# Microscopic bosonization of band structures: X-ray processes beyond the Fermi edge

## TL;DR

This paper extends bosonization to microscopic lattice models, enabling analysis of band structures and x-ray processes across all energy scales, including strong interactions, by representing fermionic kinetic energy in the energy domain.

## Contribution

It introduces a microscopic bosonization formalism that overcomes low-energy limitations, applicable to entire band structures and capable of handling nonlinear bosonic interactions.

## Key findings

- Successfully describes x-ray edge singularity at emission threshold.
- Quantitative agreement with exact diagonalization results.
- Reveals correlation signatures beyond the band bottom.

## Abstract

Bosonization provides a powerful analytical framework to deal with one-dimensional strongly interacting fermion systems, which makes it a cornerstone in quantum many-body theory. Yet, this success comes at the expense of using effective infrared parameters, and restricting the description to low energy states near the Fermi level. We propose a radical extension of the bosonization technique that overcomes both limitations, allowing computations with microscopic lattice Hamiltonians, from the Fermi level down to the bottom of the band. The formalism rests on the simple idea of representing the fermion kinetic term in the energy domain, after which it can be expressed in terms of free bosonic degrees of freedom. As a result, one- and two-body fermionic scattering processes generate anharmonic boson-boson interactions, even in the forward channel. We show that up to moderate interaction strengths, these nonlinearities can be treated analytically at all energy scales, using the x-ray emission problem as a showcase. In the strong interaction regime, we employ a systematic variational solution of the bosonic theory, and obtain results that agree quantitatively with an exact diagonalization of the original one-particle fermionic model. This provides a proof of the fully microscopic character of bosonization on all energy scales for an arbitrary band structure. Besides recovering the known x-ray edge singularity at the emission threshold, we find strong signatures of correlations even at emission frequencies beyond the band bottom.