Many-body ground states conjecture for frustration-free FBI Hamiltonians

Let a frustration-free FBI Hamiltonian be a many-body Hamiltonian whose ground-state energy is attained when each positive-semidefinite local term is minimized, and let its Hartree–Fock ground states be the ground states within the Hartree–Fock variational class. For TBG-2 with spin and valley, let $\{\lvert\Psi_i\rangle\}_{i=1}^5$ be the ferromagnetic Slater determinant states generating the Hartree–Fock ground-state manifold, and let $\mathrm{U}(4)\times\mathrm{U}(4)$ act on these states. Many-body ground states conjecture. Any many-body ground state of a frustration-free FBI Hamiltonian can be written as a linear combination of its Hartree–Fock ground states. In particular, for TBG-2 with spin and valley, any many-body ground state can be written as

$$\lvert\Psi\rangle=\int_{\mathrm{U}(4)\times\mathrm{U}(4)}\sum_{i=1}^5\alpha_{\mathfrak g,i}\,\mathfrak g\lvert\Psi_i\rangle\,\mathrm{d}\mathfrak g,$$

where $\alpha_{\mathfrak g,i}\in\mathbb{C}$. This would extend the Hartree–Fock characterization of the ground-state manifold to the full many-body problem; the paper motivates it by the simplicity of the FBI model and the low electron correlation observed in numerical studies, but does not establish it.