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Parent Hamiltonians of Jastrow Wavefunctions
by Mathieu Beau, Adolfo del Campo
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Submission summary
| Authors (as registered SciPost users): | Mathieu Beau |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202107_00025v1 (pdf) |
| Date submitted: | July 14, 2021, 1:36 p.m. |
| Submitted by: | Beau, Mathieu |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to arbitrary spatial dimensions. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-8-31 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00025v1, delivered 2021-08-31, doi: 10.21468/SciPost.Report.3470
Strengths
1) In this manuscript, the authors find the family of many-body Hamiltonians with ground-state of Jastrow form in arbitrary spatial dimensions. This extends previous results known for one-dimensional and for three-dimensional systems only, extending the cases considered in Journal of Mathematical Physics 16(5), 1172 (1975) and in Phys. Rev. B 43, 3255 (1991). 2) The authors also show that, in some cases, the pair-function can be reversed engineered, finding a wave-function corresponding to a given two-body potential.
Weaknesses
1) Most results reported in this manuscript represent a generalization of previously known results.
Report
The findings presented in this manuscript are interesting and sound. However, it is fair to say that they represent a generalization of previously know results.
I find that the manuscript is definitely suitable for publication in SciPost Physics Core, provided that the comments reported in the "Requested Changes" are adequately addressed in a revised version. In order for me to give a strong recommendation for publication in the flagship journal SciPost Physics, the authors should better emphasize the relevance of the generalized results, and discuss more in depth the physics of at least one of the novel models introduced in this manuscript. Can the authors provide some interesting predictions for these models? Such predictions would highlight the relevance of the techniques discussed in the manuscript.
Requested changes
1) In the introduction, the authors state that the Jastrow wave-function with only two-body terms is suitable to describe quantum solids. It is my understanding that, in fact, this wave-function only captures the properties of fluid states. As mentioned by the authors in the conclusions, the Nosanov-Jastrow wave-function is instead suitable to describe the solid state.
2) In the introduction, the authors write "Slater determinants of such Jastrow functions are also widely used in and quantum chemistry." First, there is perhaps a typo ("...in and..."). More importantly, this statement is not clear. In fact, electronic systems are often described via products of Jastrow functions and Slater determinants of single-particle wave-functions. Fermionic (i.e., antisymmetric) wave-functions can also be built starting from pair orbitals, but using, in general, Pfaffian wave-functions (PRL 96, 130201 (2006), J. Chem. Theory Comput. 16.10, 6114-6131 (2020)).
3) In the conclusion, the authors mention the Nosanov-Jastrow wave-function for bosonic solid states. However, it is worth mentioning that the original model does not satisfy the bosonic symmetry. In order to account for Bose-Einstein statistics, various approaches have been introduced, including symmetrized wave-functions (J. Stat. Mech. P07003 (2005), NJP 11 013047 (2009)), shadow wave-functions (PRB 38, 4516 (1988), PRL 60, 1970 (1988), PRB 71 140506 (2005)) and permutation-sampling methods (PRB 17 1070 (1978), PRL 108, 155301 (2012)).
4) The authors use the term quasi-exactly solvable models. The meaning is elucidated only at the end of the manuscript, and it refers to models for which only part of the spectrum is obtained. This definition should be given earlier. Also, it is not clear if, in fact, in most cases only the ground-state energy is known, and if its evaluation requires additional computations (e.g., Monte Carlo sampling of the Jastrow wave-function.)
Report #1 by Anonymous (Referee 1) on 2021-8-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00025v1, delivered 2021-08-16, doi: 10.21468/SciPost.Report.3399
Strengths
Weaknesses
Report
I shall be happy to recommend their revised version of this submission for publication in SciPost.
Requested changes
At least, the authors should discuss physical understanding and a possible application of such constructed Hamiltonians in more details. They should also anticipate how such kind of wave functions can be used to calculate the correlation functions.
Author: Mathieu Beau on 2021-09-23 [id 1775]
(in reply to Report 1 on 2021-08-16)See attached pdf
Author: Mathieu Beau on 2021-09-23 [id 1776]
(in reply to Report 2 on 2021-08-31)See attached pdf
Attachment:
Ref2.pdf