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

Abstract

We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with random hopping amplitudes decaying with the distance as $|t_l|\sim l^{-\alpha}$, and address the question how far an analogue of the random-singlet 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 $\alpha\le 2$, 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. Besides 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.