Covariance matrix spectrum of correlated charge insulators reveals hidden connections to coupled cluster, matrix product, and Rokhsar-Kivelson states

Charge ordering induced by strong short-range repulsion in itinerant fermion systems typically follows a two-site alternation pattern. However, the covariance matrix spectrum of the one-dimensional, half-filled, spinless $t\ensuremath{-}V$ model reveals a post-Hartree-Fock picture at strong repulsion, with emergent four-site disruptions of the underlying staggered mean-field state. These disruptions are captured in a thermodynamically extensive manner by a compact four-fermion coupled cluster (doubles) state (CCS). Remarkably, all properties of this state may be computed analytically by combinatorial means, and also derived from an exactly solvable correlated hopping Hamiltonian. Furthermore, this coupled cluster state can be reexpressed as a low-rank matrix product state (MPS) with bond dimension exactly four. In addition, we unveil a hidden connection between this coupled cluster ansatz and a Rokhsar-Kivelson state (RKS), which is the ground state of a solvable parent quantum tetramer model. The broad picture that we uncover here thus provides deep connections between several core concepts of correlated fermions and quantum chemistry that have previously enjoyed limited synergy. In contrast to a recent perturbative treatment on top of Hartree-Fock theory, our approach asymptotically captures the correct correlations in the $t\ensuremath{-}V$ model at small $t/V$, and remains a qualitatively accurate approximation even outside the perturbative regime. Our results make the case for further studies of the covariance matrix for correlated electron systems in which ground states have nontrivial unit-cell structure.