Quantum
dimer model
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by p ...
s were introduced to model the physics of
resonating valence bond In condensed matter physics, the resonating valence bond theory (RVB) is a theoretical model that attempts to describe high-temperature superconductivity, and in particular the superconductivity in cuprate compounds. It was first proposed by an Am ...
(RVB) states in
lattice spin systems
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an ornam ...
. The only degrees of freedom retained from the motivating spin systems are the valence bonds, represented as dimers which live on the lattice bonds. In typical dimer models, the dimers do not overlap ("hardcore constraint").
Typical phases of quantum dimer models tend to be
valence bond crystal
Valence or valency may refer to:
Science
* Valence (chemistry), a measure of an element's combining power with other atoms
* Degree (graph theory), also called the valency of a vertex in graph theory
* Valency (linguistics), aspect of verbs re ...
s. However, on non-bipartite lattices, RVB liquid phases possessing
topological order
In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian ge ...
and
fractionalized spinon
Spinons are one of three quasiparticles, along with holons and orbitons, that electrons in solids are able to split into during the process of spin–charge separation, when extremely tightly confined at temperatures close to absolute zero. The e ...
s also appear. The discovery of topological order in quantum dimer models (more than a decade after the models were introduced) has led to new interest in these models.
Classical dimer models have been studied previously in
statistical physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxi ...
, in particular by P. W. Kasteleyn (1961) and
M. E. Fisher (1961).
References
Exact solution for classical dimer models on planar graphs:
*
*
Introduction of model; early literature:
*
*
Topological order in quantum dimer model on non-bipartite lattices:
* ;
*
*
Topological order in quantum spin model on non-bipartite lattices:
*
*
Quantum lattice models
Condensed matter physics
Statistical mechanics
Matching (graph theory)
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