Temporal structures of the X-ray photoemission problem
Theoretical studies of X-ray photoemission from simple metals have traditionally focused on the frequency domain, aiming to reproduce experimental spectra. Here, we investigate the same problem in the time domain in search of physical insight and methodological advances. Our results reveal prominent aspects of the problem that are inconspicuous in the frequency domain. The calculated $\mathcal{F}(t)$ exhibits a weakly damped harmonic oscillation that modulates the Doniach-Sunjic power-law decay of the photoemission rate, a behavior arising from the coherent interference between two classes of particle-hole excitations. From a methodological perspective, we advance the time-dependent numerical renormalization-group (NRG) approach by exploring the eNRG method, a real-space variant that is more flexible than Wilson's construction. Giving special attention to strong core-hole potentials, we compare the photocurrents obtained from two complementary time-dependent eNRG algorithms with (i) an analytical expression for $\mathcal{F}(t)$ that becomes highly accurate at moderately long times and (ii) results from numerical diagonalization of a tight-binding Hamiltonian, which covers the time interval in which our analytical expression is less precise. Anticipating extensions to correlated-impurity models, we identify the sources of deviation and discuss the virtues and drawbacks of the two algorithms.