Non-Bloch Self-energy

The non-Hermitian skin effect describes the exponential localization of single-particle eigenstates near the boundaries of the system. We tackle its generalization to the many-body regime by investigating interacting fermions in open quantum systems, and present an exact integral formula to calculate the self-energy based on the non-Bloch band theory. We employ a bi-base mapping to represent the fermionic Liouvillian by two types of quasi-particle excitations from the vacuum (steady state). We then utilize the perturbation theory to calculate the self-energy of these quasi-particles and develop a Feynman-diagrammatic representation. By imposing complex momentum conservation on the vertices, we simplify the formula to a double integral over the Brillouin zone. We demonstrate the high accuracy of our formula by comparing it to numerical results. Our integral formula provides a quantitative tool for investigating interacting fermions in open quantum systems with the non-Hermitian skin effect.