Entanglement spectrum of random-singlet quantum critical points

The entanglement spectrum (i.e., the full distribution of Schmidt eigenvalues of the reduced density matrix) contains more information than the conventional entanglement entropy and has been studied recently in several many-particle systems. We compute the disorder-averaged entanglement spectrum in the form of the disorder-averaged moments $\stackrel{̲}{\mathrm{Tr}{\mathrm{\rho }}_{\mathrm{A}}^{\mathrm{\alpha }}}$ of the reduced density matrix ${\rho }_A$ for a contiguous block of many spins at the random-singlet quantum critical point in one dimension. The result compares well in the scaling limit with numerical studies on the random $\mathit{XX}$ model and is also expected to describe the (interacting) random Heisenberg model. Our numerical studies on the $\mathit{XX}$ case reveal that the dependence of the entanglement entropy and spectrum on the geometry of the Hilbert space partition is quite different than for conformally invariant critical points.

Figure 1

Ab initio Renyi entropies for a disordered $\mathit{XX}$ chain of 1024 spins. The average is over 73 000 realizations. The variation of the color shows results from $\alpha =1$ (upper line) to $\alpha =2.9$ (bottom line). The yellow line is the asymptotic von Neumann entropy ($\alpha =1$) obtained by Laflorencie.[25]

Figure 2

Ab initio Rènyi entropies for a disordered $\mathit{XX}$ chain of 1024 spins minus the RSP value. The averages are over the same sample of 73 000 realizations.

Figure 3

The finite-size scaling function for the entanglement entropy $Y(x)$ in Eq. (29). Main: RSP data averaged over 1 440 000 disorder realizations for $L=1024$. The continuous (red) curve is the proposed phenomenological formula (31) describing perfectly the data points. Inset: The same plot for different values of $L$ showing the collapse on a single scaling function.

Figure 4

Top: RSP results for ${\mathrm{Ŝ}}_{\alpha }$ as a function of $S_{\alpha }$ for a chain of $1024$ spins and 1 440 000 disorder realizations. Bottom: Even-odd average of ${\mathrm{Ŝ}}_{\alpha }$ eliminating leading corrections to the scaling. In both panels the continuous lines are the analytic RSRG result for $A_t$.

Figure 5

The universal constant $A_t$ obtained from RSP data for $L=1024$ (1,440,000 disorder realizations) and $L=10\hspace{0.5em}000$ (320 000 realizations). Main plot: For $\alpha ⩾3.5$ finite-size effects are negligible and the RSRG prediction (continuous line) describes the data. Inset: Crossover to the nonuniversal Poissonian behavior (green continuous line) for larger $\alpha$.

Figure 6

Ab initio ${\mathrm{Ŝ}}_{\alpha }$ as a function of $S_{\alpha }$ for a spin-chain of $1024$ spins and 73 000 disorder realizations. The continuous lines represent the RSRG prediction for the slope. The additive terms are different from those in Fig. 4.

Figure 7

Scaling functions for the correction to the scaling $f_t(S)$ in Eq. (32) obtained as the difference of $s_{\alpha }^{\mathrm{odd}}(\ell )$ and $s_{\alpha }^{\mathrm{even}}(\ell )$. Full and dashed lines correspond to uniform and exponential distributions of disorder, respectively.

Figure 8

${\mathrm{Ŝ}}_{\alpha }$ for two disorder distributions. The RSP data are for chains of 10 000 spins and averaged over 320 000 configurations.

Figure 9

${\mathrm{Ŝ}}_{\alpha }$ as function of $S_{\alpha }$ for $\alpha =2.9$ and for two disorder distributions (RSP data with $L=10\hspace{0.5em}000$ and 320 000 configurations). The scaling function is disorder independent.

Figure 10

The quantity $\Delta$ defined in Eq. (38) vs $S_{\alpha }$ for uniform and exponential distributions of disorder. With varying $\alpha$ the two differences are the same, showing the universality of the coefficient $B_{t(\alpha )}$.