Exact Solutions to Special High Dimensional O(n) Models, Dimensional Reductions, gauge redundancy, and special Frustrated Spin and Orbital models

Zohar Nussinov

Published 2001-03-08, updated 2002-08-22Version 5

This work addresses models (e.g. potential models of directed orbital systems- the manganates) in which an effective reduction dimensionality occurs as a result of a new symmetry which is intermediate between that of global and local gauge symmetry. This path towards dimensional reduction is examined in simple O(n) spin models and lattice gauge theories. A high temperature expansion is employed to map special anisotropic high dimensional models into lower dimensional variants. We show that it is possible to have an effective reduction in the dimension without the need of compactifying some dimensions. These models are frustrated and display a symmetry intermediate between local and global gauge symmetries. Some solutions are presented. Our dimensional reductions are a generlization of the trivial dimensional reduction that occur in pure two dimensional gauge theories. It will be further seen that the absence of a ``phase interference'' effect plays an important role in high dimensional problems. By identifying another (``permutational'') symmetry present in the large n limit, we will further show how to generally map global high dimensional spin systems onto a one dimensional chain and discuss implications.