# Instabilities of quantum critical metals in the limit $N_f\rightarrow0$

### Submission summary

 As Contributors: Petter Säterskog Arxiv Link: https://arxiv.org/abs/2010.03077v2 (pdf) Date accepted: 2021-03-04 Date submitted: 2021-01-28 13:56 Submitted by: Säterskog, Petter Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We study a model in 2+1 dimensions composed of a Fermi surface of $N_f$ flavors of fermions coupled to scalar fluctuations near quantum critical points (QCPs). The $N_f\rightarrow0$ limit allows us to non-perturbatively calculate the long-range behavior of fermion correlation functions. We use this to calculate charge, spin and pair susceptibilities near different QCPs at zero and finite temperatures, with zero and finite order parameter gaps. While fluctuations smear out the fermionic quasiparticles, we find QCPs where the overall effect of fluctuations leads to enhanced pairing. We also find QCPs where the fluctuations induce spin and charge density wave instabilities for a finite interval of order parameter fluctuation gaps at $T=0$. We restore a subset of the diagrams suppressed in the $N_f\rightarrow0$ limit, all diagrams with internal fermion loops with at most 2 vertices, and find that this does not change the long-range behavior of correlators except right at the QCPs.

Published as SciPost Phys. 10, 067 (2021)

### List of changes

* Added some intuitive description of the Nf->0 limits under consideration to the introduction.
* Added a summary of the main results to the Introduction
* Added the complete derivation of Eq. 60, 61 instead of referring to previous work.
* Added a comment about quantum/thermal contributions and some more references to earlier works
* Added some references to earlier works that were missed.
* Added some discussion of the IR divergence in ref 41.
* Expanded on introduction to Section 4
* Added some comments on Fermi surface curvature corrections in Section 4 and in the Discussion.
* Added a comment about temperature dependent boson gap and reference to earlier works.
* Added some physical explanation of the h^\pm functions to the summary of results in the introduction, to Section 2, and to the Discussion.
* Reformulated the discussion of what can be expected at finite Nf in the Discussion.

### Submission & Refereeing History

Resubmission 2010.03077v2 on 28 January 2021
Submission 2010.03077v1 on 9 October 2020

## Reports on this Submission

### Anonymous Report 5 on 2021-2-25 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2010.03077v2, delivered 2021-02-25, doi: 10.21468/SciPost.Report.2609

### Report

The author has addressed my concerns satisfactorily. Regarding the innovation of the work, the author is correct that while instabilities have been studied extensively in other formalisms, it was not done in the real-space formulation. So, I retract my previous objection to publication in SciPost and am happy to recommend publication.

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### Report 1 by Ipsita Mandal on 2021-2-12 (Invited Report)

• Cite as: Ipsita Mandal, Report on arXiv:2010.03077v2, delivered 2021-02-12, doi: 10.21468/SciPost.Report.2546

### Report

The author has made sufficient efforts to address all referee comments. Although I do not quite agree with all his arguments, I believe the work merits publication.

In particular, the author comments:"All of: Ref 10, Ref 21, Phys. Rev. B 91, 125136 (2015), Eur. Phys. J. B (2016) 89: 278, Annals of Physics, 376, 89 (2017), Phys. Rev. B 98, 024510 (2018), and arXiv:2006.10766 have departed from the theory of interest here by changing the dimensionality in different ways such that it is no longer strongly coupled. This allows the authors to study the model perturbatively and find the renormalization group flows. My work has also departed from the theory of interest by taking Nf->0 but it is crucially still strongly coupled and an infinite set of complex diagrams has to be summed to calculate correlation functions. The epsilon expansions can find new fixed points but strong coupling makes it impossible to tell if they persist down to 2+1 dimensions and what their scalings would be there. Similarly my work gives correlation functions at Nf=0, but it is ultimately unclear what resemblance these bear to the Nf=1 case. The advantage of the small Nf approach is that it actually works at strong coupling and additionally that it gives explicit correlation functions at 0 and finite T. I also think the intuition is a bit clearer in the small Nf case than in the case of generalized dimensions of the Fermi surface or its embedding space."

I totally disagree with these arguments. Dimensional regularization is a well-defined technique backed by the concrete proofs that as the spatial dimensions of a system are increases, quantum fluctuations weaker, and as such thee is a definition of upper critical dimension. This was the main idea of Wilson's \phi^4 theory where this was implemented. But the author's method involves taking N_f->0, which is a singular limit, and there cannot me a concrete mathematical justification for this singular limit. This reminds me of the replica trick where people take number of replicas n ->0 at the end, which gives correct results in most cases, but this step is not mathematically justified.

With this grain of salt, I accept this as a method which happens to work for some systems.

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