Towards a Quantum Fluid Theory of Correlated Many-Fermion Systems from First Principles

Dear Editors,

we thank both referees for their in-depth review of our manuscript and for providing constructive feeback.
We believe we have addressed all concerns and comments raised by the referees and have revised our manuscript accordingly.

We hope our revised manuscript is now suitable for publication.

Sincerely,
Attila Cangi

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Response to Referee 2
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>This paper deals with the Quantum Hydrodynamics (QHD) approach for many-fermions systems. The main aim of the paper is at >clarifying the validity of the Bohm potential. The claim of the authors is that this widely used framework fails for strong perturbations. >The authors derive a many-fermion Bohm potential, showing that it yields different results as compared with the Bohm potential.

>This paper deals with a difficult problem, as no effective computational method for simulating many-fermion systems out of >equilibrium exists, especially for two or higher-dimensional systems. Thus, I believe that the paper deserves to be published.}

We thank the referee for reviewing our manuscript and for recommending publication. We address the referee's comment below.

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Comment 1:

>As far as I understand in using eq. (5) and eq. (6) to simulate the out-of-equilibrium dynamics one has to determine or find some >suitable approximations for $v_{xc}$ and $P_e$. So it is not clear a priori whether having a many-fermion potential is sufficient to >obtain trustable results for the out-of-equilibrium dynamics. I think it would be nice if the authors could comment on that in the manuscript.

We agree that choosing reliable approximations for the terms $v_{xc}$ and $P_e$ is important.
Recent developments in the construction of exchange-correlation functionals are highly relevant. For example, the parametrization of the interacting electron gas at finite temperature by Groth et al. [Phys. Rev. Lett. 119, 135001 (2017)] would be a suitable extension to improve upon $v_{xc}$.
The electronic pressure $P_e$ can in principle be evaluated exactly according to its definition in terms of an average over orbital contributions. While this is beyond the scope of this manuscript, it would be an interesting analysis. It would give quantitative insight into the common approximation to $P_e$ in terms of the ideal Fermi gas.

We have revised our manuscript below Eq. (6) accordingly:
``Proven approximations to the exchange-correlation energy, i.e., $v_{xc}$ can be employed where recent developments such as the parametrization of the interacting electron gas at finite temperature~(36) provide a solid basis for an accurate inclusion of exchange-correlation effects into the QHD equations.
...
Commonly, the electronic pressure $P_e$ is approximated by the ideal Fermi gas.''
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Response to Referee 1
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>The present work concerns the investigation of a new approach to the quantum hydrodynamics (QHD) of many-fermion systems. >Specifically, the key development consists in considering a many-fermion Bohm potential (MFBP), in contrast to the standard Bohm >potential (SBP) that is typically used in current applications.
>
>The appropriateness of the MFBP in the present context is first argued on theoretical grounds, by providing a first-principles derivation >starting from the formally exact Kohn-Sham equations. Subsequently, considering a harmonically perturbed electron gas, it is shown >that the MFBP significantly differs from the SBP in the presence of large density perturbations. Differences in the resulting forces are of >comparable order of magnitude to other terms in the QHD equations, so that significant deviations in the corresponding dynamics can >be expected.
>
>The manuscript is interesting and the results presented in it appear sound, in light of the theoretical derivations and the benchmarking >provided for the density calculations. Furthermore, the developments of this manuscript appear timely and relevant, due to the great >interest in many-fermion systems originating from different fields and the intrinsic limitations of existing numerical approaches, such >as Monte Carlo and DMRG. Developments in this direction could have many applications, as argued in the introduction and conclusion.
>
>However, while the results presented in this work are valuable, I believe that the manuscript could benefit from a more definite >presentation of what precisely constitutes a new development. For instance, the derivations of the Theory section closely mirror the >developments of a previous work of two of the authors, Ref. [11]; see esp. Eq. (37-38) and (42-47) therein. In fact, the MFBP already >appears in [11] (although this terminology is not used), where it is argued that the difference between the SBP and MFBP was neglected >in previous derivations, and that this can be expected to lead to significant differences when orbital amplitudes are not identical. >Furthermore, I think that claims such as “illustrating the failure of the standard Bohm potential” or “this enables more accurate QHD >simulations” would be more strongly justified in the presence of an explicit study of the resulting dynamics.
>
>This work’s main advancement is explicitly showing the difference between the MFBP and SBP for a physically relevant scenario, and >arguing that this can be expected to significantly affect the resulting dynamics. I believe this to be in itself a relevant result, and I >therefore recommend the publication of this manuscript provided the following points are addressed (i.e. either implemented or >convincingly rejected):

We thank the referee for reviewing our manuscript and for providing valuable feedback. We address the individual points below.

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Comment 1:
>As discussed above, theoretical derivations significantly draw on earlier developments, esp. those of Ref. [11], which should be more >thoroughly referenced throughout the Theory section.}

We agree with this. In the revised manuscript we now point to Ref. [11] at various places where appropriate.
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Comment 2:
>In order to make the manuscript more self-contained and to facilitate a comparison to earlier methods, it would be beneficial to >explicitly provide additional details on how QHD has been performed in earlier works. Specifically: do all mentioned earlier works simply >make use of Eqs. (5-6), but with the MFBP being replaced by the SBP? If so, this should be more explicitly stated. If not, any additional >differences should be highlighted. (Of course, different approximation methods for the stress tensor etc. could also be used, which are >not presently relevant; what would be useful here is to elucidate what exactly is the starting point for earlier approaches vs. the present >work.)

In earlier works, the momentum equation was used along with the Bohm potential containing the mean density (SBP), Eq. (1). There have been a number of different derivations of this result [see Refs. (17; 24; 25), all cited before Eq. (1)].
The bottom line is that indeed in prior works indeed only Eq. (1) has been used, whereas the primary point of this paper is to employ the many-fermion Bohm potential, Eq. (2).

In response to the referee, we implemented the following change in our manuscript. At the end of Sec. 2 we now point out the relation to the traditional QHD approach:

"We stress that in traditional QHD used in prior works, the many-fermion Bohm potential is approximated by the standard Bohm potential in terms of the mean density as defined in Eq. (1)."
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Comment 3:
>When Eq. (1) is given, a few Refs could be included where this particular form was used.

We agree and added a few relevant references (Refs. [17, 24, 25]) before Eq. (1).
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Comment 4:
>When Eq. (7) is introduced, it would also be useful to provide some Refs in which this Hamiltonian was previously considered.

We agree and added several references above Eq. (7):

[37] S. Moroni, D. M. Ceperley and G. Senatore, Static response from quantum Monte Carlo calculations, Phys. Rev. Lett.69, 1837 (1992), doi:10.1103/PhysRevLett.69.1837.

[38] S. Moroni, D. M. Ceperley and G. Senatore, Static response and local field factor of the electron gas, Phys. Rev. Lett.75, 689 (1995), doi:10.1103/PhysRevLett.75.689.

[39] T. Dornheim, M. Böhme, Z. A. Moldabekov, J. Vorberger and M. Bonitz, Density response of the warm dense electron gas beyond linear response theory: Excitation of harmonics, Phys. Rev. Research 3, 033231 (2021), doi:10.1103/PhysRevResearch.3.033231.
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Comment 5:
>When mentioning a “breakdown” of the SBP, it would be appropriate to clarify that this is argued on the grounds of the SBP yielding >significantly different forces compared to the theoretically motivated MFBP, rather than based on a direct comparison of the resulting >dynamics. In fact, while the expected difference in dynamics is convincingly argued by comparing the resulting forces to other relevant >terms, it is not explicitly shown, and I think this point should be made clear.

We have softened the wording, instead of "breakdown" we refer to our observation as "significant differences".
Nevertheless, we believe, that our analysis in terms of forces should be sufficient to claim that the dynamics using the MFBP will significantly differ from the common approach of using the standard Bohm potential. Preliminary results for actual dynamics of shock propagation taking into account the standard Bohm potential have just become available in a preprint (Ref. 44). These results give a good insight into the expected changes when the MFBP would be used. Our revised manuscript now refers to this recent results. More details are given in our reply to the comment below.
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Comment 6:
>Also, the significance of this discrepancy for strong density perturbations is only explicitly shown for one particular system, whereas the >wording of the final part of Section 1 might suggest that this is a more general finding, then illustrated with one example.

While we have shown results for a given perturbation based on one harmonic in Eq. (7), we believe that the presented results are of general character. The reason is that any perturbation can be expressed as a linear combination of harmonic perturbations.
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Comment 7:
>Indeed, since the ultimate aim of the present work is improving a numerical method for time evolution, the paper would greatly benefit >from an explicit benchmarking of the dynamics induced by the SBP vs. the MFBP (even for some special case where at least some results >are available for comparison), since at present no examples of dynamics are shown.

We agree with the referee that our manuscript lays the basis for solving for the dynamics in terms of the QHD equations. Presently, fully time-dependent numerical solutions that employ the MFBP are absent. This is work in progress. Preliminary results (including some of the authors of this manuscript) on employing the SBP for shock propagation have just become available in a preprint which we now cite.

In the conclusions of our original mansucript, we had already discussed the expected effects of the MFBP on the shock propagation in WDM. Based on the recent results described in the new prepint, we can now make our predictions more specific (see 4th paragraph in the conclusion):

"Particularly, the effect of the standard Bohm potential has very recently been assessed in hydrodynamics simulations (45). These clearly demonstrate the different dynamics obtained and indicate the importance the many-fermion Bohm potential might have on the dynamics of the shock formation."

[45] F. Graziani, Z. Moldabekov, B. Olson and M. Bonitz, Shock physics in warm dense matter – a quantum hydrodynamics perspective (2021), arxiv:2109.09081
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Comment 8:
>In addition, I think the manuscript could potentially benefit from the following minor comments and suggestions:

>1. In the abstract, it could be useful to clarify "strong density perturbations".
>2. On p. 1, when discussing QHD, it could be helpful to clarify that the "surge of activities" concerns existing applications of QHD.
>3. It would be useful to define all quantities as soon as they are first introduced, e.g. $n_i$, $f_i$ in Eq. (2). Explicit formulae could also >be provided, e.g. the equation for $f_i$ as done in [11].
>4. On p. 4, when defining $q=n q_{min}$, $n$ could be confused with the density; a different letter could be used. It would also be >helpful to explicitly indicate what $n$ is set to in the various cases.
>5. In the caption of Fig. 2, the meaning of the red vs orange lines could also be briefly explained.
>6. In the discussion of Fig. 3 on p. 6, following "We infer that…", forces are improperly referred to as potentials. I would suggest using a >more precise wording (e.g. "many fermion / standard Bohm force").
>7. Since the MFBP and SBP are mentioned frequently in the manuscript, the authors could consider using an abbreviation (e.g. using an >acronym as done in this report).

We thank the referee for the thorough reading of our manuscript that resulted in these comments and suggestions. We have revised the paper taking into account all items, except the last. We prefer to avoid using these abbreviations in order to keep our manuscript accessible to a broader audience.
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Revision 1:

We have revised our manuscript below Eq. (6) accordingly:
``Proven approximations to the exchange-correlation energy, i.e., $v_{xc}$ can be employed where recent developments such as the parametrization of the interacting electron gas at finite temperature~(36) provide a solid basis for an accurate inclusion of exchange-correlation effects into the QHD equations.
...
Commonly, the electronic pressure $P_e$ is approximated by the ideal Fermi gas.''
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Revision 2:
At the end of Sec. 2 we now point out the relation to the traditional QHD approach:

"We stress that in traditional QHD used in prior works, the many-fermion Bohm potential is approximated by the standard Bohm potential in terms of the mean density as defined in Eq. (1)."
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Revision 3:
We added several references above Eq. (7):

[37] S. Moroni, D. M. Ceperley and G. Senatore, Static response from quantum Monte Carlo calculations, Phys. Rev. Lett.69, 1837 (1992), doi:10.1103/PhysRevLett.69.1837.

[38] S. Moroni, D. M. Ceperley and G. Senatore, Static response and local field factor of the electron gas, Phys. Rev. Lett.75, 689 (1995), doi:10.1103/PhysRevLett.75.689.

[39] T. Dornheim, M. Böhme, Z. A. Moldabekov, J. Vorberger and M. Bonitz, Density response of the warm dense electron gas beyond linear response theory: Excitation of harmonics, Phys. Rev. Research 3, 033231 (2021), doi:10.1103/PhysRevResearch.3.033231.
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Revision 4:
4th paragraph in the conclusion:

"Particularly, the effect of the standard Bohm potential has very recently been assessed in hydrodynamics simulations (45). These clearly demonstrate the different dynamics obtained and indicate the importance the many-fermion Bohm potential might have on the dynamics of the shock formation."

[45] F. Graziani, Z. Moldabekov, B. Olson and M. Bonitz, Shock physics in warm dense matter – a quantum hydrodynamics perspective (2021), arxiv:2109.09081
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