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Absence of Luttinger's theorem for fermions with powerlaw Green functions
by Kridsanaphong Limtragool, Zhidong Leong, Philip W. Phillips
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Submission summary
Authors (as registered SciPost users):  Kridsanaphong Limtragool 
Submission information  

Preprint Link:  https://arxiv.org/abs/1708.08460v4 (pdf) 
Date accepted:  20181102 
Date submitted:  20180717 02:00 
Submitted by:  Limtragool, Kridsanaphong 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the validity of Luttinger's theorem (or Luttinger sum rule) in two scaleinvariant fermionic models. We find that, in general, Luttinger's theorem does not hold in a system of fermions with powerlaw Green functions which do not necessarily preserve particlehole symmetry. However, Ref. \cite{Blagoev1997,Yamanaka1997} showed that Luttinger liquids, another scaleinvariant 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: particlehole 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.
Author comments upon resubmission
We would like to resubmit ``Absence of Luttinger's theorem for fermions with powerlaw 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.

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 particlehole 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.

We added a new appendix (now Appendix B) to address a more physical example of nonFermi liquid. We examined the validity of Luttinger's theorem of the system with selfenergy 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 powerlaw Green function.

We modified this sentence in the abstract to ``However, Ref. [1,2] showed that Luttinger liquids, another scaleinvariant 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 scaleinvariant 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 particlehole 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 nonFermi liquid.
Published as SciPost Phys. 5, 049 (2018)