Semiclassical theory of nonlocal statistical measures: Residual Coulomb interactions

In a recent paper [Phys. Rev. Lett. 100, 164101 (2008)] and within the context of quantized chaotic billiards, random plane-wave and semiclassical theoretical approaches were applied to an example of a relatively new class of statistical measures, i.e., measures involving both complete spatial integration and energy summation as essential ingredients. A quintessential example comes from the desire to understand the short-range approximation to the first-order ground-state contribution of the residual Coulomb interaction. Billiards, fully chaotic or otherwise, provide an ideal class of systems on which to focus as they have proven to be successful in modeling the single-particle properties of a Landau-Fermi liquid in typical mesoscopic systems, i.e., closed or nearly closed quantum dots. It happens that both theoretical approaches give fully consistent results for measure averages, but that somewhat surprisingly for fully chaotic systems the semiclassical theory gives a much improved approximation for the fluctuations. Comparison of the theories highlights a couple of key shortcomings inherent in the random plane-wave approach. This paper contains a complete account of the theoretical approaches, elucidates the two shortcomings of the oft-relied-upon random plane-wave approach, and treats non-fully-chaotic systems as well.

Figure 1

Drawing of the desymmetrized cardioid and stadium billiard boundaries. The dashed lines illustrate the paths of the shortest periodic orbits for each system. Whereas this orbit is isolated in the cardioid billiard, in the stadium billiard it is a member of a continuous one-parameter family of identical orbits, indicated by the gray-shaded rectangular region.

Figure 2

Comparison of ${\mathcal{S}}_i$ and the approximation given in the last line of Eq. (17). The difference ${\mathcal{S}}_i-⟨{\mathcal{S}}_i⟩$ is plotted versus the eigenstate number $i$. In (a), the results for the odd-parity eigenstates of the cardioid billiard using Dirichlet boundary conditions are shown. In (b), the same results for the even-even eigenstates of the stadium billiard with Dirichlet boundary conditions are shown, except that the mean overall constant term has been numerically subtracted. Note the significant increase in the scale of the fluctuations about the mean for the stadium billiard. Ahead in Sec. 5B, a rudimentary theory is given for bouncing ball modes that leads to Eqs. (75, 76) shown as the dashed and solid lines, respectively, in (b).

Figure 3

Variance of ${\mathcal{S}}_i$ for the cardioid (odd parity only) and quarter stadium (even-even symmetry only) billiards using Dirichlet boundary conditions. In (a), the discrete points are the cardioid billiard results, the dotted line is the result of the random plane-wave model, i.e., Eq. (25), and the long-dashed line is the semiclassical theory, Eqs. (52, 53), given in Sec. 4A ahead. In (b), the discrete points are the stadium billiard results, the dotted line is the result of the random plane-wave model [Eq. (25)] and the long dashed line is the prediction [Eq. (77)] from the semiquantitative semiclassical theory developed in Sec. 5B ahead for the bouncing ball modes.

Figure 4

Correlation function $\text{Cor}\left[{\mathcal{S}}_i{\mathcal{S}}_{i+j}\right]=\text{Covar}\left[{\mathcal{S}}_i{\mathcal{S}}_{i+j}\right]/\text{Var}\left[{\mathcal{S}}_i\right]$ for the cardioid and stadium billiard examples. In both cases the correlation function (or covariance) of the billiard is consistent with zero to within sample size fluctuations as assumed in the random plane-wave model and as implied by the semiclassical theory ahead leading to Eq. (51).

Figure 5

Sketch of the four orbits which, as $x\to 0$, coalesce into the same [nearly] periodic orbit. The top row corresponds to two (nearly) periodic orbits such that $\mathbf{r}$ lies on the trajectory either (a) just before or (b) just after the bounce off the boundary. The bottom row corresponds to two nonperiodic orbits $\left(p_x^{\prime }=-p_x\right)$ such near $\mathbf{r}$ (c) one of them does not touch the boundary and (d) the other one touches twice.

Figure 6

Fourier transform of the density of states and ${\mathcal{S}}_i-⟨{\mathcal{S}}_i⟩$ of the cardioid billiard using Dirichlet boundary conditions. In (a), the solid line is for the density of states using the first 2000 odd-parity levels of the cardioid billiard. The dashed line is for the density of states using the corresponding form of the Gutzwiller trace formula. In (b), the solid line is the Fourier transform of ${\mathcal{S}}_i-⟨{\mathcal{S}}_i⟩$ for the first 2000 ${\mathcal{S}}_i$ and the dashed line is the result of Eq. (45) calculated for the shortest periodic orbits as the one shown in Fig. 1 and the insets (along with their retracings). The agreement is quite good.

Figure 7

Fourier transform of ${\mathcal{S}}_i-⟨{\mathcal{S}}_i⟩$ and the density of states from the even-even stadium eigenstates using Dirichlet boundary conditions. The dashed line is for the density of states (which has a different vertical scale shown at the right side).