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Journal of the Physical Society of Japan

Exact Dimer State of Quantum Spin Chain with Long-Range Interactions

Hiroki Nakano, Minoru Takahashi

Published Aug 15, 1995• 6 citations
Publisher

Abstract

Quantum Heisenberg antiferromagnets in one dimension with long-range interactions are studied; a Hamiltonian which has dimer states composed of nearest-neighbor singlet pairs as eigenstates of the system is obtained. We also obtain sufficient conditions under which the eigenstates are the ground states. We propose a model which is very similar to the Haldane-Shastry model but has the dimer ground states.

DOI
10.1143/jpsj.64.2762

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Quantum Heisenberg antiferromagnets in one dimension with long-range interactions are studied; a Hamiltonian which has dimer states composed of nearest-neighbor singlet pairs as eigenstates of the system is obtained. We also obtain sufficient conditions under which the eigenstates are the ground states. We propose a model which is very similar to the Haldane-Shastry model but has the dimer ground states.
One-dimensional S =1/2 and S =1 spin systems with interactions between not only nearest-neighbor pairs of spins but also longer-range pairs are studied. We find the necessary and sufficient conditions between the coupling constants of the interactions under which the dimer state is an eigenstate of the system. We present a sufficient condition of the dimer ground state for extracted Hamiltonians. Properties of the ground state outside the dimer phase are also discussed by means of an analytical way and numerical diagonalizations.
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 present a theory of the realization of a ferromagnetic Haldane state in a spin-2 bilinear-biquadratic spin system on an orthogonal-dimer chain. The coexistence of a ferromagnetic state and a Haldane state is due to the rigorous correspondence between the eigenstates of a spin-2 model and a spin-12\frac{1}{2}21​ Heisenberg model, i.e., ``eigensystem embedding.'' Numerical exact-diagonalization calculations indicate that the ground state in the model is a fractionally magnetized M=3/4M=3/4M=3/4 Haldane state. Moreover, a ferromagnetic-dimer multiplet state is an exact ground state on a lattice, where the direct product of dimer singlet states is the ground state in a spin-12\frac{1}{2}21​ Heisenberg model that includes one-, two-, and three-dimensional orthogonal-dimer lattices. Eigensystem embedding demonstrates that a quantum ferromagnet can be obtained for an arbitrary spin $S\ensuremath{\geqq}2$ in any dimension and for any lattice in which anomalous ground states are realized in a spin-12\frac{1}{2}21​ Heisenberg model.
The ground state of a spin-1/2 Heisenberg chain with both frustration and long-range interactions is studied using Lanczos exact diagonalization. The evolution of the well-known dimerization transition of the system with short-range frustrated interactions (the J1-J2 chain) is investigated in the presence of additional unfrustrated interactions decaying with distance as 1/r(alpha). It is shown that the continuous (infinite-order) dimerization transition develops into a first-order transition between a long-range ordered antiferromagnetic state and a state with coexisting dimerization and critical spin correlations at wave number k=pi/2. The relevance of the model to real systems is discussed.