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

Nilanjan Sasmal, Adolfo del Campo

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.

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

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.
Paper Structure (13 sections, 48 equations, 7 figures, 8 tables)

This paper contains 13 sections, 48 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Quantum Many-body Wavefunctions on Graphs
  3. Parent Hamiltonian of Graph Jastrow Wavefunctions
  4. Parent Hamiltonian of a Graph-Jastrow Wavefunction Under Confinement
  5. Parent Hamiltonian of Graph Jastrow wavefunctions in 1D
  6. $K_N$ complete graph models: All-to-all pairwise interactions
  7. Path $P_N$ and Cycle $C_N$ Graph Nodes with Nearest Neighbor Interactions
  8. $2r$ regular graph models: Interactions with truncated range
  9. Generating many-body quantum models by graph operations
  10. Graph joins and mixtures
  11. Graph products
  12. Discussion
  13. Conclusions

Figures (7)

  • Figure 1: Top: Regular graphs with $N=7,8$ and open boundary conditions and truncated pairwise interactions for $r=1,2,3,4$. Bottom: Circulant graphs with $N=12$ and periodic boundary conditions for $r=1,2,3,4$, the $2r$-regular graph interpolates between the cycle graph $C_{12}$ ($r=1$) and the complete graph $K_{12}$ ($r=6$).
  • Figure 2: Complete bipartite graphs: (a) $K_{8,8}$ and (b) $K_{8,5}$.
  • Figure 3: Star graphs with $N=10,60$; describing a quantum many-body system of central-spin or impurity system.
  • Figure 4: Ladder graph $L_7=P_7 \square P_2$ (top) and flattened ladder graph (bottom) realized as a 1D model with two particles bound per unit cell, which can be visualized as $x_i = (x_i^A,x_i^B)$.
  • Figure 5: Quantum many-body systems represented in the form of a Prism graph $Y_{25}=C_{25} \square P_2$. See table \ref{['Grph_prod_ex']}.
  • ...and 2 more figures