Bond Solid State Research Articles

Matrix product representations for all valence bond states

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.

Valence Bond Solid Phases in a Cubic Antiferromagnet

We report on a valence bond projector Monte Carlo simulation of the cubic lattice quantum Heisenberg model with additional higher-order exchange interactions in each unit cell. The model supports two different valence bond solid (VBS) ground states. In one of these states, the dimer pattern is a three-dimensional analogue of the columnar pattern familiar from two dimensions. In the other, the dimers are regularly arranged along the four main diagonals in 1/8 of the unit cells. The phases are separated from one another and from a Néel phase by strongly first-order boundaries. Our results strengthen the case for exotic transitions in two dimensions, where no discontinuities have been detected at the Heisenberg Néel-VBS transition driven by four-spin plaquette interactions.

Complete analysis of two spin correlations of valence bond solid chains for all integer spins

For the valence bond solid (VBS) chain ground state introduced by Affleck et al. we evaluate exactly all 2S two-point correlation functions for all integer spinsS with a simple transfer matrix method. These correlations are expectation values of 2S different two-site irreducible tensor operators also presented in this article. Each of the firstS correlations exhibits its own purely exponential decay whereas the others vanish. It turns out that in the largeS limit there are both very large and very short ranged correlations. We use the correlation functions to calculate the probabilities that two spins at distancer add up to total spin σ and also derive a formula for the probability that two spins at distancer enclose an angle Θ. This provides a complete and perspecuate description of the VBS state for arbitrary integer spinS.