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Deriving density-matrix functionals for excited states
by Julia Liebert, Christian Schilling
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Submission summary
| Authors (as registered SciPost users): | Christian Schilling |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2210.00964v2 (pdf) |
| Date submitted: | 2023-01-22 17:11 |
| Submitted by: | Schilling, Christian |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We initiate the recently proposed $\boldsymbol{w}$-ensemble one-particle reduced density matrix functional theory ($\boldsymbol{w}$-RDMFT) by deriving the first functional approximations and illustrate how excitation energies can be calculated in practice. For this endeavour, we first study the symmetric Hubbard dimer, constituting the building block of the Hubbard model, for which we execute the Levy-Lieb constrained search. Second, due to the particular suitability of $\boldsymbol{w}$-RDMFT for describing Bose-Einstein condensates, we demonstrate three conceptually different approaches for deriving the universal functional in a homogeneous Bose gas for arbitrary pair interaction in the Bogoliubov regime. Remarkably, in both systems the gradient of the functional is found to diverge repulsively at the boundary of the functional's domain, extending the recently discovered Bose-Einstein condensation force to excited states. Our findings highlight the physical relevance of the generalized exclusion principle for fermionic and bosonic mixed states and the curse of universality in functional theories.
List of changes
1) We corrected the three typos pointed out by referee 1.
2) We interchanged the second and third expression in old Eq. (5)/new Eq. (6).
3) We added the subscript $N$ in the trace in the paragraph below Eq. (1) and in old Eq. (2)/new Eq. (3).
4) We removed the term ''dispersion relation'' in the context of $t_{\boldsymbol{p}}$ below old Eq. (15)/new Eq.(16) and added the word ''relation'' to '' dispersion relation'' below old Eq. (28)/new Eq. (29).
5) We added the definition of the set $\mathcal{E}^N(\boldsymbol{w})$ as a new Eq. (2).
6) We added the term ''non-vanishing weights'' in the first paragraph of Sec. IV.E. also after old Eq. (36)/new Eq. (37) to explain the definition of $r=2$.
7) We added the definition of the set $\mathcal{S}_d$ and the dimension $d$ in the sentence below old Eq. (35)/new Eq. (36).
8) We added the explanation of the abbreviation ''GOK'' in the sentence below Eq. (1).
9) We added the normalization of the weight vector, $\sum_{i=1}^D w_i=1$, in the paragraph above the new Eq.(2).
10) There was a wrong factor of $2$ in the Fourier transform of the kinetic energy in Appendix A and we corrected Eqs. (A13)-(A19) accordingly.
11) We recall the definition of $\boldsymbol{p}^\prime$ below the old Eq. (43)/new Eq. (44).
12) We added below old Eq. (20)/new Eq. (21) an explanation of the purpose of the operators $\hat{\beta}_0, \hat{\beta}^\dagger_0$ to stress that they ensure particle-number conservation.
13) We added below old Eq. (22)/new Eq. (23) an additional explanation how old Eq. (24)/new Eq. (25) is derived and refer the reader to Ref. [51].
14) We added a further comment regarding consequences of level crossings in functional theories in Sec. IV.C and added the references [54-56].
15) We corrected a typo in old Eq. (45)/new Eq. (46).
16) We added an additional explanation how old Eq. (47)/new Eq. (48) is derived below old Eq. (47)/new Eq. (48).
17) We recall the definition of $\boldsymbol{q}^\prime$ below old Eq. (55)/new Eq. (56).
18) We added below Eq. (55)/new Eq. (56) an explanation of the term ''gradient force is collectively diverging'' in the context of the BEC force.
19) In Sec. V we replaced ''$r=2$ finite weights'' by ''$r=2$ non-zero weights''.
20) We added in the first paragraph of Sec. IV a sentence explaining that the three approaches to derive a universal functional can be applied to fermions on an equal footing and comment on the Legendre-Fenchel transform as an explicit example. We also added a comment on the validity of these different routes to derive universal functionals and of Sec. IV.C and IV. D in the last sentence of the second paragraph in Sec. V.
21) We added a comment on attractive interactions below Eq. (54)/new Eq. (55).
22) We added an explanation why the universal functional $\mathcal{F}_{\boldsymbol{w}}$ is in general not convex before applying an exact convex relaxation in the paragraph below old Eq. (2)/new Eq. (3).
23) We added an explanation that we use for simplicity the same symbol for the one-particle Hamiltonian $\hat{h}$ on the $N$-particle Hilbert space and the one-particle Hilbert space below old Eq. (2)/new Eq. (3).
Current status:
Reports on this Submission
Report #3 by Klaas Giesbertz (Referee 1) on 2023-2-13 (Invited Report)
- Cite as: Klaas Giesbertz, Report on arXiv:2210.00964v2, delivered 2023-02-13, doi: 10.21468/SciPost.Report.6737
Report
The authors have made significant improvements to the article. We only have a few points which we would still like to be seen addressed.
- The factor 2 has now been corrected in the Appendix, but the wrong factor seems still to be present in Eq. (14) and the line before.
- After eq.(4) 2nd paragraph : “The non-convexity of the set $\mathcal{E}^1_N({\bf w})$ implies that also universal functional $\mathcal{F}_{\bf w}$ is not necessarily convex.” A non-convex domain immediately implies that the functional cannot be convex, so “… also universal functional $\mathcal{F}_{\bf w}$ is not convex”. However, $\mathcal{E}^1_N({\bf w})$ can be convex as the authors explicitly use a convex example themselves, so to say “The non-convexity of the set $\mathcal{E}^1_N({\bf w})$” is not completely correct. Perhaps the sentence could be replaced by something like “Since $\mathcal{E}^1_N({\bf w})$ is not necessarily convex, the functional $\mathcal{F}_{\bf w}$ is not necessarily convex?” Or “In case $\mathcal{E}^1_N({\bf w})$ is not convex, the functional $\mathcal{F}_{\bf w}$ cannot be convex.”?
- As the minimum over convex functionals is not necessarily convex getting eq.(42) from eq.(40), and eq.(41) is not “immediate”. Consider for example the function $f(x) = \min\{x^2, (x-2)^2\}$. This is a minimimization over two convex functions, but f(x) is not convex. So using only these arguments, one only has ≤ in eq. (42) or alternatively one could take the convex hull of the current r.h.s.
Nicolas Cartier, Sarina Sutter and Klaas Giesbertz
Author: Christian Schilling on 2023-03-02 [id 3424]
(in reply to Report 3 by Klaas Giesbertz on 2023-02-13)We highly appreciate the effort made by the referees and reply to these points below.
We corrected the typos in Eqs. (14) and (15) (Eqs. (13) and (14) in the new manuscript) and the line above Eq. (14).
We understand that point by the referees. Yet, it is not completely uncommon to extend the definition of convexity of a function also to scenarios with non-convex domains (see, e.g., 'Convex functions on non-convex domains', Econmics Letters 22, 251-255, 1986). Nonetheless, as a compromise we tweaked the corresponding paragraph, below Eq. (4).
We thank the referees for this important comment and we added a lower convex envelop operation `conv' on the rhs of Eq.(40) in the revised manuscript. It actually even turns out that the minimum of the family of functions defined in new Eq. (43) is not convex. Because of this, we had to modify some statements in the subsequent sections (paragraph below Eq. (43), last paragraph of Sec. IV.E.2, first paragraph of Sec. IV.E.3 and the end of the second paragraph of Sec. V).