Saturation of number variance in embedded random-matrix ensembles

We study fluctuation properties of embedded random matrix ensembles of noninteracting particles. For ensemble of two noninteracting particle systems, we find that unlike the spectra of classical random matrices, correlation functions are nonstationary. In the locally stationary region of spectra, we study the number variance and the spacing distributions. The spacing distributions follow the Poisson statistics, which is a key behavior of uncorrelated spectra. The number variance varies linearly as in the Poisson case for short correlation lengths but a kind of regularization occurs for large correlation lengths, and the number variance approaches saturation values. These results are known in the study of integrable systems but are being demonstrated for the first time in random matrix theory. We conjecture that the interacting particle cases, which exhibit the characteristics of classical random matrices for short correlation lengths, will also show saturation effects for large correlation lengths.

Figure 1

Schematic representation of (a) spectral density for an unfolded one-particle spectrum, (b) spectral density for corresponding two-particle spectrum, and (c) spectral density for the unfolded two-particle spectrum.

Figure 2

Number variance at short correlation lengths for all three types of EEs. Agreement with Poisson result, shown by solid lines, is excellent.

Figure 3

$k\text{th}$ nearest-neighbor spacing distribution of two-particle unfolded spectra for (a) EGOE, (b) EGUE, and (c) EGSE. $k=0$ denotes the nearest neighbor spacing distribution. Solid lines show the theoretical results for the normalized Poisson distribution.

Figure 4

The number variance for the form factor given in Eq. (30) for several values of $\delta$. Inset shows the same figure for small correlation lengths.

Figure 5

Number variance for the model for interacting particle system discussed in Sec. 8 for several values of $\delta$ with $g\left(k\right)$ as that of (a) GOE, (b) GUE, and (c) GSE. Inset of each figure shows the corresponding short-range behavior.

Figure 6

The two particle form factor for EGUE in different regions of spectra denoted by $\xi =3.0,105.6$, and 225.5, respectively.

Figure 7

Number variance vs. $r$ for EGUE for three regions denoted by $\xi$. Solid lines are obtained from Eq. (38).

Figure 8

Two-particle form factor for the EGSE for three values of $\xi$.

Figure 9

Variation of number variance with $r$ for EGSE. The number variance is calculated for various values of $\xi$.

Figure 10

The two-particle form factor for EGOE for three values of $\xi$.

Figure 11

Number variance vs. $r$ for EGOE. The spectra are generated from the one-particle spectra of size $N=1000$. Number variance is evaluated for $\xi =3.0,105.6$, and 225.5.

Figure 12

Number variance for all three ensembles at $N=5000$ and $\xi =20.0$, which is equivalent to $\zeta =50000$.

Figure 13

(a) Spectral density and (b) number variance for two-particle spectrum constructed from one-particle spectra with semicircular spectral density given in (G1) with $N=1000$. Number variance is calculated in the same region on the unfolded scale where it was calculated in Fig. 7. Data points denoted by circle, square, and triangle represent the number variance around $\zeta =1500,50000$, and 100 000, respectively. Upper, middle, and lower solid lines represent the number variance in the same region for the uniform case (i.e., for $R_1\left(x\right)=1$).