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

TL;DR

This paper investigates the subleading corrections to the entanglement entropy and singlet-length distribution in random singlet phases of spin chains, revealing new power-law terms and their implications for understanding quantum entanglement in disordered systems.

Contribution

The study analytically and numerically characterizes subleading terms in singlet-length distribution and entanglement entropy, including half-integer power corrections, in random singlet phases.

Findings

Subleading terms in singlet-length distribution include integer and half-integer powers of 1/l.

Subleading correction to entanglement entropy is of order 1/ell for XX and 1/ell^{1/2} for XXX models.

SDRG accurately predicts the order but not the non-universal coefficients of subleading terms.

Abstract

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