Taxonomy of Integrable and Ground-State Solvable Models: Jastrow Wavefunctions on Graphs and Parent Hamiltonians

TL;DR

This paper introduces a new class of many-body quantum models based on graph-structured interactions, generalizing Jastrow wavefunctions and providing explicit parent Hamiltonians with novel solvable properties.

Contribution

It develops a framework for graph-based continuous-variable quantum systems with explicit ground states and Hamiltonians, extending known models and discovering new solvable cases.

Findings

Derived explicit parent Hamiltonians with two- and three-body interactions.

Mapped the landscape of models using graph theory, including known and new examples.

Provided exact ground-state wavefunctions and energy eigenvalues for various graph structures.

Abstract

We introduce a family of many-body systems of distinguishable continuous-variable particles in which interparticle interactions are set by the adjacency matrix of a graph. The ground-state wavefunction of such systems is of a generalized Jastrow form involving the product of pair-correlation functions over the edge set of the graph. These systems describe quantum fluids when the graph is complete, and the pair function has a well-defined permutation symmetry. In general, they provide the continuous-variable generalization of spin systems on graphs, with broken permutation symmetry. The corresponding parent Hamiltonian is shown to include (a) two-body interactions determined by the graph adjacency matrix and (b) three-body interactions over all possible 2-paths on the graph. Employing elements of graph theory, we chart the landscape of models, recovering known instances in the literature and providing numerous new examples of ground-state solvable models for which the system Hamiltonian, ground-state wavefunction, and corresponding energy eigenvalue are specified.