Comment on "Absence of Luttinger's Theorem", by Kiaran B. Dave, Philip W. Phillips and Charles L. Kane, arXiv:1207.4201

TL;DR

This paper argues that the absence of Luttinger's theorem in a specific SU(N) model is due to ground state degeneracy and shows that with a small perturbation, the theorem holds, clarifying previous claims of its breakdown.

Contribution

The paper demonstrates that the failure of Luttinger's theorem in the SU(N) model is caused by ground state degeneracy and that it can be restored with a perturbation, clarifying misconceptions about its breakdown.

Findings

Luttinger's theorem is valid when ground state degeneracy is lifted.

Ground state degeneracy causes the observed failure of the theorem.

The singularity in self-energy is due to ground state non-uniqueness.

Abstract

In this Comment, we first present general arguments showing that the absence of the Luttinger theorem (LT) for the SU(N) model of Dave, Phillips and Kane (DPK) is rooted in the non-uniqueness of the ground state (GS) of this model for 0 < n < N, where n denotes the number of particles in the GS; the validity of the Luttinger theorem for n = N/2, when N even, is accidental, a consequence of particle-hole symmetry. Consequently, by supplementing the Hamiltonian of the SU(N) model with a perturbation Hamiltonian that removes the GS degeneracy, the LT is to apply also for the SU(N) model in the limit of the coupling constant, \lambda, of this perturbation approaching zero, where the limit \lambda --> 0 is clearly to be taken subsequent to taking the zero-temperature limit of the thermal single-particle Green function in the expression for the Luttinger number N_L. We explicitly establish the validity of this statement for the case of N=4. The details of the relevant calculations being distinctly transparent, one can readily convince oneself that our observation is valid for arbitrary N. It follows that the issues raised by DPK, such as non-existence of the Luttinger-Ward functional and "breakdown of the elemental particle picture in strongly correlated electron matter", are all inessential to the observed failure of the LT. As regards the singularity of the self-energy \Sigma(\omega) on the real \omega-axis, observed by DPK, we demonstrate that this also is a direct consequence of the non-uniqueness of the GS of the SU(N) model for 0 < n < N. In the light of the above observations, we are in a position to state that to this date no case has come to light indicative of the failure of the LT under the conditions for which it has been deduced. [Abridged Abstract]