Fermi-Edge Singularities in the Mesoscopic X-Ray Edge Problem

We study the x-ray edge problem for a chaotic quantum dot or nanoparticle displaying mesoscopic fluctuations. In the bulk, x-ray physics is known to produce Fermi-edge singularities—deviations from the naively expected photoabsorption cross section in the form of a peaked or rounded edge. For a coherent system with chaotic dynamics, we find substantial changes; in particular, a photoabsorption cross section showing a rounded edge in the bulk will change to a slightly peaked edge on average as the system size is reduced to a mesoscopic (coherent) scale.

Figure 1

Probability distribution $P(|\Delta {|}^2)$ of the overlap $|\Delta {|}^2$ for different parameter sets $\{N,M,-v_c/d\}$. (a) Although all ${\delta }_F\approx -\pi /2$, the $P(|\Delta {|}^2)$ are visibly different as a result of different bulklike overlaps $|{\Delta }_b{|}^2$ (increasing with the legend entries, COE case). (b) The curves coincide when scaled with $|{\Delta }_b{|}^2$. The inset shows the CUE result with a noticeably different $P(|\Delta {|}^2)$ distribution for small $|\Delta {|}^2$ originating from the different wave function statistics.

Figure 2

Exact and approximate probability distributions of the overlap for (a) weak ($v_c/d=-0.25$) and (b) strong ($v_c/d=-10$) perturbation (COE, $N=100,M=50$). The exact $P(|\Delta {|}^2)$ is increasingly well reproduced by the range-1 and range-2 approximations, indicating that the fluctuations originate from levels around the Fermi energy. The dashed line denotes the range-1 analytical result.

Figure 3

X-ray photoabsorption spectra for (a) the mesoscopic (chaotic) and (b) the bulklike (nonfluctuating) case in terms of averaged cross sections (COE, $N=100,M=50,v_c/d=-10$). (a) The mesoscopic spectrum (filled circles) shows a (slightly) peaked edge. Open circles indicate the individual spin contributions for the optically active (full line) and spectator (dashed line) spins. The former arises as the sum of direct and replacement (dash-dotted line), as well as shake-up (dashed line, crosses), processes. For comparison we give the bare spectrum (thin line) without either AOC or many-body effects, and the result with AOC only (direct process, thin dashed line). (b) Comparison of the peaked mesoscopic spectrum (circles) to the bulklike results for the same optical channel (triangles, $l_0=1$) that shows a rounded $K$ edge. The mesoscopic peak is, however, less pronounced than the bulklike $L$ edge (squares, $l_0=0$).