Statistics of wave functions in disordered systems with applications to Coulomb blockade peak spacing

Despite considerable work on the energy-level and wave function statistics of disordered quantum systems, numerical studies of those statistics relevant for electron-electron interactions in mesoscopic systems have been lacking. We plug this gap by using a tight-binding model to study a wide variety of statistics for the two-dimensional, disordered quantum system in the diffusive regime. Our results are in good agreement with random matrix theory (or its extensions) for simple statistics such as the probability distribution of energy levels or spatial correlation of a wave function. However, we see substantial disagreement in several statistics which involve both integrating over space and different energy levels, indicating that disordered systems are more complex than previously thought. These are exactly the quantities relevant to electron-electron interaction effects in quantum dots; in fact, we apply these results to the Coulomb blockade, where we find altered spacings between conductance peaks and wider spin distributions than traditionally expected.

Figure 1

The mean inverse participation ratio (IPR) as a function of disorder for two energies. The mean IPR increases with disorder; the plateau in the middle of each curve, most noticeable for $1∕25$ filling, corresponds to the diffusive regime. RMT predicts a universal value of 3.0 (dotted line). The system is a $164\times 264$ rectangle, at $B=0$, with five disorder realizations and about 400 different states used in each case.

Figure 2

(Color online) Probability density of the spacing between neighboring energy levels. The calculated distribution (dashed) matches the Wigner surmise (solid) obtained from the orthogonal ensemble of RMT. Both the mean and integral are normalized to 1. (Filling is $1∕25$ and $B=0$.)

Figure 3

(Color online) Probability distribution of $\mathcal{A}{\mid \psi \left(\mathbf{r}\right)\mid }^2$ in three different regimes: semi-ballistic at $W=0.2$ (squares), diffusive at $W=0.5$ (circles), and localized at $W=1.2$ (pluses). In the diffusive case, we see excellent agreement with the Porter-Thomas distribution predicted by RMT. The integral of each curve is normalized to 1. Filling is $1∕25$ and $B=0$. (Because of the hard-wall boundary condition, the probabilities were sampled on the inner $3∕4$ of the rectangle.)

Figure 4

Spatial correlation of the wave function in the diffusive regime. The data for $⟨\psi \left(\mathbf{r}\right)\psi (\mathbf{r}+{\mathbf{r}}^{\prime })⟩$ is close to the random plane wave result, especially for small $r^{\prime }$. Filling is $1∕25$, $W=0.5$, and $B=0$. To avoid boundary interference, we used a $30\times 30$ section of the rectangle centered $1∕4$ of the side length from each boundary. Correlations were measured in the $x$ direction from points within this section ($y$ direction correlations are identical).

Figure 5

Spatial correlation of the square of the wave function in the diffusive regime. Calculated results for $⟨{\psi }^2\left(r\right){\psi }^2(r+r^{\prime })⟩$ are in reasonable agreement with the random plane wave result, especially for $r^{\prime }<20$. RMT predicts that the correlation function should rapidly approach 1. See Fig. 4 caption for parameters.

Figure 6

Distribution of the inverse participation ratio (IPR) for the three different regimes (semi-ballistic at $W=0.2$, diffusive at $W=0.5$, and localized at $W=1.2$). At low disorder, the RMT/RPW prediction of a thin, Gaussian distribution is met. With increasing disorder, and even in the diffusive regime, the distributions become clearly asymmetric and increasingly wide. The integral under each curve is normalized to 1. ($1∕25$ filling with $B=0$.)

Figure 7

(Color online) The mean of the $M_{ij}$ as a function of disorder strength. When averaged over all wave functions in our energy window (circles), the mean rises roughly quadratically with increasing disorder strength from 1, the RMT value. In contrast, for consecutive energy levels ($i=j-1$, squares), the mean is suppressed below 1 at weak disorder and shows evidence of the three regimes. (Five disorder realizations with $B=0$.)

Figure 8

(Color online) The dependence of mean $M_{ij}$ on the spacing between the two states for the three different regimes (semiballistic at $W=0.2$, diffusive at $W=0.5$, and localized at $W=0.8$). The point at which $j-i=0$, corresponding to the IPR, is omitted for clarity. The mean increases with $W$, and the effect is magnified for close eigenfunctions. Note that the curve for the diffusive system is nearly flat. ($1∕25$ filling with $B=0$.)

Figure 9

(Color online) Energy level spacing distribution in a magnetic field large enough to break time-reversal symmetry (6 ${\phi }_0$ through $2\pi \mathcal{A}$). The curves at three different energies, all in the diffusive regime, match the corresponding Wigner surmise of RMT (solid). Both the mean and integral are normalized to 1.

Figure 10

(Color online) Mean, median, and standard deviation of $M_{ij}$ as a function of the spacing between the two states. The scale for the standard deviation is on the right, and the point $j=i$ is omitted for clarity. The similarity in form of the mean and the standard deviation, as well as the increasing disparity between mean and median with closeness in energy, are striking. ($1∕100$ filling, $W=0.35$, $B=6$.)

Figure 11

Covariance of $M_{hj}$ and $M_{ij}$ (distinct $h$, $i$, $j$) in three diffusive cases as a function of the spacing between $i$ and $h$, averaged over all $j$ in the energy window. For clarity, the first ten points in each plot are shown intact, whereas the remaining points are averaged in groups of 10. Note the qualitative difference at $1∕25$ filling between the zero field and $B=6$ cases.

Figure 12

(Color online) Standard deviation of $F_j\left[n\right]$ for all four diffusive systems, averaged over all $j$ in the energy window. The magnitude is surprisingly large. Predictions that $F_j\left[n\right]$ would quickly saturate are not met; this is seen most clearly in the presence of a magnetic field where the increase is roughly linear even at $n=400$.

Figure 13

(Color online) The root-mean-square of $(F_{j+1}\left[n\right]-F_j\left[n\right])$ for all four diffusive systems, averaged over all $j$ in the energy window. This quantity is directly relevant to Coulomb blockade peak spacing. Although similar to those in Fig. 12, the curves are not simply $\sqrt{2}$ r.m.s. $\left(F_j\left[n\right]\right)$ because of correlations between $F_{j+1}\left[n\right]$ and $F_j\left[n\right]$ that are largest at low energy. As in Fig. 12, the magnitude is surprisingly large.

Figure 14

(Color online) Peak spacing distributions for the three systems with $B=6$ flux quanta and $J_s=0.4$. The integral under both the odd and even curve is normalized to 1, and both curves are shifted left a distance $J_s$, as is customary. Note the wide spacing distribution, the depression of the odd peak at $J_s$, and the lack of a strong odd/even effect, all increasingly apparent as energy is lowered.

Figure 15

(Color online) Spin distributions for the three systems with $B=6$ flux quanta and $J_s=0.4$. Note that the average net spin increases as energy is lowered, and even at $1∕75$ filling, a spin of 1 is more likely than a spin of 0. The demonstrated spins are much larger than are predicted in most theories.