Reconstructing Spin Hamiltonians of 2D Gutzwiller-Projected Wavefunctions

TL;DR

The paper tackles the challenge of identifying parent Hamiltonians that stabilize 2D Gutzwiller-projected spin-liquid wavefunctions by applying correlation-matrix reconstruction to three mean-field Ansätze (projected Fermi sea on a square lattice and $\pi$-flux states on square and triangular lattices). By building a basis of long-range spin exchanges and path-based permutation operators and analyzing the nullspace of the correlation matrix, the authors assess whether a local Hamiltonian exists that has the projected states as eigenstates (ideally ground states) on 4×4 lattices; they find no such exact Hamiltonian within this basis. The results also reveal a notable dependence on boundary conditions (gauge holonomies) and show that a soft occupancy constraint in 1D can approximate stabilization, though not exactly in 2D for the finite systems studied. The study suggests that stabilizing these 2D spin liquids likely requires more frustrated, higher-order, or nonlocal terms and points to future work with larger operator bases and advanced numerical methods (e.g., tensor networks) to explore larger systems and more complex stabilizing Hamiltonians.

Abstract

We apply the correlation matrix Hamiltonian reconstruction technique to the two-dimensional Gutzwiller-projected Fermi sea and π-flux states on finite-sized square and triangular lattices. Our results indicate no spin Hamiltonian with simple local interaction terms stabilizes such states for finite system sizes. We develop a quantitative assessment of the importance of local interactions to the stabilization of these liquid states. Lastly, we systematically assess arguments for the origin of local terms driving a Gutzwiller-projected ground state.

Figures (7)

• Figure 1: (a) The choice of nontrivial $\text{U}(1)$ holonomies around the cycles of a torus correspond to magnetic flux insertions. (b) Square $\pi$-flux ansatz flux pattern in the Landau or striped gauge. (c) Triangular $\pi$-flux ansatz flux pattern. Each pair of triangular plaquettes carries a flux $\pi$.
• Figure 2: Brillouin zones of the mean-field ansatze used in this work. (a) The projected Fermi sea with trivial (periodic) boundary conditions is degenerate. The wavevectors marked in red correspond to the degenerate choices of Fermi sea filling. (b) The projected Fermi sea with antiperiodic boundary conditions in the $y$ direction produces a non-degenerate ground state. (c) The Brillouin zone of the $\pi$-flux ansatz on the square lattice with centered boundary conditions. (d) Triangular $\pi$-flux ansatz with centered boundary conditions. The red markings indicate the Dirac points.
• Figure 3: Wavefunction overlaps and difference in two-point correlators of Gutzwiller-projected ground state and exact ground state as a function of single-occupancy constraint strength. (a) Tight-binding model in $d=1$ with $N=10$ sites, whose projected ground state is the exact ground state of the Haldane--Shastry model. (b) $\pi$-flux state in $d=2$ on $4\times 4$ square lattice. Here $\Delta\langle S_0^z S_r^z\rangle \equiv \langle S_0^z S_r^z\rangle_\text{ED} - \langle S_0^z S_r^z\rangle_\text{P}$.
• Figure 4: Difference $\Delta E \equiv \langle H_\alpha\rangle - E_0$ between exact-diagonalization ground state energy $E_0$ and expectation value $\langle H_\alpha \rangle$ of approximate Hamiltonian $H_\alpha = \alpha S_\text{tot} + \sum_i \gamma_iO_i$.
• Figure 5(a): Normalized eigenvalue decreases for each unique permutation operator.