Taming convergence in the determinant approach for x-ray excitation spectra

A determinant formalism in combination with ab initio calculations was proposed recently and has paved a way for simulating and interpreting x-ray excitation spectra in condensed-phase systems. The method systematically takes into account many-electron effects in the Mahan-Nozi\'eres-De Dominicis (MND) theory, including core-level excitonic effects, the Fermi-edge singularity, shakeup excitations, and wave function overlap effects such as the orthogonality catastrophe, all within a universal framework using many-electron configurations. A heuristic search algorithm was introduced to search for the configurations that are important for defining the x-ray spectral line shape, instead of enumerating them in a brute-force way. The algorithm has proven to be efficient for calculating $\mathrm{O}$ $K$ edges of transition metal oxides, which converge at the second excitation order (denoted as ${f}⁽ⁿ⁾$ with $n=2$), i.e., the final-state configurations with two e-h pairs (with one hole being the core hole). However, it remains unknown how the determinant x-ray spectra converge for general cases and at which excitation order $n$ should one stop the determinant calculation. Even with the heuristic algorithm, the number of many-electron configurations still grows exponentially with the excitation order $n$. In this work we prove two theorems that can indicate the order of magnitude of the contribution of the ${f}⁽ⁿ⁾$ configurations, so that one can estimate their contribution very quickly without actually calculating their amplitudes. The two theorems are based on singular-value decomposition (SVD) analysis, a method that is widely used to quantify entanglement between two quantum many-body systems. We examine the $K$ edges of several metallic systems with the determinant formalism up to ${f}⁽⁵⁾$ to illustrate the usefulness of the theorems.