Direct product of random unitary matrices: Two-point correlations and fluctuations

We study the ensembles of direct product of $m$ random unitary matrices of size $N$ drawn from a given circular ensemble. We calculate the statistical measures, viz. number variance and spacing distribution to investigate the level correlations and fluctuation properties of the eigenangle spectrum. Similar to the random unitary matrices, the level statistics is stationary for the ensemble constructed by their direct product. We find that the eigenangles are uncorrelated in the small spectral intervals. While, in large spectral intervals, the spectrum is rigid due to strong long-range correlations between the eigenangles. The analytical and numerical results are in good agreement. We also test our findings on the multipartite system of quantum kicked rotors.

Figure 1

The plot of number variance ${\mathrm{\Sigma }}^2\left(r\right)$ vs interval size $r$ for form factor in Eq. (25) of main text. Inset represents the same data for large $r$.

Figure 2

The plot of form factor $C^{2\otimes }\left(p\right)$ vs $p$ for (a) COE $\otimes$ COE, (b) CUE $\otimes$ CUE, and (c) CSE $\otimes$ CSE. The circles denote numerical data. The size of the spectrum is $N^2={100}^2$. The solid curves in (a)–(c) denote analytical results (see main text). The dashed lines indicate $C^{2\otimes }\left(p\right)=N^2$.

Figure 3

Number variance ${\mathrm{\Sigma }}^2\left(r\right)$ vs interval size $r$ for (a) COE $\otimes$ COE, (b) CUE $\otimes$ CUE, and (c) CSE $\otimes$ CSE. The numerical data is shown by circles. The size of the spectrum is $N^2={100}^2$. Data for small intervals is shown in insets, where the dashed lines denote ${\mathrm{\Sigma }}^2\left(r\right)=r$. The solid curves in both main frame and inset of panels (a)–(c) are plotted using Eqs. (30, 31, 32), respectively.

Figure 4

Number variance ${\mathrm{\Sigma }}^2\left(r\right)$ versus interval size $r$ for direct product of three independent random unitary matrices drawn from COE (circle), CUE (square), and CSE (diamond), respectively. The size of the spectrum is $N^3={100}^3$. The dashed line indicates ${\mathrm{\Sigma }}^2\left(r\right)=r$.

Figure 5

Level-spacing distribution $P_{n-1}\left(s\right)$ for different values of $n$ (see keys). The different symbols circle, square, and diamond in the same panel represent numerical data for COE $\otimes$ COE, CUE $\otimes$ CUE, and CSE $\otimes$ CSE, respectively. The size of the spectrum is $N^2={100}^2$. The solid curve in different panels denotes Poisson distribution [Eq. (33)].

Figure 6

Level-spacing distribution $P_{n-1}\left(s\right)$ for different values of $n$ (see keys). The symbols circle, square, and diamond in the same panel represent numerical data for COE $\otimes$ COE, CUE $\otimes$ CUE, and CSE $\otimes$ CSE, respectively. The size of the spectrum is $N^2={100}^2$. The dashed curves represent Gaussian distribution plotted using Eq. (34). The solid curves denote Poisson distribution [Eq. (33)].

Figure 7

Plot of number variance ${\mathrm{\Sigma }}^2\left(r\right)$ vs interval size $r$ for (a) $\lambda =0$, (b) $\lambda =0.9$. The size of the spectrum obtained from the direct product of independent matrices is $N^2={101}^2$. Numerical data for small $r$ is given in insets, where the dashed lines indicate ${\mathrm{\Sigma }}^2\left(r\right)=r$. The solid curves in main frame and inset of (a) and (b) are plotted using Eqs. (30) and (31), respectively.