Sublattice scars and beyond in two-dimensional U(1) quantum link lattice gauge theories

In this article, we elucidate the structure and properties of a class of anomalous high-energy states of matter-free U(1) quantum link gauge theory Hamiltonians using numerical and analytical methods. Such anomalous states, known as quantum many-body scars in the literature, have generated a lot of interest due to their athermal nature. Our starting Hamiltonian is H = 𝒪_kin + λ𝒪_pot, where λ is a real-valued coupling, and 𝒪_kin (𝒪_pot) are summed local diagonal (off-diagonal) operators in the electric flux basis acting on the elementary plaquette □. The spectrum of the model in its spin-1/2 representation on L_x × L_y lattices reveal the existence of sublattice scars, |ψ_s ⟩, which satisfy 𝒪_pot,□ |ψ_s⟩ = |ψ_s⟩ for all elementary plaquettes on one sublattice and 𝒪_pot,□ | ψ_s ⟩ =0 on the other, while being simultaneous zero modes or nonzero integer-valued eigenstates of 𝒪_kin. We demonstrate a “triangle relation” connecting the sublattice scars with nonzero integer eigenvalues of 𝒪_kin to particular sublattice scars with 𝒪_kin = 0 eigenvalues. A fraction of the sublattice scars have a simple description in terms of emergent short singlets, on which we place analytic bounds. We further construct a long-ranged parent Hamiltonian for which all sublattice scars in the null space of 𝒪_kin become unique ground states and elucidate some of the properties of its spectrum. In particular, zero energy states of this parent Hamiltonian turn out to be exact scars of another U(1) quantum link model with a staggered short-ranged diagonal term.