Non-Bloch self-energy of dissipative interacting fermions

Abstract

The non-Hermitian skin effect describes the phenomenon of exponential localization of single-particle eigenstates near the boundary of the system. We consider its generalization to the many-body regime by investigating a general class of interacting fermion lattice models in Markovian open quantum systems. Therein, the elementary excitations from the "vacuum" (steady state) are given by two types of dissipative fermionic modes composed of single-fermion operators, which govern the long-time nonequilibrium dynamics. We perturbatively calculate the self-energy matrix of these bare modes in the presence of interactions, and utilize the non-Bloch band theory to derive an exact integral representation. By imposing complex momentum conservation, we obtain a simplified expression for corrections to Liouvillian spectrum that agrees well with numerical calculations to high precision. We further perform perturbative analysis of Liouvillian eigenstates and identify signatures of interaction-enhanced NHSE at the quasiparticle level, manifested as renormalization of the generalized Brillouin zone. Our results establish a diagrammatic framework for dissipative interacting fermions with non-Hermitian skin effect in a description of full-fledged Lindblad master equations, which resembles Fermi liquid theory in terms of interaction-dressed quasiparticles.

Figures (4)

• Figure 1: (a) A sketch of Liouvillian spectrum. Blue (green) circles correspond to the model with (without) interactions, and $g$ ($g'$) is the gap. Only eigenvalues close to the imaginary axis are presented. The orange circle corresponds to the steady state. (b) A schematic plot about the self-energy. Interactions between fermions (black dots) with strength $U$ are mapped to the doublon-mediated hopping of a fermionic mode (red dot). (c) Feynman diagrams of the second-order self-energy. Red (blue) lines correspond to the $a(b)$-fermion, labeled by its complex momentum; wavy lines denote interactions.
• Figure 2: Self-energies obtained from Eq. \ref{['Eqn:SelfEnergyBZ']}, compared with exact diagonalization (ED) results. Parameters: $t=1,\gamma=0.5$, interaction strength $u = 0.2$. Corrections are all displayed in the unit of $u^2$. (a) Self-energy corrections to eigenvalues $\delta E(\theta)$ as a function of angle $\theta$ on GBZ. Red dots are obtained using Eqs. (\ref{['Eqn:SelfEnergyBZ']}, \ref{['Eqn:Weight']}), and the blue triangles denote ED results. System size $N=31$. (b) Finite-size scaling of the Liouvillian gap correction. The thermodynamic-limit extrapolation approaches the analytical result (the red point).
• Figure 3: Deformation of the generalized Brillouin zone induced by interactions. (a) Angular dependence of the relative modulus change, shown in logarithmic form for several $\gamma$ with fixed parameters $t=1, u=0.2$. (b) Illustration of GBZ deformation in the complex plane for $\gamma = 0.5$. Scale is amplified for visibility. The dashed (solid) curve denotes the GBZ before (after) including the self-energy. Blue arrows indicate directions of deformation.
• Figure 4: Analytical results of the self-energy for different boundary conditions, as a function of angle of $\beta$ on GBZ (OBC, red dots) and of the momentum (PBC, blue dots). Real and imaginary parts are shown respectively. $t=1,\gamma=0.5$.