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Parent Hamiltonians of Jastrow Wavefunctions

by Mathieu Beau, Adolfo del Campo

Submission summary

As Contributors: Mathieu Beau
Preprint link: scipost_202107_00025v2
Date accepted: 2021-10-20
Date submitted: 2021-09-23 12:12
Submitted by: Beau, Mathieu
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Computational
  • High-Energy Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


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.

Published as SciPost Phys. Core 4, 030 (2021)

Author comments upon resubmission

We thank the referees for the accurate summary of our contribution and for useful suggestions.

The novel models we have introduced have no precedence in the literature to the best of our knowledge. We term the models according to names that allows to quickly appreciate the relation with some other models, but the additive character of the interactions cannot be taken for granted and generally mixing terms occur.
Nonetheless, the strength of our contribution is in facilitating the systematic construction of new models: choose your favorite one- and and two-body functions in the Jastrow form and find the parent Hamiltonian through the equations we derive.
For historical accidents, the generality of the formalism and the results we report have not been presented in spite of the various related attempts documented in the introduction starting in 1975 and sustained to this day. In this sense, we close this effort by providing the complete family, which is infinite. Our examples are chosen to show how instances of known models are included in our formalism and how to generate new models in these infinite family.

In addition, we wish to draw the attention of the referee to the work by Kane et al. which established that parent Hamiltonians of Jastrow wavefunctions share the same long-wavelength behavior than that of the same Hamiltonian in the absence of three-body interactions. That makes the resulting models physically appealing: the physics is set by the two-body interactions. Further, the family is infinite ans instances can be found by reverse engineering for given interactions. In view of these results and the long-time impact of the preceding related works on the topic, it seems clear that our work is an important contribution meriting publication in SciPost Physics. In some sense, it closes a search started in 1975 by Calogero by identifying the complete family of parent Hamiltonians of Jastrow wavefunctions in any spatial dimension.
To put in perspective our results, it is important to notice that the more limited Calogero construction is actually a reference result and in a sense a textbook result (see ref. [13] and [29] among many others). In addition, as we mention in the discussion, our work has a plethora of ramifications.

List of changes

- we expanded the discussion on Jastrow functions and added two references, see [11] and [12].
- we added a sentence on the quasi-solvability of models given by equation (1).

- we have expanded the discussion in this revision to reflect various approaches in quantum solids. We added seven new references [55-61].
- We emphasized that the system are quasi-exactly solvable and added the following sentence: "The systems discussed are generally quasi-exactly solvable and thus only part of the spectrum may be derived"

- We added: "We thank P. Le Doussal for pointing out the recent reference [61] about one-dimensional fermionic ground-states and for discussing the connection of our work with his recent work [62] about diffusion of interacting particles in one dimension."

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