Entanglement entropy in the two-dimensional random transverse field Ising model

The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is numerically studied through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always linearly scales with the block size. However, besides this area law contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.

Figure 1

(Color online) Basic RG transformations (see text for details). Left: energy scale is a field and right: energy scale is an Ising coupling.

Figure 2

(Color online) (a) Finite-size scaling of magnetization ratio given in Eq. (2). (b) Entropy per surface $S\left(L\right)∕L$ vs $\ln \ln L$ at critical field $h_0^c=5.35$. The dashed line is a linear fit in $\ln \ln L$ scale. The deviation from $\ln \ln L$ is clearly seen in systems with $M⩾128$ for $40\lesssim L\lesssim 80$, where $S\left(L\right)∕L$ scales almost independently of $M$.

Figure 3

(Color online) (a) Scaling plots of $\delta S\left(L\right)$ to reveal the subleading term of $S\left(L\right)$ at critical field $h_0^c=5.35$ for different geometries of subsystem $A$: a square [in (a)] and a cross shape [in (b)]. The dashed lines are linear fits in $\ln L$ scale with $b_{\mathit{\text{square}}}=-0.019$ and $b_{\mathit{\text{cross}}}=-0.08$, respectively.

Figure 4

(Color online) (a) Scaling of the largest active cluster size $N_c$ during RG shows a signature of classical percolation at a finite energy scale. (b) Scaling of $\delta N$ at percolation threshold ${\Gamma }_{\infty }=5.27$ for $h_0=3.2$; $A$ is a line interval on the boundary. (c) Scaling of $\delta N$ at bond (upper) and site (lower) classical percolation thresholds; $A$ takes the geometry of a square. In (b) and (c), the dashed lines are linear fits in $\ln L$ scale with corresponding slope $b$ marked on the plot.