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

by Kridsanaphong Limtragool, Zhidong Leong, Philip W. Phillips

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This Submission thread is now published as SciPost Phys. 5, 049 (2018)

Submission summary

As Contributors: Kridsanaphong Limtragool
Arxiv Link: (pdf)
Date submitted: 2018-02-28 01:00
Submitted by: Limtragool, Kridsanaphong
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


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. 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: particle-hole 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.

Ontology / Topics

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Green's functions Nonlinear Luttinger liquids
Current status:
Has been resubmitted

Submission & Refereeing History

Published as SciPost Phys. 5, 049 (2018)

Resubmission 1708.08460v4 on 17 July 2018

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Submission 1708.08460v2 on 28 February 2018

Reports on this Submission

Anonymous Report 1 on 2018-4-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1708.08460v2, delivered 2018-04-12, doi: 10.21468/SciPost.Report.415


1) Addresses a question of importance to the frontier of the field
2) Solid analytic calculation
3) Well organized


1) Concludes with general statements based on too specific special cases
2) Abstract and Introduction are somewhat overselling


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 power-law dependence on frequency, where the deviation from Fermi-liquid 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 particle-hole symmetry), but is typically violated in the first example (although can be restored for 1<\alpha<2 by fine-tuning the low-energy cutoff). The breakdown of the theorem in the latter case is attributed to violation of a couple of criteria (particle-hole 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 well-defined quasi-particles, 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 self-energy has anomalous \omega-dependence. However, it appears to be a somewhat artificial, relatively simple generalization of the free-Fermion 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 self-energy 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 non-Fermi-liquid 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 power-law 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 non-Fermi-liquid
3) Modify a misleading sentence in the Abstract as detailed in the report

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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