The analytically tractable zoo of similarity-induced exceptional structures

This paper maps out the analytically tractable landscape of multifold exceptional structures in three and four dimensions under generalized similarities, revealing that spectral symmetries induced by these similarities can reduce the dimensionality of exceptional point manifolds beyond naive constraint counting, thereby unifying diverse physical scenarios from optics to open quantum systems.

The Gist

Imagine you are a detective trying to solve a mystery in a strange, invisible world called Non-Hermitian Physics. In this world, things don't always behave like normal objects; they can gain energy, lose energy, or even disappear.

In this world, there are special "meeting points" called Exceptional Points (EPs). Think of these as cosmic traffic jams where two or more distinct paths (eigenvalues) and their travelers (eigenvectors) crash into each other and merge into a single, confused blob.

Usually, finding these traffic jams is incredibly hard. It's like trying to balance a house of cards in a hurricane; you need to tune many knobs perfectly to make them appear. But, this paper introduces a new set of rules called "Similarities." These are like secret shortcuts or hidden symmetries in the universe that make these traffic jams appear much more easily and in more interesting shapes.

Here is the breakdown of their discovery, using simple analogies:

1. The "Zoo" of Traffic Jams

The authors realized that scientists were only looking at the biggest, most dramatic traffic jams (called EP4s, where 4 paths merge). They were ignoring the smaller, surrounding chaos.

The paper maps out the entire "zoo" of these structures. They found that the big EP4s don't just appear out of nowhere; they are the "kings" sitting on top of a hierarchy of smaller structures:

• EP2s: Simple two-way crashes (forming surfaces or sheets).
• EP3s: Three-way crashes (forming lines or arcs).
• EP4s: Four-way crashes (forming single points).

The Analogy: Imagine a mountain range. The peaks are the EP4s. But you can't have a peak without the slopes (EP3s) and the foothills (EP2s). The authors mapped out the entire mountain range, showing how the peaks are connected to the valleys by winding roads.

2. The Magic of "Similarity"

In normal physics, to get a specific traffic jam, you have to twist 10 different dials. But if you apply a "Similarity" (a specific rule about how the system behaves), the universe does some of the work for you.

• The "Pseudo-Hermitian" Rule: This is like a mirror. If you look at the system in a mirror, it looks the same but flipped. This rule forces the traffic jams to appear in specific patterns, often creating "twin" jams or forcing them to stay on a straight line (the real number line).
• The "Self-Skew" Rule: This is like a spinning top that looks the same when spun upside down. This rule forces the traffic jams to appear right at the center (zero energy) or in specific pairs.

The Surprise: The authors found that sometimes, these rules prevent certain types of traffic jams from forming. For example, in a 4-band system with the "Self-Skew" rule, you can get a 4-way crash, but you cannot get a 3-way crash. It's like a bouncer at a club who says, "You can bring 4 friends, or 2 friends, but never 3."

3. The "Double Trouble" (Multiple Similarities)

What happens if you apply two or three of these rules at the same time? The paper shows that the system becomes even more exotic.

• The Effect: The traffic jams become even more stable and appear in even higher dimensions.
• The Result: In a 6-band system, they found EP6s (six paths merging). These aren't just floating points; they are connected by a complex web of lower-level crashes (EP4s, EP3s, and EP2s).
• The Topology: The authors used a concept called "Winding Numbers" to classify these. Imagine walking around a traffic jam. If you walk in a circle around it, does your path twist once? Twice? This "twist" is a topological fingerprint that proves the traffic jam is a stable, unbreakable feature of the system, not just a fluke.

4. Why Should You Care? (The Real World)

You might think this is just abstract math, but the authors explain that this is happening in real labs right now:

• Optics (Light): Lasers and fiber optics use these principles to control how light behaves.
• Electrical Circuits: Engineers are building circuits that act like these "traffic jams" to create new types of sensors.
• Quantum Computers: Understanding these points helps in building more stable quantum systems.
• Acoustics: Even sound waves in special materials can form these structures.

The Big Picture

Think of this paper as the Google Maps for a previously unexplored territory. Before this, scientists knew where the "cities" (the big EP4s) were, but they didn't know about the "roads," "bridges," and "tunnels" (the EP2s and EP3s) connecting them.

The authors discovered that:

1. Rules change the landscape: Applying specific "Similarity" rules reshapes the terrain, creating new roads and blocking old ones.
2. Everything is connected: The big, rare events are always supported by a network of smaller, more common events.
3. It's predictable: Even though the system is chaotic, the "Similarity" rules allow scientists to predict exactly where these strange structures will appear and how they are connected.

In short, this paper turns a confusing mess of mathematical possibilities into a clear, organized map, showing us how to build and control these exotic "traffic jams" in the real world.

Technical Summary

1. Problem Statement

Exceptional Points (EPs) are non-Hermitian spectral degeneracies where eigenvalues and eigenvectors coalesce simultaneously. While $n$-fold EPs ($EP_n$) are generally known to require the tuning of $2n-2$ real parameters (codimension), the presence of symmetries or generalized similarities can reduce this codimension, making $EP_n$ stable in lower dimensions.

However, a critical gap exists in the literature:

• Isolation vs. Hierarchy: Previous studies have largely treated high-order $EP_n$ ($n>2$) as isolated objects. In reality, stable $EP_n$ points generically emerge as special points on manifolds of lower-order exceptional structures ($EP_m$ where $m < n$).
• Incomplete Mapping: The complex interplay between these hierarchical structures (e.g., how $EP_4$ points connect to $EP_3$ arcs and $EP_2$ surfaces) under the influence of generalized similarities has not been systematically mapped, particularly in 3D and 4D systems.
• Constraint Counting Fallacy: Simply counting the number of constraints imposed by a similarity is insufficient to predict the dimensionality of the resulting exceptional manifolds. Spectral symmetries can force constraints to be satisfied by lower-order manifolds in unexpected ways, or forbid certain degeneracies entirely.

2. Methodology

The authors employ a combination of analytical algebraic methods and minimal toy models to map the "zoo" of exceptional structures.

• Generalized Similarities: Instead of focusing on specific unitary symmetries (like PT-symmetry), the authors utilize three generalized similarity relations defined by invertible generators:

  1. Pseudo-Hermiticity: $H = \eta H^\dagger \eta^{-1}$ (Spectral symmetry: $\{\lambda\} = \{\lambda^*\}$).
  2. Pseudo-anti-Hermiticity: $H = -\Gamma H^\dagger \Gamma^{-1}$ (Spectral symmetry: $\{\lambda\} = \{-\lambda^*\}$).
  3. Self-skew-similarity: $H = -S H S^{-1}$ (Spectral symmetry: $\{\lambda\} = \{-\lambda\}$).
     Note: Unitary symmetries are shown to be special cases of these similarities.

• Minimal Frobenius Companion Models: The authors construct generic minimal models using Frobenius companion matrices. By imposing the constraints derived from the similarity relations on the coefficients of the characteristic polynomial, they derive the exact conditions for the emergence of $EP_n$.

• Dimensional Reduction & Mapping:

  • For Self-skew-similarity, they map $2n$-band and $(2n+1)$-band systems onto unconstrained $n$-band non-Hermitian models. This reveals that adding a flat band at $\lambda=0$ promotes degeneracies at the origin by one order.
  • For Multiple Similarities, they demonstrate that imposing two similarities automatically implies the third, further reducing the codimension of $EP_n$ to $\lfloor n/2 \rfloor$.

• Topological Classification: The authors extend the classification of multifold EPs using Resultant Winding Numbers. They construct a "Resultant Hamiltonian" (a Hermitian operator built from resultants of the characteristic polynomial and its derivatives) to map the non-Hermitian eigenvalue topology to Hermitian eigenvector topology (Altland-Zirnbauer classes).

3. Key Contributions and Results

A. Pseudo-Hermiticity in 3D (4-band systems)

• Hierarchy: $EP_4$ points do not appear in isolation. They emerge as intersections of $EP_3$ arcs, which in turn lie on $EP_2$ surfaces.
• Spectral Constraints: The similarity forces $EP_2$ structures to appear either as purely real surfaces or as complex conjugate pairs.
• Codimension Anomalies: The paper identifies "special" $EP_2$ arcs (e.g., imaginary $EP_2$ pairs) that have a codimension of 2 (higher than the expected $2-1=1$ for pseudo-Hermitian systems). This occurs because the spectral symmetry forces these points to appear in pairs, effectively doubling the constraint requirement.
• Fermi Structures: The models exhibit "three-level Fermi surfaces" where three eigenvalues share the same real part, intersecting with the discriminant to form the $EP_3$ arcs.

B. Self-Skew-Similarity in 4D (4-band systems)

• Forbidden $EP_3$: Unlike pseudo-Hermitian systems, self-skew-similarity in 4-band systems forbids the emergence of $EP_3$ structures entirely.
• Structure: $EP_4$ points are connected by two distinct types of $EP_2$ surfaces:
  1. A surface at $\lambda = 0$ (where the flat band would be).
  2. A surface at finite $\lambda$.
• Dimensional Promotion: Adding a 5th band (which must be flat at $\lambda=0$ due to symmetry) promotes the $EP_4$ to $EP_5$ and the finite-$\lambda$ $EP_2$ surface to an $EP_3$ surface, while the zero-energy $EP_2$ surface remains an $EP_2$ surface (now embedded in a 5-band system).

C. Multiple Similarities (3D and 4D)

• Triple Similarity: Imposing any two similarities implies the third. This reduces the codimension of $EP_n$ to $\lfloor n/2 \rfloor$.
• 3D Six-Band Model ($EP_6$):
  • $EP_6$ pairs are connected by a complex hierarchy: $EP_4$ arcs at $\lambda=0$, $EP_3$ arcs (real or imaginary), and $EP_2$ surfaces.
  • Forbidden $EP_5$: Despite the reduced codimension suggesting $EP_5$ could exist as 1D arcs, the specific spectral symmetry of multiple similarities forbids $EP_5$ in six-band systems.
  • Generic Structures: The spectrum also hosts "generic" $EP_m$ structures (codimension $2m-2$) that are not induced by the similarities but are constrained by them to appear in specific quadrants of the complex plane.
• 4D Eight/Nine-Band Models: Similar hierarchies are found for $EP_8$ and $EP_9$, confirming the generalizability of the findings.

D. Topological Classification

• Resultant Winding Number: The authors classify the Abelian eigenvalue topology of $EP_n$ induced by multiple similarities using the winding number of the resultant vector $\mathbf{R}$.
• Symmetry Classes:
  • If $\lfloor n/2 \rfloor$ is odd, the topology belongs to Class A (Chern numbers).
  • If $\lfloor n/2 \rfloor$ is even, it belongs to Class AIII (Winding numbers).
  • For $n=2,3$, the classification falls into Class D ($\mathbb{Z}_2$ invariant).
• Vector Bundles: The resultant vector is identified as a section of the tangent bundle of a torus $T^{\lfloor n/2 \rfloor}$, providing a geometric interpretation of the topology.

4. Significance and Impact

• Theoretical Completeness: This work provides the "final chapter" in the analytical understanding of similarity-induced exceptional structures, moving beyond isolated points to a complete map of the hierarchical "zoo" of degeneracies in 3D and 4D.
• Experimental Relevance: The predictions are directly applicable to various physical platforms:
  • Optics/Photonics: PT-symmetric systems and topological photonics.
  • Electrical Circuits: Topolectrical circuits with balanced gain/loss.
  • Quantum Systems: Ultracold atoms, Liouvillian descriptions of open systems, and nitrogen-vacancy centers.
• Design Principles: By understanding how spectral symmetries constrain the dimensionality and existence of specific $EP_m$ manifolds, researchers can engineer systems to host specific high-order degeneracies or avoid unwanted ones.
• Universality: The results demonstrate that the "zoo" of structures is universal across different physical realizations, governed fundamentally by the algebraic properties of the similarity relations rather than the specific details of the dynamical matrix.

In conclusion, the paper establishes that similarity relations do not merely lower the codimension of high-order EPs but fundamentally restructure the entire spectral landscape, creating intricate, interconnected manifolds of lower-order degeneracies that are strictly constrained by the underlying spectral symmetries.