Entropy of an acoustic black hole in Bose-Einstein condensates

We compute the entanglement entropy associated to the Hawking emission of a ($1+1$)-dimensional acoustic black hole in a Bose-Einstein condensate. We use the brick wall model proposed by ’t Hooft, adapted to the momentum space, in order to tackle the case when high frequency dispersion is taken in account. As expected, we find that in the hydrodynamic limit the entropy only depends on the size of the box in the near-horizon region, as for gravitational ($1+1$)-dimensional black holes. When dispersion effects are considered, we find a correction that depends on the square of the size of the near-horizon region measured in units of healing length, very similar to the universal correction to the entropy found in the case of spin-$1/2$ Heisenberg XX chains.

Figure 1

Normalized entropy versus the temperature for small temperatures. The bottom curve is obtained by subtracting the function $a/x$ from the integrand of Eq. (22) and setting $x_{\min }=0$. The top curve is obtained by setting $x_{\min }={10}^{-12}$.