Parity-induced generalized Brillouin zone without non-Hermitian skin effect

This paper demonstrates that spectral sensitivity to boundary conditions and generalized Brillouin zone features, typically associated with the non-Hermitian skin effect, can also emerge as parity-induced even-odd effects in non-Hermitian systems where wavefunctions remain delocalized.

The Gist

Imagine you are listening to a choir. In a normal, perfectly balanced choir (what physicists call a "Hermitian" system), the sound waves travel evenly. If you change the rules at the edges of the room—say, by putting up a wall instead of an open window—the song changes slightly, but the singers still stand in their usual spots, spread out across the stage.

Now, imagine a strange, "non-Hermitian" choir where the singers have microphones that either amplify their voices (gain) or silence them (loss). In many of these strange systems, something dramatic happens called the Non-Hermitian Skin Effect. It's like a sudden, chaotic rush where every single singer in the choir abandons the center of the stage and piles up tightly against one specific wall. The song changes completely depending on whether the wall is there or not. Physicists have long believed that if the song changes drastically based on the walls, and if the singers pile up, it must be this "Skin Effect."

The Paper's Big Discovery
This paper, by Alexander Felski, says: "Wait a minute. That's not always true."

The author found a special setup where the song changes drastically based on the walls, and the mathematical description of the song requires "imaginary" numbers (a complex map), yet the singers do not pile up against the wall. They stay spread out across the stage, just like in a normal choir.

Here is how the paper explains this using simple analogies:

1. The "Odd vs. Even" Parity Trick

The key to this discovery is the number of singers in the choir.

• Odd Number of Singers: If you have 5, 7, or 9 singers, the system behaves "normally." The song is stable, and the singers stay spread out.
• Even Number of Singers: If you have 4, 6, or 8 singers, something weird happens. The song becomes unstable and changes its pitch (energy) drastically.

The paper calls this a "Parity-Induced Effect." It's like a seesaw. If you have an odd number of people, the balance is different than if you have an even number. In this specific non-Hermitian model, having an even number of "sites" (singers) breaks a hidden symmetry. This breakage forces the math to use a "Generalized Brillouin Zone"—a fancy way of saying the map of the song has to be drawn in a complex, twisted space rather than a simple straight line.

2. The "Ghost Map" vs. The Real Stage

Usually, when physicists see a song that requires a twisted, complex map (Generalized Brillouin Zone), they assume the singers must be piling up against the wall (the Skin Effect).

• The Old Belief: Twisted Map = Piled-up Singers.
• The New Finding: Twisted Map = Piled-up Singers OR Just a weird even/odd number trick.

In this specific model (called the SSH* model), the math looks like it needs a twisted map to explain the song, but the singers are actually standing perfectly still in the middle of the stage. They are delocalized. The "twisted map" is just a mathematical artifact caused by the even number of singers, not a physical pile-up of people.

3. Why Does This Matter?

The author compares this to a "false alarm."
Imagine you hear a siren (the strange song) and see smoke (the complex math). You usually assume there is a fire (the Skin Effect). But this paper shows that sometimes, the siren and smoke are just caused by a specific type of machine turning on and off based on whether it's an even or odd hour. There is no fire; the building is safe.

The paper emphasizes that:

• This effect only happens in finite systems (small choirs with a specific number of singers).
• If you make the choir infinitely large (the "thermodynamic limit"), the even/odd difference disappears, and the singers return to normal behavior.
• This effect can even happen alongside a real Skin Effect, acting as a separate, distinguishable feature.

Summary in a Nutshell

The paper reveals that drastic changes in a system's behavior and the need for complex mathematical maps do not automatically mean the system is "skinning" (piling up states at the edges).

Sometimes, it's just a parity effect—a subtle quirk that happens when you have an even number of components versus an odd number. The singers are still spread out, but the song sounds different because of the count, not because they are huddled in a corner. This forces physicists to be more careful: just because the math looks like a "Skin Effect," it doesn't mean the physical states are actually localized.

Technical Summary

Technical Summary: Parity-induced generalized Brillouin zone without non-Hermitian skin effect

Problem Statement
In non-Hermitian physics, the non-Hermitian skin effect (NHSE) is characterized by the macroscopic accumulation of bulk states at boundaries and an acute sensitivity of the energy spectrum to boundary conditions. This phenomenon is typically described by a non-trivial generalized Brillouin zone (GBZ), where quasimomenta become complex, deviating from the conventional real-valued Brillouin zone. A prevailing assumption in the field is that the emergence of a complex GBZ and the resulting spectral sensitivity to boundary conditions are inextricably linked to the formation of localized skin states. This paper challenges that assumption by investigating whether such features can arise in non-Hermitian systems that do not exhibit the skin effect, specifically in the absence of the thermodynamic limit.

Methodology
The author analyzes a specific non-Hermitian tight-binding model, denoted as $H_{SSH^*}$, which is a variant of the Su-Schrieffer-Heeger (SSH) model. The Hamiltonian features alternating complex conjugate coupling strengths, effectively representing imaginary dimerization ($g \to i\delta$). The model is defined as:
$$H_{SSH^*} = \sum_{n} [-t + (-1)^n i\delta](c^\dagger_n c_{n+1} + c^\dagger_{n+1} c_n)$$
The study employs the generalized Brillouin zone formalism to analyze the system's spectrum under open boundary conditions (OBC) compared to periodic boundary conditions (PBC). Crucially, the analysis is conducted on finite chains, distinguishing between chains of odd length ($N$) and even length ($N$). The symmetry properties of the Hamiltonian are examined under the ramified Altland-Zirnbauer classification, specifically focusing on time-reversal symmetry ($\mathcal{TRS}^\dagger$) and parity reflection. The author derives the eigenvalue equations and constructs standing-wave solutions to determine the nature of the quasimomenta ($k$) and the spatial profile of the wavefunctions.

Key Contributions and Results

1. Parity-Dependent Spectral Sensitivity:
   The study reveals a distinct even-odd effect dependent on the chain length $N$.

   • Odd Length ($N$): The system preserves $\mathcal{TRS}^\dagger$ symmetry (class BDI$^\dagger$). The eigenvalues remain confined to the real and imaginary axes, and the OBC spectrum aligns with the PBC spectrum. The quasimomenta are real, and wavefunctions are fully delocalized.
   • Even Length ($N$): The reflection symmetry around a central site is broken, reducing the symmetry class to AI$^\dagger$. In this regime, the eigenvalues move into the complex plane, exhibiting acute sensitivity to boundary conditions that mimics the NHSE.

2. Non-Trivial GBZ without Skin Effect:
   For even-length chains, the analysis demonstrates that the discrepancy between OBC and PBC spectra is captured by a non-trivial generalized Brillouin zone with complex quasimomenta ($k \in \mathbb{C}$). The condition for the quasimomenta is given by $\tan[k(N+1)] = -i(\delta/t)\tan k$. While this yields complex $k$ values with non-zero imaginary parts (typically associated with exponential localization), the resulting eigenstates are not exponentially localized at the boundaries. Instead, they remain completely delocalized and exhibit (skew-)symmetry across the chain.

3. Finite-Size Origin:
   The complex GBZ and the associated spectral sensitivity are shown to be finite-size effects. As the chain length $N \to \infty$ (thermodynamic limit), the imaginary part of the quasimomenta is suppressed ($\propto 1/N$), and the spectrum converges to the behavior of the odd-length case and the conventional PBC description. Thus, the effect vanishes in the thermodynamic limit, distinguishing it from the robust NHSE.

4. Coexistence with Skin Effect:
   The paper further demonstrates that this parity-induced effect can coexist with a genuine non-Hermitian skin effect. When directional coupling (Hatano-Nelson terms) is added to the $H_{SSH^*}$ model, the system exhibits both the constant imaginary offset characteristic of the skin effect and the parity-dependent deformation of the GBZ. In this combined scenario, the GBZ deviations in the thermodynamic limit are rooted in the skin effect, while the even-odd distinction remains a separate, distinguishable feature.

Significance and Claims
The paper claims to resolve an apparent conflict in non-Hermitian topology: the existence of a non-trivial generalized Brillouin zone and complex quasimomenta does not necessarily imply the non-Hermitian skin effect or the localization of bulk states. The author argues that these features can arise as parity-induced even-odd effects in finite non-Hermitian systems, provided the system is away from the thermodynamic limit.

The significance lies in decoupling the concept of the generalized Brillouin zone from the physical phenomenon of state localization. The work highlights that spectral sensitivity to boundary conditions and complex quasimomenta can be artifacts of finite-size symmetry breaking (specifically the breaking of reflection symmetry in even-length chains) rather than topological invariants associated with skin modes. This distinction is crucial for correctly interpreting numerical simulations of finite non-Hermitian systems and for understanding the limits of non-Bloch band theory in regimes where the thermodynamic limit is not reached. The paper concludes that while the GBZ description is essential for capturing the system's behavior, the presence of complex $k$ values alone is insufficient evidence for the non-Hermitian skin effect.