Corrections to the singlet-length distribution and the entanglement entropy in random singlet phases

We consider random singlet phases of spin-$\frac{1}{2}$, random, antiferromagnetic spin chains, in which the universal leading-order divergence $\frac{\ln 2}{3}\ln\ell$ average entanglement entropy a block $\ell$ spins, as well closely related leading term $\frac{2}{3}l^{-2}$ distribution lengths are known by strong-disorder renormalization group (SDRG) method. Here, we address question how large subleading terms above quantities are. By an analytical calculation performed along special SDRG trajectory XX chain, identify series integer powers $1/l$ singlet-length with $\frac{4}{3}l^{-3}$. Our numerical analysis shows that, for fixed point, is generally $O(l^{-3})$ non-universal coefficient and also reveals half-integer powers: $l^{-7/2}$ $l^{-5/2}$ XXX points, respectively. present originating approach can be interpreted calculated chain from one-particle states equivalent free-fermion model. These results imply that next to logarithmic one $O(\ell^{-1})$ point $O(\ell^{-1/2})$ coefficients. For model, where comparison exact diagonalization possible, order confirmed but find fails provide correct coefficient.