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arXiv:cond-mat/9411085 · Journal of Physics A Mathematical and General

Temperley-Lieb words as valence-bond ground states

P F Arndt, T Heinzel, C M Yung

Published Jun 21, 1995• 3 citations
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Abstract

Based on the Temperley-Lieb algebra we define a class of one-dimensional Hamiltonians with nearest- and next-nearest-neighbour interactions. Using the regular representation we present ground states of this model as words of the algebra. Two-point correlation functions can be computed employing the Temperley-Lieb relations. Choosing a spin- 1/2 representation of the algebra we obtain a generalization of the (q-deformed) Majumdar-Ghosh model. The ground states become valence-bond states.

DOI
10.1088/0305-4470/28/12/001

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Based on the Temperley-Lieb algebra we define a class of one-dimensional Hamiltonians with nearest and next-nearest neighbour interactions. Using the regular representation we give ground states of this model as words of the algebra. Two point correlation functions can be computed employing the Temperley-Lieb relations. Choosing a spin-1/2 representation of the algebra we obtain a generalization of the (q-deformed) Majumdar-Gosh model. The ground states become valence-bond states. (orig.)

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Based on the Temperley-Lieb algebra we define a class of one-dimensional Hamiltonians with nearest- and next-nearest-neighbour interactions. Using the regular representation we present ground states of this model as words of the algebra. Two-point correlation functions can be computed employing the Temperley-Lieb relations. Choosing a spin- 1/2 representation of the algebra we obtain a generalization of the (q-deformed) Majumdar-Ghosh model. The ground states become valence-bond states.
We present a systematic method to construct the spin models of Heisenberg type in higher dimensions with nearest and non-nearest neighbour interactions. These models constructed in this way are of Temperley-Lieb-Jones (TLJ) algebraic structures and -invariances. The TLJ algebra is generalized to adjusting the lattice spin models. The Hamiltonians of Heisenberg spin models in one dimension (including second nearest neighbour interactions) and in two-dimensional triangular lattice (with nearest interaction) are constructed explicitly. The hidden symmetries are shown to be the group, and the terms in Hamiltonian for different lattice cells are explicitly shown to be the representations of elements of (TLJ) algebra.
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.
Using the matrix product formalism, we define a multiparameter family of spin models on one-dimensional chains, with nearest- and next-nearest-neighbor interactions for which exact analytical expressions can be found for its doubly degenerate ground states. The family of Hamiltonians depends on five continuous parameters and the Majumdar-Ghosh model is a particular point in this parameter space. The model can also be represented as a zigzag spin ladder in which there are extra three-particle couplings between spins on vertices of triangles. The doubly degenerate ground states of these models models have a very simple structure, they are the product of entangled (dimer) states on adjacent sites (or consecutive rungs of the ladder in the zigzag ladder picture). At the Majumdar-Ghosh point, these entangled states become the spin-singlets pertaining to this model. We will also calculate, in closed form, the two point correlation functions, both for finite sizes of the chain and in the thermodynamic limit.
We investigate the Temperley–Lieb–Jones algebraic structures in isomorphic higher spin chain models with nearest neighbour interactions and symmetry. The Temperley–Lieb–Jones algebraic constructions for such chains of spins from 1 to are presented. For the case of spin-1 we also give the Temperley–Lieb–Jones algebraic representations related to the anisomorphic spin chain with -deformed symmetry. Their related spin chain Hamiltonians with and symmetries as well as the corresponding solutions of Yang–Baxter equation are also discussed.
We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence bond states proposed in the Affleck--Kennedy--Lieb--Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest-neighbor interactions, and the partially dimerized or AKLT/valence bond solid states, which are constructed from them by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions.