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 an arbitrary spatial dimension. 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.
Highlights
We provide explicitly the complete family of parent Hamiltonian in arbitrary spatial dimension d with ground state of Jastrow form including one and two-body pair functions, i.e., Ψ0(r1, . . . , rN ) = i g(ri) i
We have identified the complete family of Hamiltonians with a ground-state of Jastrow form, involving one and two-body functions
For arbitrary d our results provide the complete family of parent Hamiltonians of Jastrow wavefunction without restriction to Calogero-like models associated with SU(1, 1) symmetry [34] or the nonlocal momentum dependent terms [35]
Summary
Solvable models play a prominent role in many-body physics. Their study has guided the exploration of strongly correlated states of matter, superconductivity, and other rich phenomena. Beyond the one-dimensional case, restricting the Jastrow form to the pair-wise product, Calogero and Marchioro [27] identified the family of parent Hamiltonians with a ground state of the form (1) in three spatial dimensions. The latter generally include two-body and threebody interactions. Gambardella used a group theoretical approach to identify the family of parent Hamiltonian of Jastrow ground-state wavefunctions in translationally invariant systems with SU(1, 1) symmetry [34] The latter applies to Calogero-like models with inverse-square interactions but it is rather restrictive and excludes relevant cases involving, e.g., contact and Coulomb interactions. Our results pave the way to the systematic construction of quasi-solvable models in an arbitrary spatial dimension
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