Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong-disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite-randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite-randomness fixed point in the diluted transverse Ising model in higher dimensions.

Figure 1

An example of the typical ground state in the random quantum Ising model (a) in 1D, and (b) in 2D; it contains a collection of spin clusters of various sizes, which are formed and decimated during the RG. The entanglement of a block (shaded area) is give by the number of decimated clusters (indicated by red loops) that connect the block with the rest of the system.

Figure 2

The distribution of the last decimated effective log-fields $\ln {\tilde{h}}_{\infty }$, and the distribution of the last decimated effective log-bonds $\ln {\tilde{J}}_{\infty }$ in the RG calculations. At $h_0=1.175$, the distributions, shown in (a) and (b), get broader with increasing system sizes, indicating the RG flow towards infinite randomness; i.e., the system is critical. A scaling plot of the data in (a) using energy-length scaling $\ln {\tilde{h}}_{\infty }\sim L^{\psi }$ with $\psi =0.55$ is presented in (c). The solid line is just a guide to the eye. The subfigures (d) and (e) show the log-field distribution at $h_0=1.18$ and the log-bond distribution at $h_0=1.17$, respectively; the distributions show a power-law decaying tail in the low-energy region, which is clear evidence that the system is in the Griffiths phases.

Figure 3

Left panel: The disorder-averaged block entropy per surface unit ${\overline{S}}_{\ell }/\ell$ vs the linear size of the block $\ell$ for a system size $L=64$ for various values of $h_0$. We observe that the entropy for $\ell =L/2$ reaches its maximum at the critical point $h_c=1.175$ (cf. Fig. 2). Right panel: The block entropy per surface area vs $\ln \ell$ on a log-scale for different system sizes $L$ at the critical point. The data show a straight line (guided by the dashed line), corresponding to the scaling obeying the area law with a double-logarithmic correction, as given in Eq. (4).

Figure 4

The entropy per surface area ${\overline{S}}_{\ell }/\ell ={\overline{s}}_{\ell }$ vs $\ell$ near the percolation threshold $p_c=0.5$ for the 2D bond-diluted Ising model at small transverse fields for $L=512$. The curves converge to finite values for $\ell \to \infty$, corresponding to the area law. The inset shows ${\overline{s}}_{\ell }-{\overline{s}}_{\infty }$ as a function of $\ell$. ${\overline{s}}_{\infty }$ is estimated from ${\overline{s}}_{L/2}$ at $L=512$. The dashed line corresponds to ${\ell }^{-1}$.