Parent Hamiltonians of Jastrow Wavefunctions

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.

1) Most results reported in this manuscript represent a generalization of previously known results.

The authors find the family of many-body Hamiltonians with ground-state of Jastrow form in arbitrary spatial dimensions, finding that, in general, these Hamiltonians include two and three body interactions. This finding extends the Calogero-Marchioro result, which was valid only for 3D, to arbitrary dimensions. The authors also discuss the case of wave-function with one-particle terms, leading to long-range interactions in the parent Hamiltonian. Several models are discussed. Interestingly, they discuss the reverse-engineering 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.

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.)

The authors of this manuscript presented a systematic description of the recent well-established method to construct a 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. In section 2, the parent Hamiltonians with 2-body & 3-body pairwise potentials in d-spatial dimensions are constructed. In section 3, the 1-body 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 long-range contributions are involved by mixing the 2- and 1-body 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.

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.

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 2-body interaction (Eq.11) and external potential (Eq.17). while the 2-body 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.

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.