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Non-canonical degrees of freedom

by Eoin Quinn

Submission summary

As Contributors: Eoin Quinn
Preprint link: scipost_202102_00002v1
Date submitted: 2021-02-01 09:53
Submitted by: Quinn, Eoin
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
Approach: Theoretical

Abstract

Non-canonical degrees of freedom provide one of the most promising routes towards characterising a range of important phenomena in condensed matter physics. Potential candidates include the pseudogap regime of the cuprates, heavy-fermion behaviour, and also indeed magnetically ordered systems. Nevertheless it remains an open question whether non-canonical algebras can in fact provide legitimate quantum degrees of freedom. In this paper we survey progress made on this topic, complementing distinct approaches so as to obtain a unified description. In particular we obtain a novel exact representation for a self-energy-like object for non-canonical degrees of freedom. We further make a resummation of density correlations to obtain analogues of the RPA and GW approximations commonly employed for canonical degrees of freedom. We discuss difficulties related to generating higher-order approximations which are consistent with conservation laws, which represents an outstanding issue. We also discuss how the interplay between canonical and non-canonical degrees of freedom offers a useful paradigm for organising the phase diagram of correlated electronic behaviour.

Current status:
Editor-in-charge assigned


Author comments upon resubmission

Dear Editor,

We hereby submit the revised manuscript, addressing the points made by the referees. We hope this is ready for publication in SciPost Physics.

List of changes

- a footnote has been added in Sec. 2.2 to clarify that the splitting of the electron falls outside the classification of elementary particles coming from high-energy physics.
- in Sec. 3 it is clarified that the hopping and interaction parameters are taken to be real.
- in Sec. 3 the value of \lambda corresponding to the Heisenberg model of Eq. (31) is now specified.
- at the end of Sec. 5.2, and throughout the text, it is clarified that the closed equations specifying the self-energy-like object for non-canonical degrees of freedom take a functional differential form, and that it is through perturbative expansion of these equations that systematic approximations for the Green's function are obtained.

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Comments

Claudio Attaccalite on 2021-02-23
Category:
correction

Dear Eoin Quinn

I read your manuscript and I found it very interesting, in particular the clear introduction to the non-canonical degrees of freedom (DOF) and their relation to different physical problems. The manuscript present also different way to derive self-energy in a closed form use non-canonical DOF. I found the manuscript clear and useful for future applications of this approach, for this reason I suggest the publication in SciPost Physics. However I have some questions and comments that you may address:

1) At page 8 you discuss the violation of the Luttinger sum rule. Is it proved that the Luttinger sum rule is violated in Hubbard Fermi liquid? And in which circumstances? May you add some references to this discussion.

2) If you have a system with spin-orbit coupling the spin and charge degree of freedom are already coupled, do the canonical and non-canonical degrees of freedom became similar in this case? It should be nice if you can add a comment on the effect of spin-orbit in the present formulation.

3) At page 20 you discuss the violation of Eq.84 that is related to the conserving approximation in the sense of Kadanoff-Baym, and related to charge, momentum and energy conservation. From the discussion it seems to me that it is difficult to impose these conservations with non-canonical DOF, however you proposed to use a more general external field to “force” these conversations. Is this just a proposition, or you can prove that deriving equation of motion with a more general external source lead to a conservative approximation for the non-canonical DOF?

Author Eoin Quinn on 2021-03-01
(in reply to Claudio Attaccalite on 2021-02-23)

Dear Claudio Attaccalite,

Thank you for the report on the manuscript, and we are glad that you find it interesting, clear and useful. Let us address your questions/comments in turn.

1) At page 8 you discuss the violation of the Luttinger sum rule. Is it proved that the Luttinger sum rule is violated in Hubbard Fermi liquid? And in which circumstances? May you add some references to this discussion.

That employing Hubbard operators to organise electronic correlations generically leads to a violation of the Luttinger sum rule can be seen already from the original work of Hubbard e.g. Ref. [13], and is discussed in some detail for example in Ref. [43]. We now add these citations to the discussion in the text. The key point, as described in the relevant paragraph of the manuscript, is that the Hubbard interaction acts linearly on the non-canonical formulation of the electronic DOF, see Eq. (16), in a manner which induces a splitting of the electron (to see this compare the action of \eta^z and \theta on the c-operators through Eq. (18)), with the consequence that the resulting single-particle modes generically violate the Luttinger sum rule for U != 0.

Perhaps it is worthwhile to emphasise that violation of the Luttinger sum rule is a 'feature not a bug' of the Hubbard approach. While historically this violation has been used to marginalise the approach, it can instead be invoked as evidence that the Hubbard Fermi liquid is not adiabatically connected to the more conventional Landau Fermi liquid, corroborating the Discussion section of the manuscript. We now mention this at the end of the relevant paragraph.

2) If you have a system with spin-orbit coupling the spin and charge degree of freedom are already coupled, do the canonical and non-canonical degrees of freedom became similar in this case? It should be nice if you can add a comment on the effect of spin-orbit in the present formulation.

We do not know how to make a general statement on this, but it is certainly something which would be interesting to explore, and we now include this case in our discussion of where the non-canonical formulation of the electron may be useful. When the effect of spin-orbit coupling is to induce additional interactions in the tight-binding description of a system we do not expect this to modify which degrees of freedom can be used for organising correlations. In general, it is not possible to tell a priori which are the relevant degrees of freedos, but one can compare the different known possibilities to see which, if any, are appropriate. A comment is added on page 7 to emphasise this.

3) At page 20 you discuss the violation of Eq.84 that is related to the conserving approximation in the sense of Kadanoff-Baym, and related to charge, momentum and energy conservation. From the discussion it seems to me that it is difficult to impose these conservations with non-canonical DOF, however you proposed to use a more general external field to “force” these conversations. Is this just a proposition, or you can prove that deriving equation of motion with a more general external source lead to a conservative approximation for the non-canonical DOF?

Yes we found it difficult to systematically generate conserving approximations, as outlined in Sec. 5.3. Let us emphasise however that we do not see a reason why this is an insurmountable challenge, and are hopeful that this issue may somehow be resolved by an appropriate shift of perspective. In Sec. 5.4 we change focus to making a resummation of density induced correlations within our formulation, achieved by introducing a general external field. To our knowledge, such an effort has not been outlined before for non-canonical DOFs. Unfortunately this does not overcome the issues related to generating conserving approximations, and we have edited the first paragraph of this section to clarify this. Nevertheless, it may be hoped that the resulting Eqs. (103)-(105) will provide a useful step towards understanding screening effects in systems whose behaviour is governed by non-canonical DOFs.