Asymptotic non-Hermitian degeneracy phenomenon and its exactly solvable simulation

This paper explains the impossibility of regularizing intrinsic-exceptional-point singularities in non-Hermitian quantum models like the PT-symmetric imaginary cubic oscillator by constructing an exactly solvable finite-dimensional matrix toy model that mimics the asymptotic degeneracy of these systems while demonstrating that, unlike conventional exceptional points, such singularities cannot be resolved through small perturbations.

The Gist

The Big Problem: A Quantum Machine That Breaks Itself

Imagine you are trying to build a quantum machine (a model of how particles behave) using a specific type of energy field called the "Imaginary Cubic Oscillator."

Think of this machine like a car engine. Usually, engines run smoothly, and you can predict exactly how they will behave. But this specific engine has a fatal flaw. As the car goes faster (representing higher energy levels), the engine parts start to fuse together.

In physics terms, this is called an Intrinsic Exceptional Point (IEP).

• The Analogy: Imagine a choir where, as the song gets louder, every singer's voice starts to merge into a single, indistinguishable hum. Eventually, you can't tell who is singing what anymore. The individual voices (quantum states) lose their identity and become "degenerate" (identical).
• The Result: Because the voices merge, the mathematical rules that usually govern the universe (specifically, the ability to calculate probabilities) break down. The machine is "unphysical"—it's a broken engine that can't exist in our reality.

For a long time, physicists (including the author of this paper) thought this machine was a dead end. You couldn't fix it because the problem was built into the very fabric of the equation.

The New Idea: Build a Lego Model Instead

The author, Miloslav Znojil, decided to try a different approach. Instead of trying to fix the broken, infinite engine directly, he decided to build a Lego model of it.

• The Metaphor: Imagine you want to study a massive, continuous waterfall. It's too big to hold in your hands. So, instead, you build a small, discrete model using a bucket of water and a series of steps. You can't perfectly copy the infinite waterfall, but you can capture its essence using a finite number of steps.

In this paper, the author replaces the continuous, broken equation with a finite grid (a matrix).

• He takes the "infinite" problem and chops it up into a manageable number of blocks (let's say $N$ blocks).
• He creates a simplified "toy model" using a two-parameter switch (let's call them A and B) to control the energy.

The Discovery: The "Kink" in the Road

When the author played with his Lego model, he found something fascinating.

In the real, broken machine, the voices merge at the very end (infinity). But in his Lego model, the voices merge at a specific, reachable point on the road. This point is called a Kato's Exceptional Point (EP).

• The Analogy: Think of driving a car. In the broken model, the road just disappears into a fog that you can never reach. In the Lego model, the road hits a specific "kink" or a sharp turn where the lanes merge. You can actually drive up to that kink and study it.

The author realized that even though the Lego model isn't exactly the same as the broken infinite machine, it mimics the behavior perfectly.

• When the Lego model hits the "kink" (the EP), the energy levels merge, just like the voices in the choir.
• Crucially, because the Lego model is finite (it has a limited number of blocks), it doesn't break completely. It just gets "degenerate."

The Solution: A "Safe Zone" Around the Crash

The most important part of the paper is what happens near that kink.

In the broken infinite machine, you cannot get close to the problem without the whole thing collapsing. But in the Lego model, there is a "Safe Zone" (a physical domain) right next to the kink.

• The Analogy: Imagine a cliff edge. The broken machine falls off the cliff immediately. The Lego model, however, has a safety railing. You can walk right up to the edge of the cliff (the Exceptional Point) and look over, but as long as you stay on the path (the "Safe Zone"), you don't fall.

The author showed that by tweaking the parameters (A and B), you can keep the system in this safe zone. This proves that even though the original "Imaginary Cubic Oscillator" is theoretically broken, we can understand its behavior by studying these finite, solvable Lego models.

Why This Matters

1. It Solves a Mystery: It explains why the imaginary cubic oscillator is unphysical. It's not just "broken"; it's a limit where the rules of quantum mechanics stop working because the states merge too perfectly.
2. It Offers a Workaround: Even if we can't use the broken machine, we can use these finite Lego models to simulate what happens near the disaster. This is useful for things like quantum computing or understanding how light behaves in special materials.
3. It's a "Toy" That Works: The author used a very simple, two-parameter model. He didn't need a supercomputer to solve the whole universe; he just needed a small, cleverly designed grid to reveal the deep truth about the infinite problem.

Summary in One Sentence

The author took a broken, infinite quantum machine that defies the laws of physics, built a finite "Lego" version of it, and discovered that by studying where the Lego model's parts merge, we can finally understand and simulate the mysterious behavior of the broken machine without actually breaking the laws of reality.

Technical Summary

1. Problem Statement

The paper addresses the theoretical crisis surrounding the Imaginary Cubic Oscillator (ICO), defined by the Hamiltonian $H_{ICO} = -d^2/dx^2 + ix^3$.

• The Issue: While the ICO model is $PT$-symmetric, recent work by Siegl and Krejčiřík established that its eigenstates do not form a Riesz basis. This implies the Hamiltonian is non-diagonalizable due to an Intrinsic Exceptional Point (IEP) at infinite excitation.
• Consequence: Because of this non-diagonalizability and the operator's unboundedness, the ICO cannot be considered a valid quantum mechanical Hamiltonian in the standard sense.
• The Challenge: Direct perturbative regularization of the continuous differential operator $H_{ICO}$ is impossible because the "corridor of unitarity" (the region where the spectrum is real) becomes exponentially narrow as the system size approaches infinity ($N \to \infty$). The paper seeks a way to simulate and understand the IEP singularity using a tractable, finite-dimensional model.

2. Methodology

The author proposes a discretization-based simulation strategy to mimic the IEP behavior using finite $N \times N$ matrices.

• Discretization: The continuous coordinate line is replaced by a discrete grid of $N$ points. The kinetic energy term is represented by a discrete Laplacian matrix $T^{(N)}$ (tridiagonal with 2 on the diagonal and -1 on the off-diagonals).
• Toy Model Hamiltonian: A simplified, two-parametric non-Hermitian matrix Hamiltonian is constructed:
  $$H^{(N)}(A, B) = T^{(N)} + V^{(N)}(A, B)$$
  The potential $V^{(N)}$ is purely imaginary and $PT$-symmetric, with parameters $A$ and $B$ placed at the boundaries of the matrix (and potentially $C, D, \dots$ in more complex versions, though this paper restricts to two parameters).
• The Simulation Logic:
  1. The continuous IEP singularity (occurring at $N \to \infty$) is mimicked by Kato's Exceptional Points (EPs) in finite $N$ matrices.
  2. Specifically, the paper investigates Exceptional Points of order $M$ (EPM), where $M$ eigenvalues and eigenvectors coalesce.
  3. The goal is to find the parameter values $(A, B)$ where the central eigenvalues of the spectrum merge (degeneracy), effectively simulating the "parallelization" of eigenvectors seen in the ICO.
• Analytical Approach:
  • The problem is reduced to finding the roots of secular polynomials $P^{(N)}(E)$.
  • The author derives closed-form expressions for the coefficients of these polynomials for both even ($N=2K$) and odd ($N=2K+1$) dimensions.
  • The condition for degeneracy (EPM) is reduced to solving a system of coupled algebraic equations derived from the polynomial coefficients.

3. Key Contributions

• Exact Solvability of the Toy Model: The paper demonstrates that despite the complexity of non-Hermitian matrices, the specific two-parametric model $H^{(N)}(A, B)$ is exactly solvable. The conditions for EPMs can be reduced to finding roots of auxiliary polynomials of fixed low degree (degree 4), regardless of how large $N$ becomes.
• Derivation of Auxiliary Polynomials:
  • For even $N=2K$, the condition for a quadruple degeneracy ($M=4$) is reduced to finding roots of polynomials $Z^{(\pm 2K)}(x)$ of degree 4.
  • For odd $N=2K+1$, the condition for a quintuple degeneracy ($M=5$) is reduced to finding roots of polynomials $Z^{(2K+1)}(y)$ (where $y=B^2$) of degree 4.
• Asymptotic Analysis: The author performs an asymptotic expansion for large $N$ (large $K$). By introducing a small parameter $g = 1/(K-1)$, the roots of the auxiliary polynomials are expanded in a series, allowing for the precise determination of the EP parameters as $N \to \infty$.

4. Results

• Convergence of EP Parameters: The study confirms that as $N$ increases, the parameters $(A, B)$ defining the Exceptional Points converge to specific limits.
  • For even $N$, the critical parameter $A$ converges to $\pm \sqrt{2}$ as $N \to \infty$.
  • The numerical data (Tables 1–4) shows rapid convergence of the roots of the auxiliary polynomials to these limits.
• Validation of the Simulation: The finite-$N$ EPMs successfully mimic the asymptotic behavior of the ICO. The "parallelization" of eigenvectors at the EPM in the finite model corresponds to the loss of the Riesz basis in the continuous IEP limit.
• Feasibility of Regularization: The paper demonstrates that while the continuous IEP is unphysical, the finite-$N$ models possess a well-defined "physical domain" $D$ (where the spectrum is real). The boundary of this domain is the EPM. Crucially, unlike the continuous case, the singularity at the EPM in the finite model can be regularized via perturbation theory, restoring unitarity in the vicinity of the degeneracy.

5. Significance

• Conceptual Clarification: The paper provides a "partial remedy" to the ICO puzzle. It suggests that the unphysical nature of the IEP is an artifact of the infinite-dimensional limit. By viewing the system as the limit of a sequence of regularizable finite matrices, one can understand the IEP as an asymptotic accumulation of Kato's EPs.
• Bridging Theory and Computation: It offers a constructive framework for studying non-Hermitian degeneracies that are otherwise intractable in continuous differential operators. The use of exactly solvable matrix models allows for rigorous testing of perturbation theories that fail in the continuous limit.
• Physical Interpretation: The work reinforces the idea that the "non-Rieszian mode behavior" of the ICO is a manifestation of the system approaching an exceptional point of infinite order. The finite-$N$ simulation proves that such singularities are not inherent to the physics of the potential itself but are a result of the specific unbounded, infinite-dimensional limit.

In summary, Znojil successfully constructs an exactly solvable finite-dimensional matrix model that replicates the asymptotic degeneracy of the imaginary cubic oscillator. This model proves that the IEP singularity can be understood as the limit of regularizable Kato exceptional points, offering a pathway to analyze and potentially regularize these otherwise unphysical quantum systems.