J3{\mathit{J}}_{3}J3​=2J4{\mathit{J}}_{4}J4​ hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-j2{\mathit{j}}_{2}j2​ around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.
We discuss spin-SSS antiferromagnetic Heisenberg chains with three-spin interactions, next-nearest-neighbor interactions, and bond alternation. First, we prove rigorously that there exist parameter regions of the exact dimerized ground state in this system. This is a generalization of the Majumdar-Ghosh model to arbitrary SSS. Next, we discuss the ground-state phase diagram of the models by introducing several effective field theories and the universality classes of the transitions are described by the level-2S2S2S SU(2)\mathrm{SU}(2)SU(2) Wess-Zumino-Witten model and the Gaussian model. Finally, we determine the phase diagrams of S=1S=1S=1 and S=3/2S=3/2S=3/2 systems by using exact diagonalization and level spectroscopy.