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

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 in Scipost Physics Core.

My main criticism is the following: 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.

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):

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.
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.)
3. When Eq. (1) is given, a few Refs could be included where this particular form was used.
4. When Eq. (7) is introduced, it would also be useful to provide some Refs in which this Hamiltonian was previously considered.
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.
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.
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.

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).

Please refer to the above report.