Measurement of entanglement entropy in the two-dimensional Potts model using wavelet analysis

A method is introduced to measure the entanglement entropy using a wavelet analysis. Using this method, the two-dimensional Haar wavelet transform of a configuration of Fortuin-Kasteleyn (FK) clusters is performed. The configuration represents a direct snapshot of spin-spin correlations since spin degrees of freedom are traced out in FK representation. A snapshot of FK clusters loses image information at each coarse-graining process by the wavelet transform. It is shown that the loss of image information measures the entanglement entropy in the Potts model.

Figure 1

Demonstration of the 1D Haar wavelet transform. (a) Coarse-graining steps for each level $m$ are shown. Black (white) squares at $m=0$ denote that the correspondent bits are zero (unity). The intensity of a gray rectangle denotes its coarse-grained value. (b) Plot of the logarithm of the entropy emission demonstrated in (a). The orange line and symbols denote an ideal dependence on $m$, ${log}_2{\mathcal{E}}_m={log}_{2}L-m$.

Figure 2

(a) The $m$ dependence of normalized entropy emissions of the 2D $q$-state Potts models. Curves denote the Calabrese-Cardy formula (21). (b) Magnified view of (a). Error bars are smaller than the size of the symbols.

Figure 3

Normalized entropy emission of the 2D $q=2$ Potts model for $L=512$ at $T=0.95T_c$, $T_c$, and $1.05T_c$. The growth rates of normalized entropy emissions at off-critical points are lower than that at the critical point and they continuously decrease as $m$ is increased. Error bars are smaller than the size of the symbols.

Figure 4

Normalized entropy emission of the 2D $q=8$ Potts model for $L=1024$ at $T=0.95T_c$, $T_c$, and $1.05T_c$. Normalized entropy emission at the critical point is larger than those at off-critical points. The growth rate of normalized entropy emissions at the critical point is suppressed in $m\gtrsim 7$.