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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:  20210714 13:36 
Submitted by:  Beau, Mathieu 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We find the complete family of manybody quantum Hamiltonians with groundstate of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a twobody pairwise potential as well as a threebody potential. We thus generalize the CalogeroMarchioro construction for the threedimensional case to arbitrary spatial dimensions. The resulting family of models is further extended to include a onebody term representing an external potential, which gives rise to an additional longrange twobody interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of wellknown systems such as the CalogeroSutherland and CalogeroMoser models. We also introduce novel models, generalizing the McGuire manybody quantum bright soliton solution to higher dimensions and considering groundstates which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverseengineered to construct models with a given potential, such as a pairwise Yukawa potential, and to identify models governed exclusively by threebody interactions.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021831 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202107_00025v1, delivered 20210831, doi: 10.21468/SciPost.Report.3470
Strengths
1) In this manuscript, the authors find the family of manybody Hamiltonians with groundstate of Jastrow form in arbitrary spatial dimensions. This extends previous results known for onedimensional and for threedimensional 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 pairfunction can be reversed engineered, finding a wavefunction corresponding to a given twobody potential.
Weaknesses
1) Most results reported in this manuscript represent a generalization of previously known results.
Report
The authors find the family of manybody Hamiltonians with groundstate of Jastrow form in arbitrary spatial dimensions, finding that, in general, these Hamiltonians include two and three body interactions. This finding extends the CalogeroMarchioro result, which was valid only for 3D, to arbitrary dimensions. The authors also discuss the case of wavefunction with oneparticle terms, leading to longrange interactions in the parent Hamiltonian. Several models are discussed. Interestingly, they discuss the reverseengineering of the Jastrow pair function corresponding to a parent Hamiltonian.
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 wavefunction with only twobody terms is suitable to describe quantum solids. It is my understanding that, in fact, this wavefunction only captures the properties of fluid states. As mentioned by the authors in the conclusions, the NosanovJastrow wavefunction 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 singleparticle wavefunctions. Fermionic (i.e., antisymmetric) wavefunctions can also be built starting from pair orbitals, but using, in general, Pfaffian wavefunctions (PRL 96, 130201 (2006), J. Chem. Theory Comput. 16.10, 61146131 (2020)).
3) In the conclusion, the authors mention the NosanovJastrow wavefunction for bosonic solid states. However, it is worth mentioning that the original model does not satisfy the bosonic symmetry. In order to account for BoseEinstein statistics, various approaches have been introduced, including symmetrized wavefunctions (J. Stat. Mech. P07003 (2005), NJP 11 013047 (2009)), shadow wavefunctions (PRB 38, 4516 (1988), PRL 60, 1970 (1988), PRB 71 140506 (2005)) and permutationsampling methods (PRB 17 1070 (1978), PRL 108, 155301 (2012)).
4) The authors use the term quasiexactly 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 groundstate energy is known, and if its evaluation requires additional computations (e.g., Monte Carlo sampling of the Jastrow wavefunction.)
Report #1 by Anonymous (Referee 1) on 2021816 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202107_00025v1, delivered 20210816, doi: 10.21468/SciPost.Report.3399
Strengths
The authors of this manuscript presented a systematic description of the recent wellestablished method to construct a complete family of manybody quantum Hamiltonians with groundstate of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. In section 2, the parent Hamiltonians with 2body & 3body pairwise potentials in dspatial dimensions are constructed. In section 3, the 1body term serving as external potential in the Hamiltonian is introduced by adding one particle term to the Jastrow form of the wave function, Consequently, the longrange contributions are involved by mixing the 2 and 1body couplings in the Hamiltonians. Some simple examples were mentioned in this section. The section 4 was devoted to construct 9 models by using the method discussed in previous sections, among which the first 4 have been well studied in literatures, for example, see “PHYSICAL REVIEW RESEARCH 2, 043114 (2020)” etc., and the last 5 ones are newly constructed. However, these are in fact not totally new. They can be a kind of combinations of previous 4 models. In the section 5, they illustrated how to construct the explicit Jastrow form wave function once the interaction is known.
Weaknesses
The weakest part of this paper is their method. It is not a new method and such kind of Hamiltonians have been studied before. In addition, the physical meaning of these models was not explained.
Report
In view of the systematic construction of such kind of systems, I see that the manuscript was well organized and written. The key merit of this paper is that the parent Hamiltonian construction was systematically generalized to any spatial dimensions. In the 1d case, the newly constructed Hamiltonians involve 2body interaction (Eq.11) and external potential (Eq.17). while the 2body interaction and external potential are seen in higher dimensions (d>1) case.
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 20210923 [id 1775]
(in reply to Report 1 on 20210816)See attached pdf
Author: Mathieu Beau on 20210923 [id 1776]
(in reply to Report 2 on 20210831)See attached pdf
Attachment:
Ref2.pdf