Non-perturbative approach to Luttinger's theorem in one dimension

TL;DR

This paper extends the Lieb-Schultz-Mattis theorem to various 1D electron and spin models, establishing a non-perturbative connection to Luttinger's theorem and clarifying the role of localized spins in Fermi momentum calculations.

Contribution

It generalizes the theorem to a broad class of models and provides a non-perturbative proof of Luttinger's theorem in one dimension.

Findings

Low-energy states exist except at special fillings.

The crystal momentum of low-energy states is $2k_F$, consistent with Luttinger's theorem.

In the Kondo lattice, localized spins should be treated as electrons for Fermi momentum calculation.

Abstract

The Lieb-Schultz-Mattis theorem for spin chains is generalized to a wide range of models of interacting electrons and localized spins in one-dimensional lattice. The existence of a low-energy state is generally proved except for special commensurate fillings where a gap may occur. Moreover, the crystal momentum of the constructed low-energy state is $2k_F$, where $k_F$ is the Fermi momentum of the non-interacting model, corresponding to Luttinger's theorem. For the Kondo lattice model, our result implies that $k_F$ must be calculated by regarding the localized spins as additional electrons.