Non-Bloch self-energy of dissipative interacting fermions

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.

💡 Research Summary

The paper tackles the long‑standing problem of how the non‑Hermitian skin effect (NHSE), originally discovered for single‑particle non‑Hermitian lattices, manifests in many‑body open quantum systems with interactions. The authors consider a one‑dimensional fermionic lattice subject to Markovian dissipation described by a Lindblad master equation. The Liouvillian is split into a quadratic (non‑interacting) part L₀ and an interaction part L_I consisting of density‑density terms. By vectorizing the density matrix, the Liouvillian becomes a non‑Hermitian many‑body Hamiltonian acting on a doubled Hilbert space. A non‑unitary Bogoliubov transformation diagonalizes L₀ into two independent sets of fermionic modes, labelled a‑fermions (right‑acting) and b‑fermions (left‑acting). Each mode is characterized by a complex momentum β that lives on the generalized Brillouin zone (GBZ), the closed contour in the complex plane that replaces the ordinary Brillouin zone under open boundary conditions.

Interactions are rewritten in terms of the a/b‑fermions and generate processes that create or annihilate a pair of a‑ and b‑fermions conditioned on local density – a “doublon‑mediated” hopping. Treating the interaction perturbatively to second order, the authors derive an effective Liouvillian L_eff(E) that contains a self‑energy Σ^{(2)}(E,β). The self‑energy corresponds to a three‑particle propagator: a bare a‑fermion emits a doublon, three bare modes propagate, and the doublon is re‑absorbed. Because the system lacks translational invariance under OBC, the propagators are naturally expressed as integrals over the GBZ rather than over ordinary momenta.

A key technical advance is the deformation of the integration contours from the GBZ to the ordinary Brillouin zone (BZ). By moving two of the three complex momenta (β₁,β₂) onto the BZ and the third (β₃) onto a circle of radius |β|, the authors enforce a δ‑function that implements complex‑momentum conservation at the interaction vertices. This reduces the original triple integral to a double integral over ordinary momenta (Eq. 10), which can be evaluated analytically or numerically with high precision. The resulting “non‑Bloch self‑energy” retains the dependence on the original β through the external legs, while the internal loop respects ordinary momentum conservation.

The self‑energy is then used to correct the non‑interacting dispersion E₀(β)=X(β). For weak interactions, it suffices to evaluate Σ^{(2)} on the unperturbed energy, yielding E(β)=E₀(β)+Σ^{(2)}(E₀(β),β). Under OBC each eigenmode is a superposition of 2M non‑Bloch wave components (M is the hopping range). The dominant correction to a given eigenvalue comes from the two components with the largest magnitudes (β_M,β_{M+1}); their weighted average gives the final shift (Eq. 9). Finite‑size analysis shows that corrections scale as 1/N, confirming the perturbative approach’s validity in the thermodynamic limit.

To benchmark the theory, the authors apply it to a Liouvillian version of the Hatano–Nelson model with nearest‑neighbor hopping t, asymmetric gain/loss γ, and a small density‑density interaction u. The non‑interacting damping matrix X(β) and interaction polynomial U(β) are simple Laurent polynomials, and the GBZ is a circle of radius R=√{(t+γ)/(t−γ)}. Using Eq. 10 they compute Σ^{(2)} and compare the resulting eigenvalue corrections δE(θ) (θ being the angle on the GBZ) with exact diagonalization for N=31. The agreement is excellent; the Liouvillian gap correction extrapolates to the analytical value as N→∞.

A striking physical insight emerges: interactions renormalize the GBZ, effectively changing its radius and thus the degree of skin localization. This “interaction‑enhanced NHSE” shows that the skin effect is not solely a property of the non‑reciprocal hopping but can be amplified (or suppressed) by many‑body correlations. The authors interpret their framework as an open‑system analogue of Fermi‑liquid theory: the bare a‑ and b‑fermions are dressed by interaction‑induced self‑energies into quasiparticles that retain a well‑defined complex momentum on a renormalized GBZ.

The paper concludes by emphasizing the generality of the diagrammatic non‑Bloch self‑energy approach. It can be extended to higher dimensions, longer‑range interactions, and possibly to strong‑coupling regimes where resummations would be required. Moreover, the method provides a systematic way to incorporate quantum jumps (the full Lindblad dynamics) rather than relying on effective non‑Hermitian Hamiltonians that neglect jumps. This work therefore bridges non‑Hermitian topology, many‑body physics, and open‑quantum‑system dynamics, opening a path toward a comprehensive theory of interacting non‑Hermitian matter.