Hodge Theory for Geometrically Frustrated Magnetism

Abstract

We present an analytical formalism based on the concept of discrete differential forms and Hodge theory as a framework for understanding geometrically frustrated magnetism. The primary insight is to treat spins as a 1-form field, for which we define the Helmholtz Hodge decomposition of spin configurations into rotational, irrotational, and harmonic components. As a physical example of its application, we demonstrate how nearest neighbor spin ice---the canonical model of a geometrically frustrated magnet---and its generalizations extended and dipolar spin ice fall neatly within this framework. Our framework clarifies analytically the dumbbell approximation of dipolar spin ice and its relation to projective equivalence, and allows us to write down the first order correction which captures the lifting of rotational mode degeneracy by the dipole interactions We demonstrate how polarized neutron diffraction cross sections can be interpreted within this framework, and in particular how it can be utilized to isolate correlations of the rotational modes. We then construct a new frustrated magnetic model which is complementary to nearest-neighbor spin ice, but whose low temperature spin liquid phase can be described as a 2-form gauge theory with 1-dimensional string-like excitations. Finally, we consider anisotropic Heisenberg models and show that Coulombic physics can be observed in the spin wave fluctuations of a frustrated magnet in a long range ordered phase.