Extending Luttinger's theorem to Z(2) fractionalized phases of matter

TL;DR

This paper extends Luttinger's theorem to Z(2) fractionalized phases, establishing non-perturbative relations between particle filling and intrinsic properties in exotic quantum states.

Contribution

It generalizes momentum balance arguments to fractionalized insulators, superfluids, and Fermi liquids, revealing new relations between filling and quantum phase properties.

Findings

Derived relations between filling and intrinsic properties in fractionalized phases

Identified distinctions between fractionalized states from half and integer fillings

Provided tools for experimental identification of fractionalized states

Abstract

Luttinger's theorem for Fermi liquids equates the volume enclosed by the Fermi surface in momentum space to the electron filling, independent of the strength and nature of interactions. Motivated by recent momentum balance arguments that establish this result in a non-perturbative fashion [M. Oshikawa, Phys. Rev. Lett. {\bf 84}, 3370 (2000)], we present extensions of this momentum balance argument to exotic systems which exhibit quantum number fractionalization focussing on $Z_2$ fractionalized insulators, superfluids and Fermi liquids. These lead to nontrivial relations between the particle filling and some intrinsic property of these quantum phases, and hence may be regarded as natural extensions of Luttinger's theorem. We find that there is an important distinction between fractionalized states arising naturally from half filling versus those arising from integer filling. We also note how these results can be useful for identifying fractionalized states in numerical experiments.