# Absence of Luttinger's theorem for fermions with power-law Green functions

### Submission summary

 As Contributors: Kridsanaphong Limtragool Arxiv Link: https://arxiv.org/abs/1708.08460v4 Date accepted: 2018-11-02 Date submitted: 2018-07-17 Submitted by: Limtragool, Kridsanaphong Submitted to: SciPost Physics Discipline: Physics Subject area: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We investigate the validity of Luttinger's theorem (or Luttinger sum rule) in two scale-invariant fermionic models. We find that, in general, Luttinger's theorem does not hold in a system of fermions with power-law Green functions which do not necessarily preserve particle-hole symmetry. However, Ref. \cite{Blagoev1997,Yamanaka1997} showed that Luttinger liquids, another scale-invariant fermionic model, respect Luttinger's theorem. To understand the difference, we examine the spinless Luttinger liquid model. We find two properties which make the Luttinger sum rule valid in this model: particle-hole symmetry and $\mathrm{Im} G(\omega=0,-\infty)=0$. We conjecture that these two properties represent sufficient, but not necessary, conditions for the validity of the Luttinger sum rule in condensed matter systems.

### Ontology / Topics

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Published as SciPost Phys. 5, 049 (2018)

We would like to resubmit Absence of Luttinger's theorem for fermions with power-law Green functions" which we have revised based on the referee's comments and suggestions. We thank the referees for carefully reviewing the manuscript. We start by responding to the the first comment.

1. We agree that a rigorous proof is necessary to establish that the two properties (the vanishing of $\mathrm{Im}G(\omega)$ at $\omega = 0,-\infty$ and particle-hole symmetry) are necessary but not sufficient conditions for the Luttinger sum rule to hold. We now put this statement as a conjecture. We also softened the conclusion regarding the criteria for the validity of Luttinger's theorem.

2. We added a new appendix (now Appendix B) to address a more physical example of non-Fermi liquid. We examined the validity of Luttinger's theorem of the system with self-energy of the form, $\Sigma \sim \lambda (\omega - \varepsilon_p)^\alpha$. We found that the Luttinger's theorem doesn't hold in general like the result we obtained with the power-law Green function.

3. We modified this sentence in the abstract to However, Ref. [1,2] showed that Luttinger liquids, another scale-invariant fermionic model, respect Luttinger's theorem."

### List of changes

1. We modified the sentence “This contrasts with the result by Ref. [1,2]…” to "However, Ref. [1, 2] showed that Luttinger liquids, another scale-invariant fermionic model, respect Luttinger’s theorem." in abstract.

2. We put the two properties (the vanishing of $\mathrm{Im}G(\omega)$ at $\omega = 0,-\infty$ and particle-hole symmetry) are sufficient but not necessary conditions as a conjecture in abstract, introduction, and conclusion.

3. We softened the conclusion regarding the applicability of the result.

4. In section II.D and (now) appendix B, we addressed a more physical example of non-Fermi liquid.

### Submission & Refereeing History

Resubmission 1708.08460v4 on 17 July 2018
Submission 1708.08460v2 on 28 February 2018