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Physical review. B, Condensed matter

Long-ranged interactingS=1 spin chain with the exact valence-bond-solid state

Hiroki Nakano, Minoru Takahashi

Published Oct 1, 1996• 7 citations
Publisher

Abstract

A one-dimensional S=1 spin system with bilinear and biquadratic interactions between not only nearest-neighbor pairs of spins but also longer-range pairs is studied. We find the necessary and sufficient conditions between the coupling constants of the interactions under which the valence-bond-solid state is an eigenstate of the system. We consider in detail the interactions up to the next-nearest neighbors and report numerical results for the region where the valence-bond-solid state is the ground state. In a special case, some exact excited states are constructed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.9000

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A one-dimensional S=1 spin system with bilinear and biquadratic interactions between not only nearest-neighbor pairs of spins but also longer-range pairs is studied. We find the necessary and sufficient conditions between the coupling constants of the interactions under which the valence-bond-solid state is an eigenstate of the system. We consider in detail the interactions up to the next-nearest neighbors and report numerical results for the region where the valence-bond-solid state is the ground state. In a special case, some exact excited states are constructed. \textcopyright{} 1996 The American Physical Society.
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For special coupling ratios, the one-dimensional (1D) s=1/2 Heisenberg model with antiferromagnetic nearest and next-nearest neighbor interactions has a pure dimer ground state, and the 1D s=1 Heisenberg model with antiferromagnetic bilinear and biquadratic interactions has an exact valence-bond-solid ground state. The recursion method is used to calculate the T=0 spin dynamic structure factor for both models and, for the s=1/2 model, also the dimer dynamic structure factor. New results for line shapes and dynamically relevant dispersions are obtained.
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We investigate the ground-state phase diagram of the quantum mixed spin chain in which two S ⁣= ⁣12S\!=\!\frac{1}{2}S=21​ and two S =1 spins are arranged alternatively and isotropic exchange interactions between nearest-neighboring pairs of spins are assumed. Guided by our previous results [J. Phys. Soc. Jpn. 67 (1998) 1000] for a few limiting cases of the exchange interactions, we carry out a density-matrix renormalization-group calculation to determine the ground-state phase diagram in detail. From the gapless points of the singlet-triplet energy gap, five boundary lines are determined, which divide the phase diagram into six regions. In three regions the ground states with the Haldane-phase character appear, while in the remaining three regions those with the dimer character appear. We also present a characterization of the phases concerning how the ground state in each phase is represented in the framework of the valence-bond-solid (VBS) picture. Making use of the matrix-product wave function, we classify the VBS pictures and interpret the ground-state phase diagram in terms of symmetries schematically related to the VBS pictures. We investigate the ground-state nearest-neighboring two-spin correlation functions and a kind of string order parameters, by which we confirm the validity of the characterization.
Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.
We introduce a family of Jastrow pair product states for quasi-one-dimensional spin systems. Depending on a parameter they interpolate between the resonating valence-bond ground state of the Haldane-Shastry model describing a spin liquid and the (dimerized) valence-bond solid ground states of the Majumdar-Ghosh spin chain. These states are found to form an excellent basis for variational studies of Heisenberg chains with next-nearest-neighbor interactions and bond alternation as realized in the spin-Peierls system CuGeO3{}_{3}3​.