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

Abstract

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 $H={\mathcal{O}}_{\mathrm{kin}}+\lambda {\mathcal{O}}_{\mathrm{pot}}$, where $\lambda$ is a real-valued coupling, and ${\mathcal{O}}_{\mathrm{kin}}$ (${\mathcal{O}}_{\mathrm{pot}}$) are summed local diagonal (off-diagonal) operators in the electric flux basis acting on the elementary plaquette $\square$. The spectrum of the model in its spin-$\frac{1}{2}$ representation on $L_x\times L_y$ lattices reveal the existence of sublattice scars, $|{\psi }_s⟩$, which satisfy ${\mathcal{O}}_{\mathrm{pot},\square }|{\psi }_s⟩=|{\psi }_s⟩$ for all elementary plaquettes on one sublattice and ${\mathcal{O}}_{\mathrm{pot},\square }|{\psi }_s⟩=0$ on the other, while being simultaneous zero modes or nonzero integer-valued eigenstates of ${\mathcal{O}}_{\mathrm{kin}}$. We demonstrate a ``triangle relation'' connecting the sublattice scars with nonzero integer eigenvalues of ${\mathcal{O}}_{\mathrm{kin}}$ to particular sublattice scars with ${\mathcal{O}}_{\mathrm{kin}}=0$ 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 ${\mathcal{O}}_{\mathrm{kin}}$ 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.

18 pages, 10 figures

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