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arXiv:0801.1247

Exact symmetry breaking ground states for quantum spin chains

S. Alipour, Sima Baghbanzadeh, Vahid Karimipour

Published Jan 8, 2008• 3 citations
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Abstract

We introduce a family of spin-1/2 quantum chains, and show that their exact ground states break the rotational and translational symmetries of the original Hamiltonian. We also show how one can use projection to construct a spin-3/2 quantum chain with nearest neighbor interaction, whose exact ground states break the rotational symmetry of the Hamiltonian. Correlation functions of both models are determined in closed form. Although we confine ourselves to examples, the method can easily be adapted to encompass more general models.

DOI
10.48550/arxiv.0801.1247

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We investigate the isotropic two-leg S=S=S=12\frac{1}{2}21​ ladder with general bilinear and biquadratic exchange interactions between spins on neighboring rungs, and determine the Hamiltonians that have a matrix-product wave function as an exact ground state. We demonstrate that a smooth change of parameters leads one from the S=S=S=12\frac{1}{2}21​ dimer and Majumdar-Ghosh chains to the S=1S=1S=1 chain with biquadratic exchange, which proves that these model systems are in the same phase. The family of Hamiltonians also includes a set of models of isotropic S=S=S=12\frac{1}{2}21​ chains with only bilinear nearest- and next-nearest-neighbor interactions. The ground state for these models becomes unstable at a line of first-order phase transitions to the ferromagnetic state.
We construct an integrable quantum spin chain that includes the nearest-neighbor, next-nearest-neighbor, chiral three-spin couplings, Dzyloshinsky–Moriya interactions and unparallel boundary magnetic fields. Although the interactions in bulk materials are isotropic, the spins nearby the boundary fields are polarized, which induce the anisotropic exchanging interactions of the first and last bonds. The U(1) symmetry of the system is broken because of the off-diagonal boundary reflections. Using the off-diagonal Bethe ansatz, we obtain an exact solution to the system. The inhomogeneous T–Q relation and Bethe ansatz equations are given explicitly. We also calculate the ground state energy. The method given in this paper provides a general way to construct new integrable models with certain interesting interactions.
We use the matrix product approach to construct all optimum ground states of general anisotropic spin-2 chains with nearest-neighbour interactions and common symmetries. These states are exact ground states of the model and their properties depend on up to three parameters. We find three different antiferromagnetic Haldane phases, one weak antiferromagnetic and one weak ferromagnetic phase. The antiferromagnetic phases can be described as spin liquids with exponentially decaying correlation functions. The variety of phases found with the matrix product ansatz also gives insight into the behaviour of spin chains with arbitrary higher spins.