Scaling of entanglement entropy in the random singlet phase

We present numerical evidences for the logarithmic scaling of the entanglement entropy in critical random spin chains. Very large scale exact diagonalizations performed at the critical $XX$ point up to $L=2000$ spins $\frac{1}{2}$ lead to a perfect agreement with recent real-space renormalization-group predictions of [Refael and Moore Phys. Rev. Lett. 93, 260602 (2004)] for the logarithmic scaling of the entanglement entropy in the random singlet phase with an effective central charge $\tilde{c}=c\times \ln 2$. Moreover, we provide the first visual proof of the existence of the random singlet phase with the help of quantum entanglement concept.

Figure 1

(Color online) Schematic picture for the entanglement entropy of a subsystem of length $x$ in the random singlet phase. The entanglement is just due to the singlets connecting the subsystem with the rest of the chain. In this picture, $\mathcal{S}\left(x\right)=5\times \ln 2$.

Figure 2

(Color online) Entanglement entropy of a subsystem of size $x$ embedded in a closed ring of size $L$, shown vs $x$ in a log-linear plot. Numerical results obtained by exact diagonalizations performed at the $XX$ point. For clean nonrandom systems with $L=500$ and $L=2000$ (open circles), $\mathcal{S}\left(x\right)$ is perfectly described by Eq. (3) (red and blue curves). The data for random systems have been averaged over ${10}^4$ samples for $L=500$, 1000, 2000, and $2\times {10}^4$ samples for $100⩽L⩽400$. The expression $0.8595+(\ln 2∕3)\ln x$ (dashed line) fits the data in the regime where finite size effects are absent.

Figure 3

(Color online) Probability distribution for the entanglement entropy in the random singlet phase for the disordered $XX$ spin-$\frac{1}{2}$ chain obtained by exact diagonalizations on $L=100$ spin chains. The light histograms correspond to an even subsystem size with 50 sites $\mathbf{(}{P}^{\text{even}}=P[\mathcal{S}(100,50)∕\ln 2]\mathbf{)}$ and the dark histograms corresponds to an odd subsystem with 49 sites $\mathbf{(}{P}^{\text{odd}}=P[\mathcal{S}(100,49)∕\ln 2]\mathbf{)}$. For each disorder strength, $\delta =1$ (a), $\delta =2.5$ (b), $\delta =5$ (c), $\delta =10$ (d), ${10}^5$ different random samples have been diagonalized. The insets show the combined distributions $P^{\text{even}}+P^{\text{odd}}$. Note that all the distribution functions have been normalized to unity.