Beau and del Campo (2021) constructed parent Hamiltonians — Hamiltonians for which a given trial state is an exact eigenstate, typically the ground state — for many-body wavefunctions of Jastrow form in arbitrary spatial dimension, finding that in d≥2d\ge 2 the parent Hamiltonian generically includes a coordinate-dependent three-body potential. They identified special choices of the pair function for which the three-body term vanishes, becomes constant, or reduces to an effective two-body description, and they left open whether these Hamiltonians are integrable or merely quasi-exactly solvable and the identification of excited eigenstates and their energies. Subsequent work resolved the one-dimensional case — all 1D systems with Jastrow-form ground states are integrable — but the higher-dimensional spectral core, together with the source's envisaged extension program (spin, multiple species, supersymmetry, anyons, explicit three-body Jastrow factors), remains open.

Background

A Jastrow wavefunction is a many-body ansatz of the form Ψ(r1,…,rN)=∏i<jf(ri,rj)\Psi(\mathbf{r}_1,\dots,\mathbf{r}_N)=\prod_{i<j} f(\mathbf{r}_i,\mathbf{r}_j) (possibly multiplied by one-body factors), in which correlations enter through a product of pairwise functions; such forms underlie variational descriptions of superfluids, quantum Hall states, and nuclear matter. A parent Hamiltonian is a Hamiltonian for which the given wavefunction is an exact eigenstate; constructing one inverts the usual diagonalization problem and certifies the physical relevance of the ansatz. Beau and del Campo (2021) systematically constructed parent Hamiltonians for Jastrow-form ground states in arbitrary spatial dimension dd. In their family the pair function depends only on the interparticle distance, Ψ0=∏ig(ri)∏i<jf(rij)\Psi_0=\prod_i g(r_i)\prod_{i<j} f(r_{ij}) with rij=∥ri−rj∥r_{ij}=\|\mathbf{r}_i-\mathbf{r}_j\|, and the parent Hamiltonian with zero ground-state energy takes the explicit local form H^0=T^+V2+V3\hat H_0=\hat T+V_2+V_3, where T^=−(ℏ2/2m)∑iΔi\hat T=-(\hbar^2/2m)\sum_i\Delta_i,

V2=ℏ2m∑i<j[f′′(rij)f(rij)+(d−1)f′(rij)rij f(rij)],V3=ℏ22m∑i∑j≠k≠irijrij⋅rikrik f′(rij)f(rij) f′(rik)f(rik),V_2=\frac{\hbar^2}{m}\sum_{i<j}\left[\frac{f''(r_{ij})}{f(r_{ij})}+(d-1)\frac{f'(r_{ij})}{r_{ij}\,f(r_{ij})}\right],\qquad V_3=\frac{\hbar^2}{2m}\sum_i\sum_{j\ne k\ne i}\frac{\mathbf{r}_{ij}}{r_{ij}}\cdot\frac{\mathbf{r}_{ik}}{r_{ik}}\,\frac{f'(r_{ij})}{f(r_{ij})}\,\frac{f'(r_{ik})}{f(r_{ik})},

and the three-body term does not vanish in general unless ff is constant. Including a one-body factor gg to describe, e.g., an external trap supplements H^0\hat H_0 with a one-body potential V1V_1 and a long-range mixed two-body term V2LLV_{2LL} (Eqs. (17)–(18) and (22)–(23) of the source). In d=1d=1 the resulting Hamiltonians involve only one- and two-body potentials and connect to the Calogero–Moser–Sutherland family of exactly solvable long-range models. For d≥2d\ge 2 the parent Hamiltonian generically acquires a genuine coordinate-dependent three-body potential V3V_3, consistent with the classic observation of Calogero and Marchioro (1973) that certain three-dimensional NN-body systems with two- and three-body forces admit exact bound states. For special choices of the pair function ff the three-body term vanishes, becomes a constant, or reduces to an effective two-body term, and Beau and del Campo connected its long-wavelength behavior to the analysis of Jastrow–Laughlin quantum Hall states by Kane, Kivelson, Lee, and Zhang (1991). Two solvability notions frame the open questions: a system is quasi-exactly solvable (QES) when only a finite part of its spectrum can be obtained in closed form, while it is integrable when it admits a complete set of mutually commuting conserved quantities, from which the full spectrum is in principle accessible (for example via the Bethe ansatz, an exact parametrization of eigenstates by particle rapidities, or exchange-operator methods, which build the Hamiltonian and its commuting invariants from operators that permute particle coordinates). The construction extends a broader reverse-engineering program: parent Hamiltonians have also been reconstructed for Jastrow–Gutzwiller wavefunctions (Turkeshi 2020), for Jastrow wavefunctions on graphs (Sasmal and del Campo 2026), and for generalized Calogero–Sutherland models via conformal-field-theory null-vector techniques (Borutta, Feuerpfeil, and Iqbal 2026). Yang and del Campo (2022) proved that every one-dimensional quantum system with a ground state of Jastrow form is integrable, exhibiting the invariants of motion and linking the family to Polychronakos's exchange-operator formalism (a construction of commuting conserved quantities from coordinate-permutation operators); that work explicitly leaves higher spatial dimensions, spin degrees of freedom, species mixtures, and higher-order correlations open, and the 2026 follow-ups still list the excitation spectrum and quasi-exact solvability of the parent Hamiltonians as undetermined.

Problem Statement

For the family of parent Hamiltonians of Jastrow-form ground states constructed by Beau and del Campo — in spatial dimension d≥2d\ge 2, where the potential generically contains a coordinate-dependent three-body term — the following spectral questions posed in the source paper remain open after the one-dimensional case was settled. They concern members of the family whose three-body potential V3V_3 is genuinely coordinate-dependent: for the special choices of the pair function ff for which V3V_3 vanishes, becomes constant, or reduces to an effective two-body term (including the trivial case of constant ff), the construction yields already-known solvable models, and answering the questions below for such degenerate members does not resolve the open problem.

(1) Solvability structure. Are these Hamiltonians integrable, i.e. do they admit a complete set of mutually commuting conserved quantities, or are they merely quasi-exactly solvable, with only a finite exactly obtainable block of the spectrum? In particular, can integrability be established from the properties of the ground-state Jastrow wavefunction itself, as the exchange-operator argument achieves for d=1d=1?

(2) Excited states. Identify and construct excited eigenstates of these parent Hamiltonians and their energy eigenvalues, and determine the completeness, or boundedness, of the exactly obtainable part of the spectrum.

In its outlook, the source paper additionally envisions extending the construction to models that are supersymmetric, include internal spin degrees of freedom, multiple particle species, or truncated (short-range) interactions; to anyonic statistics (anyons being two-dimensional quasiparticles whose exchange phases interpolate between those of bosons and fermions) with two-body interactions involving the relative angular momentum; and to more general Jastrow wavefunctions with explicit three-body or higher-order correlation factors (including Nosanov–Jastrow ansätze, which supplement the Jastrow factor with Gaussian orbitals centered on lattice sites, and shadow wavefunctions, which couple the physical coordinates to auxiliary variables that are integrated out so that crystalline phases can be described in a translationally invariant form). These construction extensions remain open alongside the spectral core, but they are adjacent directions rather than part of it: the central open objective of this record is the solvability characterization — integrability versus quasi-exact solvability, and the excitation spectrum — of the Beau–del Campo parent Hamiltonians of Jastrow wavefunctions in d≥2d\ge 2 with genuinely coordinate-dependent three-body terms.

References

  1. Mathieu Beau and Adolfo del Campo, "Parent Hamiltonians of Jastrow wavefunctions", SciPost Phys. Core 4, 030 (2021). DOI: 10.21468/SciPostPhysCore.4.4.030; arXiv:2107.02869. https://arxiv.org/abs/2107.02869
  2. Jing Yang and Adolfo del Campo, "One-Dimensional Quantum Systems with Ground State of Jastrow Form Are Integrable", Phys. Rev. Lett. 129, 150601 (2022). DOI: 10.1103/PhysRevLett.129.150601; arXiv:2201.08282. https://arxiv.org/abs/2201.08282
  3. Nilanjan Sasmal and Adolfo del Campo, "Taxonomy of integrable and ground-state solvable models: Jastrow wave functions on graphs and parent Hamiltonians", Phys. Rev. B 113, 235151 (2026). DOI: 10.1103/g7hk-tlq1; arXiv:2602.22315. https://arxiv.org/abs/2602.22315
  4. Hari Borutta, Andreas Feuerpfeil, and Yasir Iqbal, "Parent Hamiltonian construction of generalized Calogero-Sutherland models", Phys. Rev. B 114, 115104 (2026). DOI: 10.1103/3by3-jsjw; arXiv:2604.11074. https://arxiv.org/abs/2604.11074
  5. Xu-Dan Xie, Zheng-Yuan Xue, and Dan-Bo Zhang, "Determining the non-Hermitian parent Hamiltonian from a single eigenstate", Phys. Rev. B 110, 235113 (2024). DOI: 10.1103/PhysRevB.110.235113; arXiv:2408.15781. https://arxiv.org/abs/2408.15781
  6. Jing Yang and Adolfo del Campo, "Time-dependent Jastrow ansatz: Exact quantum dynamics, shortcuts to adiabaticity, and quantum quenches in strongly correlated many-body systems", Phys. Rev. A 111, 053315 (2025). DOI: 10.1103/PhysRevA.111.053315; arXiv:2210.14937. https://arxiv.org/abs/2210.14937
  7. P. Bedaque, H. Kumar, and A. Sheng, "Neural network solutions of bosonic quantum systems in one dimension", Phys. Rev. C 109, 034004 (2024). DOI: 10.1103/PhysRevC.109.034004; arXiv:2309.02352. https://arxiv.org/abs/2309.02352
  8. C. L. Kane, S. Kivelson, D.-H. Lee, and S.-C. Zhang, "General validity of Jastrow-Laughlin wave functions", Phys. Rev. B 43, 3255 (1991). DOI: 10.1103/PhysRevB.43.3255. https://doi.org/10.1103/PhysRevB.43.3255
  9. F. Calogero and C. Marchioro, "Exact bound states of some N-body systems with two- and three-body forces", J. Math. Phys. 14, 182 (1973). DOI: 10.1063/1.1666291. https://doi.org/10.1063/1.1666291
  10. Xhek Turkeshi and Marcello Dalmonte, "Parent Hamiltonian reconstruction of Jastrow-Gutzwiller wavefunctions", SciPost Phys. 8, 042 (2020). DOI: 10.21468/SciPostPhys.8.3.042. https://doi.org/10.21468/SciPostPhys.8.3.042