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arXiv:cond-mat/9403080 · International Journal of Modern Physics B

q-DEFORMATIONS OF QUANTUM SPIN CHAINS WITH EXACT VALENCE-BOND GROUND STATES

Murray T. Batchelor, C M Yung

Published Nov 1, 1994• 30 citations
arXivHTMLPDF

Abstract

Quantum spin chains with exact valence-bond ground states are of great interest in condensed-matter physics. A class of such models was proposed by Affleck et al., each of which is su(2)-invariant and constructed as a sum of projectors onto definite total spin states at neighboring sites. We propose to use the machinery of the q-deformation of su(2) to obtain generalisations of such models, and work out explicitly the two simplest examples. In one case we recover the known anisotropic spin-1 VBS model while in the other we obtain a new anisotropic generalisation of the spin-½ Majumdar-Ghosh model.

DOI
10.1142/s021797929400155x

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Quantum spin chains with exact valence-bond ground states are of great interest in condensed-matter physics. A class of such models was proposed by Affleck et al., each of which is su(2)-invariant and constructed as a sum of projectors onto definite total spin states at neighboring sites. We propose to use the machinery of the q-deformation of su(2) to obtain generalisations of such models, and work out explicitly the two simplest examples. In one case we recover the known anisotropic spin-1 VBS model while in the other we obtain a new anisotropic generalisation of the spin-½ Majumdar-Ghosh model.
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.
Generalizing the mapping between the Potts model with nearest neighbor interaction and the six-vertex model, we build a family of “fused Potts models” related to the spin k/2 U q su (2) invariant vertex model and quantum spin chain. These Potts models still have variables taking values 1, …,Q[Formula: see text] but they have a set of complicated multispin interactions. The general technique to compute these interactions, the resulting lattice geometry, symmetries, and the detailed examples of k=2, 3 are given. For Q>4, spontaneous magnetizations are computed on the integrable first order phase transition line, generalizing Baxter’s results for k=1. For Q≤4, we discuss the full phase diagram of the spin 1 (k=2) anisotropic and U q su (2) invariant quantum spin chain [it reduces in the limit Q=4 (q=1) to the much studied phase diagram of the isotropic spin 1 quantum spin chain], Several critical lines and massless phases are exhibited. The appropriate generalization of the valence bond state method of Affleck et al. is worked out.
Inspired by the exact solution of the Majumdar-Ghosh model, a family of one-dimensional, translationally invariant spin Hamiltonians is constructed. The exchange coupling in these models is antiferromagnetic, and decreases linearly with the separation between the spins. The coupling becomes identically zero beyond a certain distance. It is rigorously proved that the dimer configuration is an exact, superstable ground-state configuration of all the members of the family on a periodic chain. The ground state is twofold degenerate, and there exists an energy gap above the ground state. The Majumdar-Ghosh Hamiltonian with a twofold degenerate dimer ground state is just the first member of the family. The scheme of construction is generalized to two and three dimensions, and illustrated with the help of some concrete examples. The first member in two dimensions is the Shastry-Sutherland model. Many of these models have exponentially degenerate, exact dimer ground states.