Exact parent Hamiltonians for 2D Gutzwiller-projected states

Construct exact two-dimensional spin Hamiltonians whose ground states are specified Gutzwiller-projected wavefunctions P_G|Ψ_MF⟩ derived from the mean-field ansatz H_MF = ∑_{⟨ij⟩} χ_{ij} c_i^† c_j + H.c. for nontrivial U(1) background gauge link configurations χ_{ij} on finite square or triangular lattices, since the corresponding 2D analogues are not known for most such configurations.

Background

Gutzwiller-projected wavefunctions are obtained by projecting mean-field parton states with U(1) gauge link variables χ_{ij} into the physical single-occupancy spin Hilbert space. In one dimension, the projected Fermi sea is the exact ground state of the Haldane–Shastry Hamiltonian, providing a benchmark parent Hamiltonian.

In two dimensions, while an exact parent Hamiltonian was proposed for the chiral spin liquid, a general exact Hamiltonian for other Gutzwiller-projected states (e.g., projected Fermi sea and π-flux states on square and triangular lattices) is not known. The present work applies correlation matrix reconstruction and finds no exact local Hamiltonian within a broad operator basis for these states on 4×4 lattices, underscoring the open problem of constructing exact 2D parent Hamiltonians beyond special cases.