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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
As Contributors:  Kridsanaphong Limtragool 
Arxiv Link:  http://arxiv.org/abs/1708.08460v2 (pdf) 
Date submitted:  20180228 01:00 
Submitted by:  Limtragool, Kridsanaphong 
Submitted to:  SciPost Physics 
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. This contrasts with the result by Ref. \cite{Blagoev1997,Yamanaka1997} that Luttinger liquids 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 $\mathrm{Im} G(\omega=0,\infty)=0$. These two properties represent sufficient, but not necessary, conditions for the validity of the Luttinger sum rule in condensed matter systems.
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Reports on this Submission
Anonymous Report 1 on 2018412 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1708.08460v2, delivered 20180412, doi: 10.21468/SciPost.Report.415
Strengths
1) Addresses a question of importance to the frontier of the field
2) Solid analytic calculation
3) Well organized
Weaknesses
1) Concludes with general statements based on too specific special cases
2) Abstract and Introduction are somewhat overselling
Report
In this manuscript, the authors examine the validity of Luttinger’s theorem in strongly correlated systems, where the Fermi liquid paradigm (used in the original proof) breaks down. They study in detail two examples for systems of interacting Fermions, where the concept of Fermi sphere (and hence “Fermi sphere volume”) is not well defined. In the first, the Fermions Green’s function assumes an anomalous powerlaw dependence on frequency, where the deviation from Fermiliquid is parametrized by an anomalous exponent \alpha, with \alpha>1 corresponding to the free Fermions limit. The second example is a 1D model, i.e. a Luttinger liquid. Based on direct analytic calculations of the Fermions density in these two cases, they conclude that Luttinger’s theorem does not hold in general but is recovered under certain restrictions. In particular, it does hold in the Luttinger liquid case (even when modified to break particlehole symmetry), but is typically violated in the first example (although can be restored for 1<\alpha<2 by finetuning the lowenergy cutoff). The breakdown of the theorem in the latter case is attributed to violation of a couple of criteria (particlehole symmetry and vanishing of Im{G(\omega=0, infinity}), which the authors conclude are necessary conditions for Luttinger’s theorem to hold.
In view of the growing interest in the behavior of electronic systems in the regime of no welldefined quasiparticles, the paper deals with an important question and is certainly relevant to this frontier. As far as I checked, the calculations are valid and I agree with the main results. However, I have several comments that the authors should address:
1) My primary concern is that the general conclusions drawn from the presented study, namely, the criteria stated as sufficient for Luttinger’s theorem to hold, are not fully substantiated. The manuscript presents a detailed derivation for two concrete cases, choosing specific functional forms of the Fermions Green’s function, and demonstrates that they are consistent with the stated criteria. However, to conclude that the criteria are sufficient, a direct proof is desired that an arbitrary G(p,\omega) satisfying them obeys Luttinger’s theorem. Unless the authors can come up with such direct proof, I believe their conclusions should be posed as a (plausible) conjecture.
2) The specific form of G(p,\omega) Eq. (3) is motivated by physical cases where the selfenergy has anomalous \omegadependence. However, it appears to be a somewhat artificial, relatively simple generalization of the freeFermion Green’s function, which indeed enables an exact analytic calculation, but is not obviously compatible with any physical example. In particular, assuming a shift of the energy \epsilon_p by some selfenergy of general form (e.g. an anomalous power of \omega) may, at best, reduce to Eq. (3) in some restricted regime of p,\omega, not the entire range required to compute the density n. Can the author address such an alternative form for the nonFermiliquid Green’s function?
3) As a minor remark, the abstract (4th line) states: “This contrasts with the result by Ref. [1,2]…”. I find it a bit misleading, since it creates the impression that the present paper contradicts the results of Ref. [1,2] while it actually agrees with them: violation of Luttinger’s theorem is shown for the powerlaw Eq. (3), not for a Luttinger liquid.
In summary, the paper is worth publishing in SciPost provided the authors address the above comments, and possibly modify some misleading statements.
Requested changes
1) Soften the general applicability of the conclusions (or provide a more convincing argument in their favor).
2) (Optional) If possible, address a more physical example of nonFermiliquid
3) Modify a misleading sentence in the Abstract as detailed in the report