Power-law corrections to entanglement entropy of horizons

We reexamine the idea that the origin of black-hole entropy may lie in the entanglement of quantum fields between the inside and outside of the horizon. Motivated by the observation that certain modes of gravitational fluctuations in a black-hole background behave as scalar fields, we compute the entanglement entropy of such a field, by tracing over its degrees of freedom inside a sphere. We show that while this entropy is proportional to the area of the sphere when the field is in its ground state, a correction term proportional to a fractional power of area results when the field is in a superposition of ground and excited states. The area law is thus recovered for large areas. Further, we identify the location of the degrees of freedom that give rise to the above entropy.

Figure 1

Plots of logarithm of GS, ES, and MS (Eq/Hi) entropies versus $\ln (R/a)$, where $R=a(n+1/2)$ radius of the hypothetical sphere (horizon), for $N=300$, $n=100–200$ and $o=30$, 40, 50 (in the ES and MS cases). The numerical precision is 0.01%.

Figure 2

Plots of ratios of GS and MS (Eq/Hi) or ES entropies and their reciprocals versus the area $\mathcal{A}$ (in units of $a^2$, $a$ being the lattice spacing) for $o=30$, 50. The plots show the asymptotic nature of the MS and ES entropies with respect to the GS entropy. The curves on the upper half (above 1) show the variation of $S_{\mathrm{XS}}/S_{\mathrm{GS}}$ with $\mathcal{A}$, where XS stands for MS(Eq/Hi) or ES, while the lower curves show the variation of $S_{\mathrm{GS}}/S_{\mathrm{XS}}$ with $\mathcal{A}$.

Figure 3

Best fit plots (solid lines) of the relative mixed state entropies ($S_{\mathrm{MS}}/S_{\mathrm{GS}}$) for equal and high mixings versus the area $\mathcal{A}$ (in units of $a^2$), for $o=30$, 40, 50. The corresponding data are shown by asterisks.

Figure 4

Plots of the distribution of entropy per partial wave $[(21+1)S_l]$ in the cases of GS, ES, and MS (Eq/Hi), for $N=300$, $n=100$, and $o=30$, 40, 50.

Figure 5

Plots of the percentage contribution $pc(q)$ to the total entropy as a function of window position $q$, for a window size $d=5$ and fixed $N=300$, $n=100$, in each of cases of GS, ES, and MS (Eq/Hi). For ES and MS (Eq/Hi) the solid curve is for $o=30$ whereas the broken curve is for $o=50$.

Figure 6

Best fit plots of the relative variation of the total entropy $S_m$ for a massive scalar field (in units of the total entropy $S_0$ corresponding to a massless scalar field) with the mass $m$ times the lattice spacing $a$, for fixed $n=100$, $o=30$, in each of cases of GS, ES, and MS (Eq/Hi). The corresponding data are shown by asterisks. The fits show an exponential damping of the ratio $S_m/S_0$ with mass.