Wiener-Hopf factorization and non-Hermitian topology for Amoeba formulation in one-dimensional multiband systems

This paper establishes Wiener-Hopf factorization as a rigorous framework to extend the Amoeba formulation and the generalized Szegö limit theorem from single-band to one-dimensional multiband non-Hermitian systems, thereby clarifying applicability criteria and enabling systematic analysis of symmetry-decomposed spectral potentials.

The Gist

Imagine you are trying to predict the weather in a giant, chaotic city. In a normal, calm city (a Hermitian system), the wind blows evenly everywhere. If you know the wind speed in the center, you can easily guess it at the edges. This is like standard physics, where we use "Bloch band theory" to predict how electrons move in a solid.

But now, imagine a city with a strange, one-way street system where the wind gets sucked violently toward the edges of the city, leaving the center almost empty. This is the Non-Hermitian Skin Effect (NHSE). In this chaotic city, the wind at the edge is totally different from the wind in the center. The old rules (Bloch theory) break down completely because they assume the wind is the same everywhere.

This paper is about building a new, better map for these chaotic cities. Here is the breakdown using simple analogies:

1. The Problem: The "Amoeba" Map Got Stuck

Scientists previously tried to map these chaotic systems using a method called the "Amoeba formulation." Think of this as a digital map that tries to find the "lowest point" in a landscape to predict where the wind (electrons) will settle.

• The Success: For simple, single-lane roads (single-band systems), this map worked perfectly. It found the lowest valley.
• The Failure: When the city got complex with multiple lanes (multiband systems), the map got confused. Imagine two different types of wind blowing in opposite directions at the same time. The map tried to find a single "lowest point," but the winds were fighting each other. The map crashed, or gave the wrong answer, especially in systems with special symmetries (like time-reversal symmetry).

2. The Solution: The "Wiener-Hopf" Toolkit

The authors introduce a powerful mathematical tool called Wiener-Hopf Factorization (WHF).

• The Analogy: Imagine you have a tangled knot of rope (the complex physics of the system). The old method tried to pull the knot apart by guessing. The WHF method is like a specialized pair of scissors that knows exactly where to cut the rope to separate the "left-moving" parts from the "right-moving" parts without breaking the knot.
• The Magic: By using this tool, the authors can separate the messy, competing winds into neat, organized bundles. This allows them to see the true structure of the city, even when the winds are fighting.

3. The "Double-Deck" Trick (Hermitian Doubling)

To make this work for the chaotic non-Hermitian systems, they use a clever trick called Hermitian Doubling.

• The Analogy: Imagine you are trying to understand a ghost (the non-Hermitian system). Ghosts are hard to study directly. So, you build a "mirror world" (the doubled Hermitian system) where the ghost has a solid reflection.
• In this mirror world, the math is much friendlier. The authors show that if you solve the puzzle in the mirror world using their new scissors (WHF), you can translate the answer back to the real, chaotic world perfectly.

4. The New Rules: When the Map Needs a "Correction"

The paper discovers a specific rule for when the old "Amoeba" map fails and needs a fix.

• The "Residual Indices": Think of these as "traffic tickets." If the scissors (WHF) cut the rope perfectly, you have zero tickets, and the old map works. But if there are leftover knots (non-zero indices), it means there are special "edge modes" (traffic jams) at the boundaries that the old map ignored.
• The Fix: The authors provide a formula to add a "correction term" to the map. It's like adding a detour sign to your GPS. Once you add this correction, the map works again, even in the most complex, multi-lane chaotic cities.

5. Why This Matters

• For Class AII† (Symmetric Systems): They proved that their new method naturally creates a "symmetry-decomposed" map. This explains why previous scientists had to guess how to split the map in half to make it work. It turns out, the math was hiding in plain sight all along!
• The "Fragile" Discovery: They found a weird, unstable state (the $\kappa=2$ phase) where the city looks like it has a special symmetry, but it's actually a "house of cards." If you nudge it slightly, the special state collapses. This helps scientists understand which exotic states are real and which are just illusions.

Summary

In short, this paper takes a broken GPS (the Amoeba formulation) that couldn't handle complex, multi-lane traffic (multiband non-Hermitian systems). They fixed it by introducing a new pair of mathematical scissors (Wiener-Hopf) and a mirror-world trick (Hermitian doubling).

Now, scientists can accurately predict where electrons will pile up at the edges of materials, even when the physics is chaotic and the old rules don't apply. It turns a messy, confusing knot into a clean, solvable puzzle.

Technical Summary

1. Problem Statement

The Non-Hermitian Skin Effect (NHSE) causes bulk eigenmodes to localize extensively at boundaries, invalidating conventional Bloch band theory under periodic boundary conditions (PBC). To address this, the Amoeba formulation was developed to compute the spectral potential (and thus the spectrum) under open boundary conditions (OBC) by optimizing the Ronkin function.

However, the current Amoeba formulation faces a critical limitation:

• Multiband Failure: While the underlying Generalized Szegö Limit Theorem applies in principle to arbitrary dimensions, its practical implementation via the Amoeba formulation has been restricted to single-band systems.
• Competing Localization: In multiband systems, different bands may have different localization lengths. When symmetry-protected degeneracies (e.g., Kramers pairs in class AII$^\dagger$) exist, contributions from different bands compete during the optimization of the Ronkin function.
• Lack of Rigorous Foundation: Previous attempts to resolve this in class AII$^\dagger$ systems relied on phenomenological, symmetry-guided decompositions of the Ronkin function. The mathematical connection between these decompositions and the Generalized Szegö Limit Theorem remained unclear, preventing extension to general multiband systems (e.g., class A without protecting symmetries).

2. Methodology

The authors establish a rigorous mathematical framework by combining Wiener-Hopf Factorization (WHF) with Hermitian Doubling.

• Wiener-Hopf Factorization (WHF): The authors utilize WHF to decompose the non-Bloch Hamiltonian symbol $\sigma(\beta)$ (a matrix-valued Laurent polynomial) into factors analytic inside and outside the unit circle: $\sigma(\beta) = \sigma_+(\beta) D(\beta) \sigma_-(\beta)$. The diagonal matrix $D(\beta)$ contains partial indices ($\kappa_j$), which are integers characterizing the topological winding of the system.
• Hermitian Doubling: To apply WHF to non-Hermitian systems, the authors map the non-Hermitian Hamiltonian to a doubled Hermitian Hamiltonian (e.g., mapping Class A to Class AIII, and Class AII$^\dagger$ to Class DIII). This allows the use of established results regarding the Modified Szegö Limit Theorem for Hermitian systems.
• Symplectic Decomposition: For systems with transpose-type time-reversal symmetry (Class AII$^\dagger$), the authors derive a Symmetric Wiener-Hopf Factorization (SWHF). This naturally splits the system into two sectors ($\tau_+^{symp}$ and $\tau_-^{symp}$), providing a rigorous origin for the previously phenomenological symmetry-decomposed Ronkin functions.

3. Key Contributions

A. Generalized Szegö Limit Theorem for Multiband Systems

The paper derives the precise criteria for when the standard Amoeba optimization (minimizing the Ronkin function) is valid.

• Residual Partial Indices: The authors define residual partial indices $K^*[\sigma]$ as the partial indices of the symbol after optimizing the shift parameter $\mu$ (which corresponds to the inverse localization length).
• Validity Condition: The standard optimization $\phi(E) = \min_\mu R_\sigma(\mu)$ holds if and only if all residual partial indices vanish ($K^* = (0, 0, \dots)$).
• Correction Terms: If residual indices are non-zero (e.g., $K^* = (+1, -1)$), the simple optimization fails. The authors derive a correction term to the spectral potential:
  $$\phi(E) = \min_\mu R_\sigma(\mu) - (\mu_{k+1} - \mu_k)$$
  where $\mu_k$ and $\mu_{k+1}$ are the logarithmic moduli of the dominant roots just inside and outside the unit circle. This term accounts for the "exchange" of root contributions across the Brillouin zone boundary caused by non-trivial topology.

B. Rigorous Foundation for Class AII$^\dagger$

For systems with transpose-type time-reversal symmetry (Class AII$^\dagger$):

• The authors prove that the WHF naturally yields the symmetry-decomposed Ronkin functions ($R^{(\pm)}_\sigma$) proposed in their previous work.
• The decomposition is not arbitrary; it arises from the symplectic structure of the WHF factors.
• This provides a rigorous proof of the Generalized Szegö Limit Theorem for these systems and explains why the $Z_2$ topological invariant flips the sign of the leading root's contribution.

C. Discovery of Fragile $\kappa=2$ Phases

The framework reveals a new topological regime in Class AII$^\dagger$ systems:

• A phase with partial index $\kappa=2$ (two Kramers pairs) was identified.
• Unlike the $Z_2$-protected $\kappa=1$ phase, the $\kappa=2$ phase is fragile: it relies on a hidden unitary symmetry and disappears under small symmetry-preserving perturbations, reverting to a trivial $\kappa=0$ phase.
• In this regime, the standard Generalized Brillouin Zone (GBZ) condition ($\mu_1 = \mu_2$) is replaced by a modified condition ($\mu_2 = \mu_3$) due to the specific root structure enforced by the hidden symmetry.

4. Results

• Numerical Verification (Class A): Using a block-diagonal model of two uncoupled Hatano-Nelson chains, the authors demonstrated that in regions where partial indices do not vanish simultaneously (e.g., $K=[-1, 1]$), the conventional Amoeba formulation fails to reproduce the OBC spectrum. Applying the derived correction term successfully restored the correct spectrum.
• Numerical Verification (Class AII$^\dagger$): The WHF-based decomposition of the Ronkin function was shown to perfectly reproduce the phenomenological symmetry-decomposed functions for $\kappa=0$ and $\kappa=1$ regimes.
• Fragile Skin Effect: The numerical analysis confirmed the existence of the $\kappa=2$ skin mode and its fragility. The OBC spectrum in this regime was accurately reproduced only by using the WHF-based optimization with the specific correction for non-vanishing residual indices.
• Edge Modes: The framework was shown to remain valid even in the presence of isolated zero-energy edge modes (Sublattice Symmetry systems), as these modes do not contribute to the thermodynamic-limit spectral potential, though the residual indices can detect their topological origin.

5. Significance

• Unified Framework: This work provides the first rigorous mathematical foundation for the Amoeba formulation in multiband non-Hermitian systems, resolving long-standing ambiguities about its applicability.
• Topological Insight: It establishes a direct link between the partial indices of the Wiener-Hopf factorization and the asymptotic behavior of the OBC determinant. This bridges the gap between algebraic factorization theory and non-Hermitian bulk-boundary correspondence.
• Generalization Pathway: By identifying the specific conditions (vanishing residual partial indices) under which the Generalized Szegö Limit Theorem holds, the paper opens a pathway for systematic generalizations to other symmetry classes and potentially higher dimensions (though higher-dimensional WHF remains a mathematical challenge).
• Correction of Phenomenology: It transforms previously phenomenological fixes (symmetry decomposition) into rigorous mathematical consequences of the WHF, ensuring the robustness of the method for complex multiband systems.