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As with the ice-type models, the eight-vertex model is a square lattice model, where each state is a configuration of arrows at a vertex.The allowed vertices have an even number of arrows pointing towards the vertex; these include the six inherited from the ice-type model (1-6), sinks (7), and sources (8).
Introduction to Mathematical Statistical Mechanics. Providence, RI: American Mathematical Society. ISBN 978-0-8218-1337-9. Friedli, Sacha; Velenik, Yvan (2017). Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge: Cambridge University Press. ISBN 978-1-107-18482-4.
An ice-type model is a lattice model defined on a lattice of coordination number 4. That is, each vertex of the lattice is connected by an edge to four "nearest neighbours". A state of the model consists of an arrow on each edge of the lattice, such that the number of arrows pointing inwards at each vertex is 2.
The existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality. [3]Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon [4] proved that the two point spin correlation of the ferromagnetics XY model in dimension D, coupling J > 0 and inverse temperature β is dominated by (i.e ...
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics , where the atoms of a crystal automatically form a lattice.
A Bethe lattice with coordination number z = 3. In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite symmetric regular tree where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature by Hans Bethe in 1935.
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. [1] By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid-state physics.
In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins. The model is notable for having nontrivial interactions, yet having an analytical solution. The model was solved by Lars Onsager for the special case that the external magnetic field H = 0. [1]