  A prime characterization of many-body localized (MBL) systems is the
entanglement of their eigenstates; in contrast to the typical ergodic phase
whose eigenstates are volume law, MBL eigenstates obey an area law. In this
work, we show that a spin-disordered Hubbard model has both a large number of
area-law eigenstates as well as a large number of eigenstates whose
entanglement scales logarithmically with system size (log-law). This model,
then, is a microscopic Hamiltonian which is neither ergodic nor many-body
localized. We establish these results through a combination of analytic
arguments based on the eta-pairing operators combined with a numerical analysis
of eigenstates. In addition, we describe and simulate a dynamic time evolution
approach starting from product states through which one can separately probe
the area law and log-law eigenstates in this system.
