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Noncanonical degrees of freedom
by Eoin Quinn
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Authors (as registered SciPost users):  Eoin Quinn 
Submission information  

Preprint Link:  https://arxiv.org/abs/2009.14755v1 (pdf) 
Date submitted:  20201009 15:14 
Submitted by:  Quinn, Eoin 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Noncanonical 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, heavyfermion behaviour, and also indeed magnetically ordered systems. Nevertheless it remains an open question whether noncanonical algebras can in fact provide legitimate quantum degrees of freedom. In this manuscript we survey progress made on this topic, complementing distinct approaches so as to obtain a unified description. In particular we obtain a novel closedform expression for a selfenergylike object for noncanonical 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 higherorder approximations which are consistent with conservation laws, which represents an outstanding issue. We also discuss how the interplay between canonical and noncanonical degrees of freedom offers a useful paradigm for organising the phase diagram of correlated electronic behaviour.
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Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2021122 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2009.14755v1, delivered 20210122, doi: 10.21468/SciPost.Report.2452
Report
The revised manuscript presents a detailed description of noncanonical degrees of freedom, and generalized Dyson equation, RPA technique etc for noncanonical degrees of freedom.
Since most of the text books currently discuss only canonical degrees of freedom, though there are many important topics in condensed matter physics and beyond, where noncanonical degrees of freedom might play crucial role, I think this work will provide a good reference in future.
I think the revised manuscript satisfies the criterion of SciPost Core and I recommend it for publication.
Report
The revised manuscript presents a detailed description of noncanonical degrees of freedom, and generalized Dyson equation, RPA technique etc for noncanonical degrees of freedom.
Since most of the text books currently discuss only canonical degrees of freedom, though there are many important topics in condensed matter physics and beyond, where noncanonical degrees of freedom might play crucial role, I think this work will provide a good reference in future.
I think the revised manuscript satisfies the criterion of SciPost Core and I recommend it for publication.
Report #1 by Anonymous (Referee 1) on 2020127 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2009.14755v1, delivered 20201206, doi: 10.21468/SciPost.Report.2257
Report
The manuscript deals with the noncanonical degrees of freedom and how to construct perturbation theory for them. The latter seems to be a nontrivial question because of the modified commutation relations the Wick's theorem is not applicable, and, consequently, the diagrammatic methods are not at hand. Nevertheless, the author finds a way to formulate Dyson's equation for singleparticle Green's function.
The paper is nicely written and for sure will be an invaluable reference in its field. I recommend this paper for publication in SciPost Physics.
I have several questions and remarks that can improve the presentation:
(i) in section 3 the author implies the hoppings and interactions to be symmetric; should they also be real?
(ii) For the algebra of the noncanonical DOF that renders the Heisenberg model into Eq. (31), what is the value of \lambda (\kappa)?
(iii) Is it possible to give a functional integral presentation for the noncanonical DOF? Or, in other words, is it known anything about the coherent states for the algebra Eq. (29)?
(iv) The author claims that Eqs. (75)(82) provide a closedform for M_p^*(\omega). I wonder if that statement could be explained a bit more in the last paragraph of section 5.2. In a sense, what are the variables, and which of Eqs. (75)(82) are definitions, which are the relations, and which are the actual equations to solve.
Author: Eoin Quinn on 20210201 [id 1195]
(in reply to Report 1 on 20201207)
We are grateful to the Referee for their assessment of our manuscript and for their remarks. Let us address each point in turn.
(i) in section 3 the author implies the hoppings and interactions to be symmetric; should they also be real?
Yes for a Hermitian Hamiltonian these parameters should be real if they are symmetric. We have modified the text to clarify that we take them to be real.
(ii) For the algebra of the noncanonical DOF that renders the Heisenberg model into Eq. (31), what is the value of \lambda (\kappa)?
We have modified the text to specify that here \lambda = 1/S.
(iii) Is it possible to give a functional integral presentation for the noncanonical DOF? Or, in other words, is it known anything about the coherent states for the algebra Eq. (29)?
In general we do not know an answer to this. While there exist textbook studies on coherent states for spin systems, we have been unable to find concrete results on coherent states for the noncanonical formulation of the electron, i.e. for Hubbard operators. We agree with the referee that this would be interesting to explore, and while we do not feel that we can add constructive comments at this point, we hope that this can be addressed in future work.
(iv) The author claims that Eqs. (75)(82) provide a closedform for M_p^*(\omega). I wonder if that statement could be explained a bit more in the last paragraph of section 5.2. In a sense, what are the variables, and which of Eqs. (75)(82) are definitions, which are the relations, and which are the actual equations to solve.
We thank the Referee for raising this. We have now split this paragraph in two to first focus on the form of the closed expression for M_p^*(\omega). We now clearly point to the closed set of equations, framed in terms of the unknown G. We have also modified the text to clarify that the closed equations take a functional differential form, whose solution requires a perturbative expansion. To highlight that this is analogous to the corresponding canonical case we have added a reference for the corresponding closed equation for the canonical selfenergy to paragraph 3 of the Introduction.
Author: Eoin Quinn on 20210201 [id 1194]
(in reply to Report 2 on 20210122)We thank the Referee for their report.