Testing the validity of random-singlet state for long-range hopping models through the scaling of entanglement entropy

We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with random hopping amplitudes decaying with the distance as |tl| similar to l-alpha, and address the question how far an analog of the randomsinglet state (RSS) conceived originally for describing the ground state of certain random spin chains is valid for this model. For this purpose, we study the effective central charge characterizing the logarithmic divergence of the entanglement entropy (EE) and the prefactor of the distribution of distances between localization centers on the two sublattices, which must fulfill a consistency relation for a RSS. For alpha > 1, we find by exact diagonalization an overall logarithmic divergence of the entanglement entropy with an effective central charge varying with alpha. The large-alpha limit of the effective central charge is found to be different from that of the nearest-neighbor hopping model. The consistency relation of RSS is violated for alpha 2, while for alpha > 2 it is possibly valid, but this conclusion is hampered by a crossover induced by the short-range fixed point. The EE is also calculated by the strong-disorder renormalization group (SDRG) method numerically. Aside from the traditional scheme, we construct and apply a more efficient minimal SDRG scheme having a linear (nearest-neighbor) structure, which turns out to be an accurate approximation of the full SDRG scheme for not too small alpha. The SDRG method is found to provide systematically lower effective central charges than exact diagonalization does, nevertheless, it becomes more and more accurate for increasing alpha. Furthermore, as opposed to nearest-neighbor models, it indicates a weak dependence on the disorder distribution.