Reduction of effective dimensionality in lattice models of superconducting arrays and frustrated magnets

Some frustrated magnets and superconducting arrays possess unusual symmetries that cause the free energy or other physics of a D-dimensional quantum or classical problem to be that of a different problem in a reduced dimension d < D. Examples in two spatial dimensions include the square-lattice p + ip superconducting array, the Heisenberg antiferromagnet on the checkerboard lattice (studied by a combination of 1/S expansion and numerical transfer matrix), and the ring-exchange superconducting array. Physical consequences are discussed both for weak dimensional reduction, which appears only in the ground state degeneracy, and strong dimensional reduction, which applies throughout the phase diagram. The strong dimensional reduction cases have the full lattice symmetry and do not decouple into independent chains, but their phase diagrams, self-dualities, and correlation functions indicate a reduced effective dimensionality. We find a general phase diagram for quantum-dimensional reduction models in two quantum dimensions with N-fold anisotropy, and obtain the Kosterlitz-Thouless-like phase transition as a deconfinement of dipoles of 3D solitons. (c) 2005 Elsevier B.V. All rights reserved.