Characterizing all non-Hermitian degeneracies using algebraic approaches: Defectiveness and asymptotic behavior

This paper presents a rigorous algebraic framework to systematically characterize the asymptotic behavior and dispersion of all types of multi-block degeneracies in non-Hermitian systems under perturbations, offering a practical tool for analyzing complex experimental scenarios.

The Gist

Imagine you are a musician tuning a complex, magical instrument. In the world of standard physics (Hermitian systems), if you tighten a string, the note changes smoothly and predictably. But in the world of Non-Hermitian physics (which describes systems that lose or gain energy, like a guitar string with a dampener or a laser), things get weird.

Sometimes, two or more notes can merge into a single, strange "super-note." In physics, we call these degeneracies. There are two main types of these super-notes:

1. The "Safe" Merge (Hermitian-like): The notes merge, but if you nudge the instrument slightly, they split apart cleanly, like two friends who were hugging but let go easily.
2. The "Tricky" Merge (Exceptional Points): The notes merge, and their identities get scrambled. If you nudge the instrument, they don't just split; they swirl around each other in complex loops before separating. This is the famous Exceptional Point (EP).

The Problem: A Messy Kitchen

For a long time, scientists knew how to handle the "Tricky" merges when they were simple (just two notes merging). But what happens when you have a chaotic kitchen where multiple types of merges are happening at once? What if you have a big block of three notes merging, and next to it, two other notes merging?

When you try to tweak the system (add a perturbation), predicting how these notes will split is incredibly hard. It's like trying to guess how a pile of tangled earbuds will untangle when you pull on one specific knot. Previous methods could only handle simple knots, not the whole tangled mess.

The Solution: A New Map (Tropical Geometry)

This paper introduces a brilliant new way to map out these tangled knots using a branch of math called Tropical Geometry.

Here is the analogy:
Imagine you are looking at a mountain range at night. You can't see the details of the trees or rocks, but you can see the silhouette of the peaks against the moon.

• The Old Way: Trying to calculate the exact height of every single tree and rock (very hard, very slow).
• The New Way (Tropical Geometry): Looking at the silhouette. You ignore the small details and focus only on the steepest slopes and the highest peaks.

In this paper, the authors use "Tropical Polynomials" to draw the silhouette of the energy levels of the system. Instead of solving complex equations for every tiny change, they look at the "slope" of the energy landscape.

• If the slope is steep, the notes will split apart quickly.
• If the slope is flat, the notes stay stuck together.
• The "shape" of the silhouette tells them exactly how the notes will behave (e.g., will they split like a square root? A cube root?).

What They Discovered

By using this "silhouette" approach, the authors systematically categorized every possible way these tangled notes can behave when you tweak the system.

1. The "Multi-Block" Mystery: They solved the puzzle of what happens when you have different sizes of merges happening at the same time (e.g., a group of 3 notes merging while a group of 2 merges nearby). They found that depending on how you push the system, the notes might split in different patterns (some fast, some slow).
2. The "Magic" of Control: They showed that by choosing the right way to push the system (the right perturbation), you can actually transform one type of merge into another. It's like being able to untangle a knot just by pulling it in a specific direction, turning a "Tricky" merge into a "Safe" one, or vice versa.

Why Should You Care? (Real World Applications)

This isn't just abstract math; it's a blueprint for building better technology.

• Super-Sensitive Sensors: Imagine a sensor that can detect a single virus or a tiny change in gravity. These sensors work best near "Exceptional Points" because the system is so sensitive that a tiny push causes a huge reaction. This paper helps engineers design sensors that use the most sensitive types of merges, making them incredibly powerful.
• Better Lasers and Circuits: Understanding how these energy levels split helps in designing lasers that don't flicker and electronic circuits that process information faster.
• Quantum Computing: In the future, quantum computers might use these "tangled" states to store information. Knowing how to untangle or control them is crucial for making them work.

The Bottom Line

Think of this paper as a universal instruction manual for untangling the universe's most complex knots.

Before, scientists had a few tricks for simple knots. Now, they have a complete map (using Tropical Geometry) that shows exactly how any knot—no matter how messy or complex—will behave when you pull on it. This allows scientists to not just observe these weird quantum phenomena, but to engineer them, creating super-sensitive devices and more stable quantum systems.

Technical Summary

1. Problem Statement

Non-Hermitian (NH) systems exhibit unique spectral singularities known as Exceptional Points (EPs), where both eigenvalues and eigenvectors coalesce. While extensive literature exists on non-derogatory EPs (where the geometric multiplicity is 1, corresponding to a single Jordan block), there is a significant gap in understanding derogatory degeneracies (where the geometric multiplicity is $>1$, corresponding to multiple Jordan blocks).

The core problem addressed is the lack of a systematic framework to characterize the asymptotic behavior (dispersion relations) of all types of NH degeneracies (both defective and non-defective, single and multi-block) under both generic and non-generic perturbations. Existing methods often rely on case-by-case studies or discriminant analysis that fail to capture the full complexity of multi-block degeneracies and their response to specific perturbation directions.

2. Methodology

The authors employ a rigorous algebraic approach rooted in Tropical Geometry and Newton Polygons to analyze the leading-order behavior of eigenvalues under perturbation.

• Perturbation Setup: The system is modeled as $A(\epsilon) = A_0 + \epsilon B$, where $A_0$ is the degenerate matrix and $\epsilon$ is a small perturbation parameter.
• Newton Polygons: The characteristic polynomial $\mathcal{F}_\lambda(\epsilon) = 0$ is analyzed. By plotting the points $(i, \alpha_i)$, where $\alpha_i$ is the lowest power of $\epsilon$ in the coefficient $a_i(\epsilon)$, a Newton polygon is constructed. The slopes of the lower boundary of this polygon determine the leading power exponents ($\beta$) of the eigenvalue splitting ($\lambda \sim \epsilon^\beta$).
• Tropical Polynomials: The authors map the characteristic polynomial into the min-plus semiring ($\mathbb{R}_{\min}$).
  • Tropical addition: $a \oplus b = \min(a, b)$.
  • Tropical multiplication: $a \odot b = a + b$.
  • The tropical polynomial $\mathcal{P}(\omega_0)$ is derived from the orders of the coefficients. The tropical roots ($\omega_0$) of this piecewise-linear function correspond exactly to the leading power exponents of the eigenvalues.
  • The multiplicity of a tropical root (determined by the change in slope) indicates the number of eigenvalues sharing that specific dispersion exponent.
• Systematic Classification: The authors apply this framework to square matrices of size $N=2, 3, 4$, analyzing all possible Jordan canonical forms (e.g., $H_{2,1}$, $H_{2,2}$, $H_4$) under various perturbation scenarios.

3. Key Contributions

1. Unified Algebraic Framework: The paper provides a self-contained, mathematically rigorous method to characterize any NH degeneracy, extending beyond the standard non-derogatory EPs to include multi-block (derogatory) degeneracies and non-defective (Hermitian-like) degeneracies.
2. Systematic Perturbation Analysis: Unlike previous works that often assume generic perturbations, this work explicitly categorizes the asymptotic behavior under non-generic perturbations (specific directions in parameter space). It demonstrates how specific symmetries or parameter choices can alter the dispersion exponents (e.g., changing a $\epsilon^{1/3}$ splitting to $\epsilon^{1/2}$ or preserving degeneracy).
3. Complete Classification for Small Matrices: The authors provide exhaustive tables and tropical polynomial derivations for $2\times2$, $3\times3$, and $4\times4$ matrices, detailing the possible tropical roots and their multiplicities for every Jordan block configuration.
4. Connection to Physical Phenomena: The method is linked to physical observables such as eigenvalue braiding (topology of the Riemann surface) and sensor sensitivity.

4. Key Results

• Dispersion Relations: The leading order behavior of eigenvalues is determined by the tropical roots $\omega_0$.
  • For a generic perturbation of an $N \times N$ matrix with a single Jordan block of size $N$ (EP$_N$), the tropical root is $\omega_0 = 1/N$, yielding $N$ eigenvalues splitting as $\epsilon^{1/N}$.
  • For multi-block degeneracies (e.g., $H_{2,1}$ in $3\times3$), the tropical polynomial reveals multiple roots. For instance, a generic perturbation of $H_{2,1}$ yields roots $\omega_0 = 1/2$ (multiplicity 2) and $\omega_0 = 1$ (multiplicity 1), indicating a mix of square-root and linear dispersion.
• Non-Generic Perturbations: The authors show that by tuning perturbation parameters (e.g., setting specific matrix elements to zero), one can:
  • Lift degeneracy partially: Some eigenvalues may remain flat ($\omega_0 = 0$) while others split.
  • Change the order of splitting: A system might exhibit $\epsilon^{1/2}$ behavior instead of $\epsilon^{1/3}$ depending on the perturbation direction.
  • Preserve degeneracy: Certain perturbations fail to lift the degeneracy entirely (all $\omega_0 = 0$).
• Lidskii-Vishik-Ljusternik Theorem: The tropical approach naturally reproduces and generalizes the Lidskii theorem, providing a visual and computational tool to predict the number of series of eigenvalues and their orders without solving the full characteristic equation.
• Physical Examples:
  • Sensing: The framework explains how higher-order EPs in cavity-mediated atomic sensors and electronic circuits enhance sensitivity. It shows how specific perturbations can shift an EP$_6$ to an EP$_4$ or alter the dispersion scaling.
  • Nonreciprocal Systems: Applied to the Hatano-Nelson model, the method correctly predicts the emergence of skin effects and the splitting of EPs based on boundary conditions.
  • Liouvillian Superoperators: The paper analyzes open quantum systems, demonstrating how a non-Hermitian Hamiltonian EP of order $N$ transforms into a multi-block EP in the Liouvillian (e.g., $J_{2N-1} \oplus J_{2N-3} \dots$), and how dissipation splits these blocks.

5. Significance

This work is significant for several reasons:

• Theoretical Completeness: It fills a major gap in non-Hermitian physics by providing a systematic tool to handle derogatory degeneracies, which are common in open quantum systems and Liouvillians but were previously difficult to analyze systematically.
• Experimental Relevance: By characterizing how degeneracies respond to specific (non-generic) perturbations, the paper offers a guide for experimentalists on how to manipulate system parameters to achieve desired spectral behaviors (e.g., maximizing sensor sensitivity or controlling topological braiding).
• Computational Efficiency: The tropical geometry approach simplifies the analysis of complex characteristic polynomials into piecewise-linear functions, making it easier to predict asymptotic behavior without heavy symbolic computation.
• Engineering NH Systems: The ability to predict and control the transformation between different types of EPs (e.g., converting a multi-block EP to a single-block EP) opens new avenues for engineering non-Hermitian devices with tailored topological and sensing properties.

In summary, the paper establishes Tropical Geometry as a powerful, unifying language for the classification and prediction of non-Hermitian spectral singularities, bridging the gap between abstract algebraic theory and practical applications in sensing, topology, and open quantum systems.