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The properties of the low-lying eigenvalues of the entanglement Hamiltonianand their relation to the localization length of disordered interactingone-dimensional many-particle system is studied. The average of the firstentanglement Hamiltonian level spacing is proportional to the ground statelocalization length and shows the same dependence on the disorder andinteraction strength as the localization length. This is the result of the factthat entanglement is limited to distances of order of the localization length.The distribution of the first entanglement level spacing shows a Gaussian-likebehavior as expected for level spacings much larger than the disorderbroadening. For weakly disordered systems (localization length larger thansample length), the distribution shows an additional peak at low levelspacings. This stems from rare regions in some samples which exhibitmetallic-like behavior of large entanglement and large particle numberfluctuations. These intermediate 'microemulsion' metallic regions embedded inthe insulating phase are discussed.

On low eigevalues of the entanglement Hamiltonian, localization length, and rare regions in disordered interacting one-dimensional systems