On Ising and dimer models in two and three dimensions

Abstract

Motivated by recent interest in 2+1 dimensional quantum dimer models, we revisit Fisher's mapping of two dimensional Ising models to hardcore dimer models. First, we note that the symmetry breaking transition of the ferromagetic Ising model maps onto a non-symmetry breaking transition in dimer language -- instead it becomes a deconfinement transition for test monomers. Next, we introduce a modification of Fisher's mapping in which a second dimer model, also equivalent to the Ising model, is defined on a generically different lattice derived from the dual. In contrast to Fisher's original mapping, this enables us to reformulate frustrated Ising models as dimer models with positive weights and we illustrate this by providing a new solution of the fully frustrated Ising model on the square lattice. Finally, by means of the modified mapping we show that a large class of three-dimensional Ising models are precisely equivalent, in the time continuum limit, to particular quantum dimer models. As Ising models in three dimensions are dual to Ising gauge theories, this further yields an exact map between the latter and the quantum dimer models. The paramagnetic phase in Ising language maps onto a deconfined, topologically ordered phase in the dimer models. Using this set of ideas, we also construct an exactly soluble quantum eight vertex model.

10 pages, 9 figures autmatically included

References in corpus (10)

• An RVB phase in the triangular lattice quantum dimer model
• Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems
• Ising models of quantum frustration
• Short-ranged RVB physics, quantum dimer models and Ising gauge theories
• Quantum dimer model on the kagome lattice: solvable dimer liquid and Ising gauge theory
• Two-dimensional periodic frustrated Ising models in a transverse field
• Classical dimers on the triangular lattice
• Pocket Monte Carlo algorithm for classical doped dimer models
• Theory of d-density wave viewed from a vertex model and its implications
• A note on the 3D Ising model as a string theory

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