An Exact Solution to a Three Dimensional Ising Model and Dimensional Reductions
Zohar Nussinov
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.