Effective description of the gap fluctuation for chaotic Andreev billiards

We present a numerical study of the universal gap fluctuations and the ensemble averaged density of states (DOS) of chaotic two-dimensional Andreev billiards for finite Ehrenfest time ${\tau }_E$. We show that the distribution function of the gap fluctuation for small enough Ehrenfest time can be related to that derived earlier for zero Ehrenfest time. An effective description based on the random matrix theory is proposed giving a good agreement with the numerical results. A systematic linear decrease of the mean field gap with increasing Ehrenfest time ${\tau }_E$ is observed but its derivative with respect to ${\tau }_E$ is in between two competing theoretical predictions and close to that of the recent numerical calculations for Andreev map. The exponential tail of the density of states is interpreted semiclassically.

Figure 1

A normal dot (N) of Sinai billiard in contact with a superconductor (S).

Figure 2

Comparison of our numerical result using the parameters $E_g$ and ${\Delta }_g$ given by Eq. (2) (short dashes) and Eq. (1) (long dashes) with the theoretical predictions of the integrated distribution $F\left(x\right)$ (solid line). In the inset the same as in the main frame is plotted for the distribution $P\left(x\right)$ with square and triangular symbols, and solid line, respectively.

Figure 3

The mean field gap $E_g∕E_g^{\mathrm{RMT}}$ as a function of ratio ${\tau }_E^{\left(1\right)}∕{\tau }_D$. The straight line is the numerical fit using Eq. (4).

Figure 4

Ensemble averaged DOS $⟨\varrho \left(E\right)⟩$ (multiplied by ${\delta }_N$) as a function of $E∕E_T$ for SA billiard (solid line), the results from our effective description $⟨{\varrho }_{\mathrm{eff}}\left(E\right)⟩$ [expressing $x$ in terms of $E$ by using Eq. (2)] (short dashes), the Bohr-Sommerfeld calculation (dotted line), and the mean field DOS (long dashes).