Non-Bloch self-energy of dissipative interacting fermions

This paper establishes a diagrammatic framework for dissipative interacting fermions exhibiting the non-Hermitian skin effect by deriving an exact integral representation of the self-energy via non-Bloch band theory and demonstrating that interactions lead to interaction-enhanced skin effects through the renormalization of the generalized Brillouin zone.

The Gist

Imagine you are watching a crowded dance floor. In a normal, closed room (a standard quantum system), the dancers move around freely, and if you look at the crowd from above, the energy is spread out evenly. Everyone has a chance to dance in the middle or near the walls.

Now, imagine this dance floor is leaky. People are constantly falling off the edge or being pulled in from the outside. This is an open quantum system. In these systems, something strange happens called the Non-Hermitian Skin Effect (NHSE). Instead of dancing everywhere, all the dancers suddenly pile up and hug the walls, refusing to go near the center. It's like a crowd that, when the doors open, all rushes to the exit at once.

This paper tackles a big question: What happens when these wall-hugging dancers start talking to each other?

The Problem: A Messy Dance Floor

Usually, physicists study these "skin effect" systems by looking at single dancers. But in the real world, dancers interact—they bump into each other, hold hands, or push away. When you add these interactions to a system where everyone is already hugging the wall, the math gets incredibly messy.

Previous methods tried to solve this by pretending the dancers were invisible ghosts that didn't interact, or by using complex math that only worked for tiny groups. The authors of this paper wanted a way to handle a huge crowd of interacting dancers that are all stuck to the wall.

The Solution: A New Map and a New Language

The authors developed a new "map" and a new "language" to describe this chaos. Here is how they did it, using simple analogies:

1. The "Double Vision" Trick (Vectorization)
To understand the whole crowd, the authors used a trick called "vectorization." Imagine looking at the dance floor through a special pair of glasses that splits your vision into two: one eye sees the dancers, and the other eye sees the "holes" where dancers could be. By combining these two views, they turned the messy, interacting crowd into a simpler problem that looks like a single, giant, non-physical dancer moving in a doubled-up world. This allowed them to use standard tools from physics (like Feynman diagrams, which are basically flowcharts for particle interactions) to solve the problem.

2. The "Ghost Map" (Generalized Brillouin Zone)
In a normal room, you can describe a dancer's position using a standard map (like latitude and longitude). But in this "skin effect" system, the standard map breaks because the dancers are stuck to the wall.
The authors used a Generalized Brillouin Zone (GBZ). Think of this as a magic map where the coordinates aren't just numbers, but complex numbers (numbers with a "real" part and a "imaginary" part). On this magic map, the "wall" isn't a straight line; it's a curve that bends into the imaginary world. This map perfectly predicts where the dancers will pile up.

3. The "Self-Energy" (The Crowd's Influence)
The core of their discovery is calculating the "Self-Energy."

• The Analogy: Imagine a single dancer trying to move. In a vacuum, they move easily. But in a crowded room, every time they try to step, they bump into someone, get pushed back, or get pulled forward. This "bumping" changes how they move. In physics, we call this change in movement "Self-Energy."
• The Breakthrough: The authors calculated exactly how the interactions (the bumping) change the "Skin Effect." They found that the interactions don't just make the crowd messier; they actually strengthen the wall-hugging effect. The dancers don't just hug the wall; they hug it tighter and more specifically than before.

The Big Discovery: The Map Changes Shape

The most exciting finding is that when the dancers interact, the Magic Map (GBZ) itself changes shape.

• Before interactions: The map was a perfect circle (or a simple curve).
• After interactions: The map gets squashed and stretched. It becomes "anisotropic," meaning it looks different depending on which direction you look.
• What this means: The interactions cause the "skin effect" to become more extreme. The dancers become even more localized at the boundaries. It's as if the crowd, by talking to each other, decided to huddle even closer to the exit.

Why This Matters

This paper is like building a Fermi Liquid Theory for Open Systems.

• Fermi Liquid Theory is a famous theory that explains how electrons behave in metals (like copper wires). It treats electrons as if they are "dressed" in a cloud of other electrons, making them easier to understand.
• This Paper does the same thing for open, leaky quantum systems. They show that even in a chaotic, leaking system with interactions, you can still describe the particles as "dressed quasiparticles" moving on a warped, complex map.

In a Nutshell

The authors created a new mathematical toolkit to understand what happens when particles that are already stuck to the edge of a system start interacting with each other. They found that:

1. You can use a special "complex map" (GBZ) to track them.
2. Interactions make the "wall-hugging" effect (Skin Effect) even stronger.
3. They provided a precise formula to predict exactly how the system changes, which matches perfectly with computer simulations.

This is a major step forward because it moves us from studying simple, lonely particles to understanding complex, interacting crowds in the weird world of open quantum systems. It's a bridge between the messy reality of interacting particles and the elegant math of non-Hermitian physics.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding many-body interactions in open quantum systems exhibiting the Non-Hermitian Skin Effect (NHSE).

• Context: The NHSE causes single-particle eigenstates to accumulate exponentially at the boundaries under Open Boundary Conditions (OBC), characterized by complex momenta forming a Generalized Brillouin Zone (GBZ).
• Gap: While single-particle NHSE is well-understood via non-Bloch band theory, extending this to interacting many-body systems in full-fledged open systems (governed by Lindblad master equations) remains elusive.
• Specific Challenges:
  • Standard perturbative techniques (like Feynman diagrams) rely on momentum conservation, which is broken in OBC due to the lack of translational invariance.
  • Previous works often relied on effective non-Hermitian Hamiltonians that neglect quantum jumps (requiring post-selection), failing to describe the full dissipative dynamics.
  • There is no systematic framework to calculate how interactions dress dissipative quasiparticles and modify the GBZ.

2. Methodology

The authors develop a diagrammatic perturbation theory based on the non-Bloch band theory for interacting fermions in Markovian open quantum systems.

• Framework:

  • Lindblad Master Equation: The system dynamics are governed by $\mathcal{L}\rho = -i[H, \rho] + \sum_\mu (2L_\mu \rho L_\mu^\dagger - \{L_\mu^\dagger L_\mu, \rho\})$.
  • Vectorization: The density matrix $\rho$ is mapped to a state vector in a doubled Hilbert space ($|\rho\rangle = \sum \rho_{mn} |m\rangle \otimes |n\rangle$). This transforms the Liouvillian superoperator $\mathcal{L}$ into a non-Hermitian many-body Hamiltonian acting on the doubled space.
  • Bare Modes: The non-interacting Liouvillian ($\mathcal{L}_0$) is block-diagonalized using a non-unitary Bogoliubov transformation. This yields two sets of decoupled fermionic operators ($a$ and $b$) labeled by complex momenta on the GBZ. The "vacuum" corresponds to the non-equilibrium steady state.
  • Interactions: Density-density interactions ($H_I = \sum U_{ij} n_i n_j$) are treated as perturbations.

• Derivation of Non-Bloch Self-Energy:

  • The authors calculate the second-order self-energy correction to the damping matrix.
  • Key Innovation: Unlike standard field theory where propagators carry real momenta, here propagators carry complex momenta on the GBZ.
  • Integral Representation: The self-energy $\Sigma^{(2)}(E, \beta)$ is derived as a triple integral over the GBZ (and its conjugate GBZ$^*$) involving the interaction potential $U(\beta)$ and the damping matrix $X(\beta)$.
  • Contour Deformation: To make the calculation tractable and restore momentum conservation, the authors deform the integration contours from the GBZ to the standard Brillouin Zone (BZ). This effectively enforces complex momentum conservation at external vertices, reducing the triple integral to a double integral over the BZ.

• Eigenstate Correction:

  • For OBC, eigenstates are superpositions of $2M$ non-Bloch modes. The authors derive a weighted average formula to determine the self-energy correction for a specific eigenvalue, dominated by the modes with the largest and smallest moduli ($\beta_M$ and $\beta_{M+1}$) that satisfy the boundary conditions.

3. Key Contributions

1. Non-Bloch Self-Energy Formalism: Established a rigorous diagrammatic framework for calculating self-energy in open interacting systems, introducing the concept of "Non-Bloch self-energy" where momentum conservation is replaced by complex momentum conservation on the GBZ.
2. Exact Integral Representation: Derived an exact, simplified double-integral formula for the self-energy corrections to Liouvillian eigenvalues, valid for generic interaction ranges.
3. Interaction-Enhanced NHSE: Demonstrated that interactions do not merely perturb the system but fundamentally alter the GBZ, leading to an enhancement of the skin effect (renormalization of the GBZ).
4. Open-System Fermi Liquid Analogy: Proposed a description where interactions dress bare dissipative modes into quasiparticles, analogous to Fermi liquid theory but within a Lindblad framework.

4. Results

• Numerical Validation: The analytical results for the self-energy corrections were benchmarked against Exact Diagonalization (ED) for a Liouvillian version of the Hatano-Nelson model with nearest-neighbor interactions.
  • The analytical predictions matched ED results with high precision.
  • Finite-size scaling confirmed that the analytical limit ($N \to \infty$) is correctly captured by the theory.
• GBZ Deformation:
  • Interactions were found to deform the GBZ, making it anisotropic.
  • For weak non-reciprocity, the modulus of the complex momentum $|\beta(\theta)|$ is suppressed near $\theta=0, \pi$ and enhanced near $\theta=\pm \pi/2$.
  • This deformation signifies a stronger localization (enhanced NHSE) for the slowest-decaying modes.
• Boundary Condition Sensitivity:
  • The theory explicitly distinguishes between OBC and Periodic Boundary Conditions (PBC).
  • Under PBC, interactions open a gap at previously gapless points (scaling as $u^2$), contrasting with the non-perturbative Luttinger liquid behavior in closed 1D systems. This is attributed to the exponential decay of steady-state correlations in open systems, which prevents infrared divergences.
• Stability Condition: The perturbative approach is valid when single-particle modes are spectrally separated from multi-particle continua (ensured by a sufficiently large Liouvillian gap).

5. Significance

• Theoretical Advancement: This work bridges the gap between non-Hermitian band theory and many-body physics in open systems. It provides the first systematic tool to analyze interaction effects in dissipative systems with NHSE without resorting to effective Hamiltonians that ignore quantum jumps.
• Experimental Relevance: The framework applies to experimental platforms like ultracold atoms and quantum dots where reservoir engineering can realize specific dissipative interactions.
• New Physics: It reveals that interactions can enhance the skin effect rather than suppress it, a counter-intuitive result compared to some closed-system studies.
• Future Directions: The paper lays the groundwork for studying non-Bloch self-energies in multi-band and higher-dimensional systems, as well as exploring localization transitions in strongly interacting regimes.

In summary, the paper successfully constructs a "Fermi liquid theory" for dissipative interacting fermions, demonstrating that interactions renormalize the Generalized Brillouin Zone and enhance the non-Hermitian skin effect, all while maintaining high agreement with numerical simulations.