Exact distribution of spacing ratios for random and localized states in quantum chaotic systems

Typical eigenstates of quantum systems, whose classical limit is chaotic, are well approximated as random states. Corresponding eigenvalue spectra are modeled through an appropriate ensemble described by random matrix theory. However, a small subset of states violates this principle and displays eigenstate localization, a counterintuitive feature known to arise due to purely quantum or semiclassical effects. In the spectrum of chaotic systems, the localized and random states interact with one another and modify the spectral statistics. In this work, a $3\times 3$ random matrix model is used to obtain exact results for the ratio of spacing between a generic and localized state. We consider time-reversal-invariant as well as noninvariant scenarios. These results agree with the spectra computed from realistic physical systems that display localized eigenmodes.

Figure 1

(a) Energy levels of stadium billiards. Scarred levels are marked in dashed (red) lines. A g-g type and g-l type spacing is shown. (b) Two consecutive generic eigenstates (state numbers 200 and 201), and two consecutive states (245 and 246) (a generic state next to a localized state). (c) Distribution of spacing ratios for g-g type spacings, (d) distribution of spacings for g-l type spacings. The red (solid) and blue (broken) lines are the standard results for $p_W\left(r\right)$ and $p_p\left(r\right)$ respectively.

Figure 2

Spacing ratio distribution, for g-l type spacings, obtained from random matrix simulations of $3\times 3$ random matrices (histogram) compared with analytical $p\left(r\right)$ (red line). See text for how g-l type spacings are identified in simulations. The $\beta =1$ and $\beta =2$ cases are shown.

Figure 3

(a) Information entropy $S$ as a function of energy $E$ for the coupled quartic oscillator system at $\alpha =90$. The eigenstates having magnitude of information entropy $\lesssim 5.5$ can be identified as localized states. For the bulk of chaotic states that form the envelope, the value of $S$ is consistent with the random matrix average for the information entropy (not shown here). (b) Enlarged view of a portion of (a), consisting of 175 states, out of which four may be considered to be localized.

Figure 4

Spacing ratio distribution, for g-l type spacings, obtained from systems whose classical limit is chaotic. Histograms are obtained from spectrum computed for (a) quartic oscillator and (b) levels of Sm from ab initio calculations. The solid (red) line is the fit obtained using the analytical relation in Eq. (7).

Figure 5

Spacing ratio distribution, for g-l type spacings, obtained from systems whose classical limit is chaotic. Histograms are obtained from spectra computed for stadium billiards, with time-reversal symmetry (a) preserved ($\beta =1$) and (b) broken ($\beta =2$). The solid (red) line is the fit obtained using the analytical relation in Eqs. (7) and (11) for (a) and (b) respectively.