Dimensional reduction in superconducting arrays and frustrated magnets

TL;DR

This paper explores how certain frustrated magnets and superconducting arrays exhibit dimensional reduction, where their physical properties in higher dimensions effectively behave as lower-dimensional systems, affecting their phase diagrams and critical phenomena.

Contribution

It introduces the concept of physical dimensional reduction in frustrated systems, analyzing both weak and strong cases, and provides a general phase diagram for quantum models with N-fold anisotropy.

Findings

Identification of weak and strong dimensional reduction effects.

Derivation of a phase diagram for N-fold anisotropic quantum models.

Discovery of a Kosterlitz-Thouless-like transition as deconfinement of 3D soliton dipoles.

Abstract

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.