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Journal of the Physical Society of Japan

Long-Ranged Interacting Quantum Spin Chain with Exact Dimer State

Hiroki Nakano, Minoru Takahashi

Published Jan 15, 1997• 8 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.66.228

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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.
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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 condition in which the dimer-monomer state is exactly the ground state of an extended diamond chain with spin magnitude 1/2. The Hamiltonian of the extended diamond chain includes next-nearest-neighbor exchange interactions and distortions, where the spin magnitude of the spin pair on a singlet dimer is not generally conserved. The method of deriving the condition is based on representing the Hamiltonian in a complete square form. The dimer-monomer ground state is found even if the Hamiltonian has no space-reflection symmetries.
Interacting spin chains and ladders are known to support a plethora of quantum phases with complex ground-state phase diagrams. In this work, we study a large family of such models and determine precise, explicit conditions under which an exact dimer state is guaranteed to be the ground state. These general conditions are validated for various generalizations of the Majumdar-Ghosh model using exact diagonalization. Our results provide exact reference points in the phase diagrams of a wide class of spin chains and ladders, including those with anisotropic and arbitrary-range interactions.