Electric polarization and magnetization in metals

As detailed in my report some elements are missing or non sufficiently clear to be able to identify the strengths and weaknesses of this work.

As detailed in my report some elements are missing or non sufficiently clear to be able to identify the strengths and weaknesses of this work.

Dear editor,

The authors propose a new definition of the electric polarization and magnetization. The authors claim that, contrary to previous definitions of these quantities (in particular, in the modern theories of polarization and magnetization), the new definitions are also well-defined for metals. My main criticism of this work is the absence of any calculation (analytical or numerical) that would support this claim and also, more generally, that would confirm the viability of the authors’ approach for practical calculations. I note that also in other recent works by the same authors (PRB 99 235140, PRR 2 033126, PRR 2 043110) in which they present the same general method I have not found any calculation. Would it be possible to add some simple calculation, e.g. tight-binding or other, to illustrate the authors’ approach? As is well-known several complications might arise when putting a new method to the test.

Please find some further questions and comments below

1) The main results of the paper seem to be the definition of the polarization in Eq. (26) and the definition of the magnetization in Eq. (27). Is this indeed the case?

2) The only thing that can be measured in experiment is the change of polarization which is fully defined in terms of the time integral of the current density (assuming there is no magnetization). This works equally well for insulators and for metals. So, what is the importance of defining an absolute polarization which is a gauge-dependent quantity? (see also point 7 below)

3) In the abstract the authors write “If instead one takes the view that the electronic polarization is more fundamentally related to the existence of a complete set of exponentially localized Wannier functions, a definition is always admitted.” I do not see what is fundamental about the existence of a complete set of exponentially localized Wannier functions (ELWF). They might be very useful in practice for numerical calculations and/or for interpretation of the results but what is fundamental about them?

4) The authors use site-based quantities such as a site density and site current density but they are never explicitly defined. Could the authors please add these definitions (or point to the appropriate equations in earlier works)? Are the definitions general or are they contingent on the choice of using ELWF?

5) In the introduction it is written that a lattice sums are performed to give the microscopic polarization and magnetization. Lattice sums are generally conditionally convergent which could lead to convergence problems in practical calculations. Could the authors comment on this?

6) Also in the introduction, the authors write “One reason is that usual calculations made in minimal coupling can require the identification of sum rules to show properly behaved results at low frequencies - especially if nonlinear optical response is calculated, which is a future direction for this work - and that is not a difficulty with the calculations presented here, since the response is calculated as due to electric and magnetic fields.” In the linear response the problem of divergencies at low frequencies have been addressed in the literature and several solutions have been proposed (PRB 82 035104, PRB 95 155203). Indeed, I think the use of sum rules is an elegant approach to avoid divergencies.

7) Just below the authors write “A second reason is that in an insulator there is a clear physical significance to the response of the polarization to applied fields, as has been demonstrated within the “modern theory,"” Could the authors explain what this physical significance is?

8) Why do the authors neglect local-field corrections which can be very important? Moreover, no calculations are performed in this work, everything is purely formal, so why neglect them?

9) The authors write “Consideration of the phenomena related to spatially-varying electromagnetic fields is left for future work.” but one page further they write “The primary difference between our approach and those is that this investigation is a limiting case of a more general framework within which spatial and temporal variation of electric and magnetic fields can be into account; that is not the case in earlier works.” If this is the primary difference it would seem important to not defer spatially-varying electromagnetic fields for future work but to include them here.

10) In the conclusions the authors write “A positive feature of the approach implemented here is that it captures the complete finite-frequency response of a metallic crystal; there is no need to implement distinct approaches to calculate the anomalous Hall and Drude contributions to J(1).” Could the authors compare to PRB 86 125139 which also seems to include all contributions. Drude contributions are also included in PRB 74 245117 but not anomalous Hall contributions because time-reversal symmetry is assumed. However, I would expect that also this contribution is present if this assumption is removed.

11) In appendix B the authors write “Although most derivations [43] work with Bloch functions from the onset, some issues related to the position operator can be avoided if one works, at least initially, in the Hilbert space containing ELWFs, the space of square-integrable functions, where the usual position operator is well-defined.” The usual position operator is not compatible with periodic boundary conditions (PBC) and, therefore, ill-defined whenever PBC are applied. Therefore, I do not understand why the authors claim that by invoking ELWFs the usual position operator is well-defined. Could the authors explain this point in more detail? I note that recently a position operator has been defined that is compatible with PBC (PRB 99 205144, PRB 105 235201).

Some minor points

12) The authors write “However, often times, and as we will take to be the case in this paper, a number of Hilbert subbundles of the Bloch bundle are trivializable, …”. What is meant by “trivializable” ?

13) The conclusion section is almost two pages long. The authors might consider to be more concise.

14) References [25] and [60] are the same.