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Longterm memory magnetic correlations in the Hubbard model: A dynamical meanfield theory analysis
by Clemens Watzenböck, Martina Fellinger, Karsten Held, Alessandro Toschi
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Authors (as registered SciPost users):  Clemens Watzenböck 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.02903v1 (pdf) 
Date submitted:  20211207 16:05 
Submitted by:  Watzenböck, Clemens 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approaches:  Theoretical, Computational 
Abstract
We investigate the onset of a notdecaying asymptotic behavior of temporal magnetic correlations in the Hubbard model in infinite dimensions. This longterm memory feature of dynamical spin correlations can be precisely quantified by computing the difference between the static isolated (or Kubo) susceptibility and the corresponding isothermal one. Here, we present a procedure for reliably evaluating this difference starting from imaginary timeaxis data, and apply it to the testbed case of the MottHubbard metalinsulator transition (MIT). At low temperatures, we find longterm memory effects in the entire Mott regime, abruptly ending at the first order MIT. This directly reflects the underlying local moment physics and the associated degeneracy in the manyelectron spectrum. At higher temperatures, a more gradual onset of an infinitelylong timedecay of magnetic correlations occurs in the crossover regime, not too far from the Widom line emerging from the critical point of the MIT. Our work has relevant algorithmic implications for the analytical continuation of dynamical susceptibilities in strongly correlated regimes and offers a new perspective for unveiling fundamental properties of the manyparticle spectrum of the problem under scrutiny.
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Anonymous Report 2 on 2022112 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.02903v1, delivered 20220112, doi: 10.21468/SciPost.Report.4171
Report
The manuscript "Longterm memory magnetic correlations in the Hubbard model: A dynamical meanfield theory analysis" analyzes the existence of nondecaying spin correlations in the Hubbard model in the vicinity of the Mott transition. Using DMFT with Quantum Monte Carlo, the authors analyze the local magnetic susceptibilities in the Hubbard model and find a finite anomalous term in the Mott insulating regime corresponding to nondecaying longterm magnetic correlations.
The article is well written, and the results are interesting. Furthermore, this analysis can be implemented for different models and correlation functions. I support the publication of this manuscript.
Before publication, I would ask the authors for a few changes listed below.
Requested changes
(1) While the thermal and realtime susceptibilities have been defined using equations, I could not find the definitions of chi^T and chi^S. These definitions should be included.
(2) While reading this manuscript, I was not sure which quantities actually have been calculated with DMFT/QMC and which quantities have been obtained by analytic continuation or are used as parameters.
For example, how was A(w) obtained in equation (44)? Was it obtained by analytical continuation from the same data set as chi^th?
In equation (45), chi^R has been obtained by analytical continuation?
It should be clarified how these quantities have been obtained!
(3) If analytical continuation has been used to calculate chi^R from which C is calculated, how large is the error on C originating in an analytical continuation?
(4) Concerning the anomalous term at finite temperatures, is there an exact expression for large temperatures? If so, it would be good to compare to this in Fig. 7.
(5) In equation (44), are K, g_b, and sigma the same functions as used in the analytical continuation section?
(6) What happened with the delta function when going from equation 16 to 17?
(7) The gray colors in the right top panel in Fig. 1 are hard to see. Maybe the authors can change this color?
(8) In Fig. 3, there are only three curves visible but the legend includes 4 parameters.
Report 1 by Andrey Katanin on 202214 (Invited Report)
 Cite as: Andrey Katanin, Report on arXiv:2112.02903v1, delivered 20220104, doi: 10.21468/SciPost.Report.4141
Strengths
The paper studies longtime contributions to the susceptibility (e.g. spin susceptibility) and investigates important example of the vicinity of Mott transition in the singleband Hubbard model
Weaknesses
The suggestions below have to be considered; some corrections are required
Report
The paper by C. Watzenboeck studies longtime contributions to the susceptibility (e.g. spin susceptibility). The important case of longtime contributions to local spin susceptibility of singleband Hubbard model is considered. In general the paper is wellwritten, and sheds new light on the longtime contributions to susceptibilities in general. The study of such contribution to local spin susceptibility of singleband Hubbard model allows deeper understand formation of local magnetic moments.
Requested changes
1. The authors use a notion of isolated (Kubo) susceptibility, which is also called by them sometimes isolated (KuboNakano) susceptibility in the abstract and main text of the paper. I am not sure that this notion is rather common (in particular, in what sense the susceptibility (or system) is isolated?). I would suggest to name this quantity as a static limit of the retarded spin correlation function, or somehow else.
2. In the second and third paragraph of the Introduction I think it is important to mention everywhere that the authors talk about slowingdown of _local_ spin fluctuations ("local" is mentioned only once in the beginning of second paragraph, but it is preferable I think to insert it further, since this is crucial to differentiate this from the critical slowing down etc.).
3. Regarding this "local" slowing down, I think it is important to cite the paper 10.1103/PhysRevLett.101.166405 which was one of the first papers which discussed this topic for Hund metals.
4. Regarding Eqs. (25), (28), as well as Eqs. (19), (20), I would like to emphasize that the "anomalous" Cterms can be viewed as a certain limit of some regular contribution to the susceptibility. One such phenomenological form of the regular part, which provides these anomalous terms, was suggested in Eq. (2) of Ref. 10.1103/PhysRevB.88.155120, and reads in the notations of the author's paper chi^R(w)=C\beta i\delta/(w+i\delta), where \delta>+0. Then Im(\chi^R(w))=C\beta w \delta/(w^2+\delta^2), Re(\chi^R(w))=C\beta \delta^2/(w^2+\delta^2), chi^{th}(iw_n)=C\beta\delta/(w_n+\delta)>C \beta \delta_{n,0} at \delta>0.
5. It would be good in my opinion to clarify how the first order transition line in Fig. 1 is determined. Is it obtained from the comparison of full energies of metallic and insulating solutions, or their free energies, or something else?
6. The definition of the Widom line is in my opinion better present in the beginning of Sect. 4 (first line of page 14).
7. In Sect. 4.1 the authors discuss "rapid decay" of local spin correlation function in the insulating phase; I think it is better to characterise it as a rapid decrease, since change in the magnitude of the correlation function is not large in that regime.
8. For the data presented in the right part of Fig. 2 and left part of Fig. 4 the authors might consider the comparison to the analytic forms discussed above in p. 4 and/or suggest their own analytic forms.
9. Regarding right part of Fig. 5: do I understand correctly that finite (although small) C in the metallic phase (M>I path) is an artefact of the numerical procedure of calculating C? If yes, I think it is worth to write this explicitly.
10. In the upper plot of the right part of Fig. 7 I think it is worth to mark the position of the Widom line.
Author: Clemens Watzenböck on 20220405 [id 2358]
(in reply to Report 1 by Andrey Katanin on 20220104)
We thank the Referee for the careful reading of our manuscript, for his appreciation of our work and for finding our results suited for being presented in SciPost Physics.
The Referee has made constructive and useful comments/observations in his report. They have helped us improving the clarity and the precision of our presentation as well as the completeness of our bibliography. A structured reply to the observations of the Referee is attached as *"reply_and_latexdiff_LTM.pdf"*
Author: Clemens Watzenböck on 20220405 [id 2357]
(in reply to Report 2 on 20220112)We thank the Referee for her/his positive assessment on both our results and presentation, and for supporting publication in SciPost Physics. In particular, we appreciate the constructive criticisms of the Referee aiming at improving the clarity of selected paragraphs.
In the resubmitted manuscript we have taken care of all the points raised in her/his report. A structured reply to the observations of the Referee is enclosed below:
Thank you for pointing this out. To make the paper selfcontained we have now included the definitions in Eq. (2).
In order to avoid possible misunderstandings, we now state explicitly in section 3.2 that all quantities in real frequencies shown in our paper are obtained by analytic continuation. We have also added some more specific statement by discussing Eqs. (44)(45). [Eqs. (45)(46) in the new version; see provided latexdiff].
The errorestimate strongly depends on the signaltonoise ratio and the temperature. On the one hand, by increasing $T$ the firstMatsubara frequency will be further away from the zeroth one. Hence, one has to extrapolate over a larger frequency distance. This problem can – at least in part – be mitigated by demanding a temperaturedependent minimal blurwidth. On the other hand, the difference between the isothermal and Kubo susceptibility $\chi^{\mathrm{th}}(\mathrm{i} \omega_n=0)  \chi^{\mathcal{R}}(\omega=0) = \beta C$ becomes smaller for larger temperatures. This will lead to a larger error on our estimate of $C$ in the hightemperature regime.
Motivated by the Referee's observation, we have now included an additional analysis in the appendix for test data that is similar to measured QMCdata. Our analysis shows that for $U=3$ the errorestimate is $<1\%$ for low temperatures and for the highest considered temperature still $<5\%$. inside of the coexistence region just before metaltoinsulator phase transition we estimate the error to be rather large ($C=0.06 \pm 0.05$). The intrinsic difficulty there is that we try to distinguish a very sharp peak at finite frequency (preformed local moment with a finite lifetime) from a formally infinitely sharp peak at $\omega=0$ (anomalous term). In any case, our error estimate is also consistent with the right part of Fig. 5. In particular, after a careful analysis of the fitloss, as outlined in Eq. (44), we have concluded that $C\approx 0$ is consistent with our QMC data.
For the sake of clarity, we now mention explicitly in the main text that our estimated value for $C$, which is finite though small, should be regarded as an artifact of the extraction method, which results from a bad signaltonoise ratio. We have also included a simple error analysis for some selected cases in appendix D.
In response to this Referee's question, we have now included a small section in the appendix E, where we explicitly address the largetemperature limit ($T \gg U, W$). A direct comparison with Fig. 7 is, however, not possible, because even the highest temperature ($T=0.1$) reported there, it is still very low compared to the other relevant energy scales of the problem (like $U$ or the bandwidth $W$). We now elaborate on this point in the new appendix section.
Yes, they are. We added a sentence below Eq. (44) to clarify this.
We used that for $\chi^c(\omega)=\mathrm{i}(\mathrm{e}^{\beta \omega}  1)\chi^<(\omega)$ the prefactor $(\mathrm{e}^{\beta \omega}  1)$ is zero for $\omega=0$. We have now added a specific footnote to make this step clearer.
Thank you for pointing this out. Indeed, we verified that in the printout of one of us the lines were hardly visible, too. In order to avoid possible printsetting related problems, we have modified the colors in Fig.~1 accordingly.
There is also a fourth curve that is almost on top of another one. We now modified some of the plotmarkers to increase visibility.
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reply_and_latexdiff_LTM.pdf