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International Journal of Modern Physics B

Multilayer Extension of Two-Dimensional Solvable Statistical Models to Three Dimensions

Vladislav Popkov

Published Jan 20, 1997• 0 citations
Publisher

Abstract

We review the method of constructing solvable models in three dimensions, by starting from two-dimensional solvable models. The solvable three-dimensional models thus constructed do possess positive Boltzmann weights. These are multilayer two-dimensional systems with interactions in the third direction which can be interpreted as nearest-neighbour interactions. The set of conditions corresponding to the general 3D multilayer extension of solvable 2D models is derived.

DOI
10.1142/s021797929700023x

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We review the method of constructing solvable models in three dimensions, by starting from two-dimensional solvable models. The solvable three-dimensional models thus constructed do possess positive Boltzmann weights. These are multilayer two-dimensional systems with interactions in the third direction which can be interpreted as nearest-neighbour interactions. The set of conditions corresponding to the general 3D multilayer extension of solvable 2D models is derived.
In this work, we first reveal the nature of three dimensions: the existence of non-local behavior. Taking the three-dimensional (3D) Ising model as an example, we analyze the transfer matrices as well as the partition function to discuss in details the origin of non-local behavior in three dimensions. We then point out that fail to deal with non-local behavior is one of the main disadvantages of several theoretical approaches (such as conventional low-temperature expansions, conventional high-temperature expansions, perturbative renormalization group and Monte Carlo methods, etc), all of which take local environments into account only. This work provides a deep understanding on the natural difference between 3D bulk materials and 2D materials, and shows that how to treat the non-local behavior and the interaction between spins along the third dimension is a key point for solving exactly the problems of the 3D systems with the nearest neighboring interactions. Finally, we will briefly review recent advance in the 3D Ising model and related topics.
I discuss many-body models for correlated fermions in two space dimensions which can be solved exactly using group theory. The simplest example is a model of a quantum Hall system: two-dimensional (2D) fermions in a constant magnetic field and a particular non-local four-point interaction. It is exactly solvable due to a dynamical symmetry corresponding to the Lie algebra gl ∞ ⊕ gl ∞ . There is an algorithm to construct all energy eigenvalues and eigenfunctions of this model. The latter are, in general, many-body states with spatial correlations. The model also has a non-trivial zero temperature phase diagram. I point out that this QH model can be obtained from a more realistic one using a truncation procedure generalizing a similar one leading to mean field theory. Applying this truncation procedure to other 2D fermion models I obtain various simplified models of increasing complexity which generalize mean field theory by taking into account non-trivial correlations but nevertheless are treatable by exact methods.
A high temperature expansion is employed to map some complex anisotropic nonhermitian three and four dimensional Ising models with algebraic long range interactions into a solvable two dimensional variant. We also address the dimensional reductions for anisotropic two dimensional XY and other models. For the latter and related systems it is possible to have an effective reduction in the dimension without the need of compactifying some dimensions. Some solutions are presented. This framework further allows for some very simple general observations. It will be seen that the absence of a ``phase interference'' effect plays an important role in high dimensional problems. A very forbidding purely algebraic recursive series solution to the three dimensional nearest neighbor Ising model will be given. In the aftermath, the full-blown three dimensional nearest neighbor Ising model is exactly mapped onto a single spin 1/2 particle with nontrivial dynamics. All this allows for a formal high dimensional Bosonization.
Solvable via Bethe Ansatz (BA) anisotropic statistical model on cubic lattice consisting of locally interacting 6-vertex planes, is studied. Symmetries of BA lead to infinite hierarchy of possible phases, which is further restricted by numerical simulations. The model is solved for arbitrary value of the interlayer coupling constant. Resulting is the phase diagram in general 3-parameter space. Exact mapping onto the models with some inhomogenious sets of interlayer coupling constants is established.