Generalization of the Luttinger Theorem for Fermionic Ladder Systems

TL;DR

This paper extends the Luttinger Theorem to fermionic ladder systems using a generalized Lieb-Schultz-Mattis approach, demonstrating the conservation of Fermi wave vectors under interactions and exploring limitations in higher dimensions.

Contribution

It provides a non-perturbative proof of Fermi wave vector conservation in fermionic ladders, generalizing the Luttinger Theorem beyond traditional settings.

Findings

Existence of low-lying excited states in fermionic ladders for most fillings.

Conservation of the sum of Fermi wave vectors under interactions.

Limitations of the theorem in the limit of infinite legs (2D limit).

Abstract

We apply a generalized version of the Lieb-Schultz-Mattis Theorem to fermionic ladder systems to show the existence of a low-lying excited state (except for some special fillings). This can be regarded as a non-perturbative proof for the conservation under interaction of the sum of the Fermi wave vectors of the individual channels, corresponding to a generalized version of the Luttinger Theorem to fermionic ladder systems. We conclude by noticing that the Lieb-Schultz-Mattis Theorem is not applicable in this form to show the existence of low-lying excitations in the limit that the number of legs goes to infinity, e.g. in the limit of a 2D plane.