Entanglement properties of correlated random spin chains and similarities with conformally invariant systems

We study the Rényi entanglement entropy and the Shannon mutual information for a class of spin-1/2 quantum critical XXZ chains with random coupling constants which are partially correlated. In the XX case, distinctly from the usual uncorrelated random case where the system is governed by an infinite-disorder fixed point, the correlated-disorder chain is governed by finite-disorder fixed points. Surprisingly, we verify that, although the system is not conformally invariant, the leading behavior of the Rényi entanglement entropies are similar to those of the clean (no randomness) conformally invariant system. In addition, we compute the Shannon mutual information among subsystems of our correlated-disorder quantum chain and verify the same leading behavior as the $n=2$ Rényi entanglement entropy. This result extends a recent conjecture concerning the same universal behavior of these quantities for conformally invariant quantum chains. For the generic spin-1/2 quantum critical XXZ case, the true asymptotic regime is identical to that in the uncorrelated disorder case. However, these finite-disorder fixed points govern the low-energy physics up to a very long crossover length scale, and the same results as in the XX case apply. Our results are based on exact numerical calculations and on a numerical strong-disorder renormalization group.

Figure 1

Finite-size gap $\mathrm{\Delta }E$ for various system sizes $L=4,6,8,\cdots ,22$, for the cases of disorder strength $D=0.5$ (blue squares) and 2.0 (black circles) and anisotropy $\mathrm{\Delta }=0.5$. The lines are the best linear fits, and the corresponding estimated values of $z$ are also shown.

Figure 2

Dynamical exponent $z$ as a function of the disorder strength $D$ for several anisotropy parameter $\mathrm{\Delta }$. The results were obtained from the fit of the finite-size gap as in Fig. 1. The estimated error bars are of the order of the symbol sizes and the lines are guides to the eyes.

Figure 3

Comparison of the dynamical critical exponent for the $\mathrm{\Delta }=0$ case computed by exact diagonalizations of small chains ($L<22$), of very large chains ($L=2^{15}$), and via the SDRG method ($L={10}^6$). Inset: The finite-size gap for different system sizes where the corrections to scaling are clear. Error bars are about the same size as symbols.

Figure 4

Entanglement measures for the correlated disordered XXZ quantum spin chain with (a) anisotropy $\mathrm{\Delta }=0.5$ and disorder $D=2.5$ and for (b) anisotropy $\mathrm{\Delta }=0$ and disorder $D=2.0$ for several indices $n$ and lattice sizes $L$. Entanglement is measured by the Rényi entanglement entropy $S_n$ ($n=1,2,\text{and}3$) and by the Shannon mutual information $I$. Data are averaged over ${10}^3$ disorder realizations and the lines are guides to the eyes.

Figure 5

The effective central charge as a function of disorder strength $D$ obtained from the von Neumann entanglement entropy $S_{\mathrm{vN}}$ and from the Shannon mutual information $I$ for various anisotropies $\mathrm{\Delta }$. For $\mathrm{\Delta }\ne 0$, these values are effective ones due to the system finite size. For extremely large systems, these values cross over to $c_{\mathrm{IRFP}}^{\mathrm{eff}}=ln2$ (see Sec. 5).

Figure 6

The scaling function $f_L^D\left(x\right)$ measured by using the Rényi entropies $S_1$ and $S_2$ [see Eq. (8)] for chains of anisotropy $\mathrm{\Delta }=0$ and lattice size $L=2^{10}$ averaged over $145000$ samples.

Figure 7

The transversal spin-spin correlation function for various disorder strengths $D=0.2$, 0.5, 1.0, 1.5, and 2.0 (as indicated) and lattice sizes $L=200$ (green), 400 (blue), 800 (red), and 1600 (black). For comparison, we also plot $C^{zz}$ for the clean system (dashed line) and for the system with uncorrelated disorder with $D=1.0$ (bottommost curves). Notice the stronger fluctuations in this latter case. The dotted-dashed line is the fitting curve for the uncorrelated case in the region $sin(\pi x/L)>1/2$ (see text). In all cases, the data are averaged over ${10}^6$ different disorder configurations, except for $L=1600$ where the number of samples used is $50000$. Error bars are not shown for clarity but are of the order of the data fluctuations.

Figure 8

Decimation procedure for (a) uncorrelated and (b) correlated disorder.

Figure 9

The anisotropy along the SDRG flow for various disorder parameters $D$ and initial conditions $\mathrm{\Delta }$, for the case of perfectly correlated disorder $\alpha =1$. In panel (a) the standard deviation ${\sigma }_{\tilde{\mathrm{\Delta }}}$ is plotted as a function of the average $\langle \tilde{\mathrm{\Delta }}\rangle$, with the arrows indicating the direction of the SDRG flow. In panel (b) $\langle \tilde{\mathrm{\Delta }}\rangle$ is plotted as a function of the SDRG length scale $\xi$. The lines are guides to the eyes and the data are averaged over only 10 disorder realizations of a large lattice size of $L={10}^6$.

Figure 10

Disorder correlation along the SDRG flow for various disorder strengths $D$ and anisotropies $\mathrm{\Delta }$. The correlation parameter $\alpha$ is plotted as a function of the SDRG length scale $\xi$. Data are averaged over 1000 disorder realizations of large chains of size ${10}^7$. Lines are guides to the eyes.

Figure 11

The relation between the energy $\mathrm{\Omega }$ and the length $\xi$ scales for the disorder strengths (a) $D=1$ and (b) $D=3$ and various anisotropies $\mathrm{\Delta }$ and correlation $\alpha$. Dashed lines are the best fits for the case $\mathrm{\Delta }=0$ and $\alpha =1$ restricted to $ln\xi \ge 7$. Inset: The same as in the main plot with the logarithmic value of the vertical axis. Dashed lines are the corresponding theoretical slopes for infinite-randomness physics.

Figure 12

The scaling relation between energy $\mathrm{\Omega }$ and length $\xi$ for the $\mathrm{\Delta }=0$ chain with the even coupling constants spatially shifted by an amount $\delta$ and disorder strengths (a) $D=0.1$ and (b) $D=0.5$. Dashed lines are the fits for the unshifted case for $ln\xi \ge 7$.

Figure 13

The correlation measure $\tilde{\alpha }$ for the $\mathrm{\Delta }=0$ chain with the even coupling constants shifted by an amount $\delta$ and disorder strengths (a) $D=0.1$ and (b) $D=0.5$.