Why finite-size non-Hermitian eigenvalues can miss asymptotic eigenstates in topological physics

summary

The gist

The gist The eigenvalue spectrum can miss most of the eigenstate spectra in non-Hermitian systems, which establishes a limitation for eigenvalue-based diagnostics and suggests that eigenstate

In short

Non-Hermitian systems show a spectral dichotomy where eigenvalues and eigenstates differ in the thermodynamic limit. The Hatano-Nelson model proves that finite-size eigenvalue calculations miss many asymptotic eigenstates. This suggests that focusing on eigenstate spectra provides a more stable and complete characterization of non-Hermitian topology than relying solely on eigenvalues.

Key concepts

Eigenvalue Spectrum: This is the set of energy values obtained from solving the characteristic equation of the Hamiltonian. In non-Hermitian systems, this spectrum can be incomplete in the thermodynamic limit, failing to capture all possible physical states that exist.
Eigenstate Spectrum: This refers to the set of actual quantum states ($| ext{ψ} angle$) that satisfy $H| ext{ψ} angle = 0$. This spectrum is generally more stable and well-defined than the eigenvalue spectrum, as it directly relates to the physical states of the system.
Hatano-Nelson Model: This is a minimal model demonstrating that eigenvalues and eigenstates differ. It acts as a mathematical archetype showing how an order-$L$ Jordan block exceptional point can have only one eigenvalue (zero) but support many more eigenstates in the limit.
Skin Effect: A phenomenon in non-Hermitian systems where boundary states become macroscopic. The paper links this effect to the proliferation of hidden modes, explaining why bulk-edge correspondence fails when using only eigenvalues.

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This episode discusses

The paper

Why finite-size non-Hermitian eigenvalues can miss asymptotic eigenstates in topological physics · Read on arXiv

Department of Physics, KTH Royal Institute of Technology, Stockholm · Univ. Grenoble Alpes, CNRS, LPMMC

Non-Hermitian systems exhibit a fundamental spectral dichotomy absent in Hermitian physics: the eigenvalue spectrum and the eigenstate spectrum can deviate significantly in the thermodynamic limit. We explain how extrapolating finite-size eigenvalues can miss important asymptotic eigenstates, with the unidirectional Hatano-Nelson model serving as a minimal realization of this phenomenon. Through exact analytical solutions, we show that this model contains not only hidden asymptotic modes but also quasi-Jordan structures that become macroscopic in the thermodynamic limit and appear more generally in systems with nontrivial bulk winding. Our framework explains how the apparent failure of the bulk-edge correspondence in models such as the non-Hermitian SSH chain can reflect the failure of finite-size eigenvalue extrapolation to detect asymptotic edge states in systems with a skin effect. These results clarify the limitations of using finite-size eigenvalues alone to infer thermodynamic-limit topology.

DOI: 10.1103/2vfv-47r1

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: Today's paper: "Why finite-size non-Hermitian eigenvalues can miss asymptotic eigenstates in topological physics".

Mira: The gist The eigenvalue spectrum can miss most of the eigenstate spectra in non-Hermitian systems,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, to recap, this paper argues that there’s a fundamental spectral dichotomy in non-Hermitian systems where the eigenvalue spectrum and eigenstate spectrum diverge in the thermodynamic limit. They use the unidirectional Hatano-Nelson model as a minimal case to show how eigenvalues can completely miss eigenstates.

Mira: The core claim is that this divergence isn't just an artifact of finite size; it’s a feature of non-Hermitian Hamiltonians with non-trivial bulk winding, forcing the existence of macroscopic hidden exceptional points.

Lev: They are essentially showing that when you have a non-trivial bulk winding number, the system must host these kinds of states that aren't captured by just tracking the eigenvalues as they approach zero in the limit.

Kai: And they link this to how bulk-edge correspondence can break down in things like the non-Hermitian SSH chain, showing that it reflects the fact that eigenvalues alone cannot detect those edge states.

Mira: Furthermore, they show an exact analytical solution for this model which reveals not only these hidden modes but also quasi-Jordan structures that become macroscopic in the thermodynamic limit.

Lev: That means these structures aren't just tiny numerical glitches; they are real features of the physics when you scale up the system size to infinity.

Kai: It’s about moving our focus from what the eigenvalues tell us to what the actual eigenstates tell us about non-Hermitian topology, and that’s where this work gets interesting for us.

Mira: The paper also connects these eigenstate properties to other spectral measures, like singular values, which are generally more stable under perturbations and underpin conventional topological classifications.

Lev: So if we're building error correction codes for these systems, focusing on the state spectrum might give us a more reliable way to define stability than relying only on the eigenvalue distance.

Kai: That’s right, so this work provides a deeper way to characterize non-Hermitian topology by looking at the states themselves.

Conclusion: Kai: So, thinking about the title again, "Why finite-size non-Hermitian eigenvalues can miss asymptotic eigenstates," it really captures that central problem they are solving here—the limitation of using finite-size data to predict the big picture in these systems.

Mira: It emphasizes that for non-Hermitian systems, you can’t just trust the eigenvalue spectrum to tell you what's happening in the thermodynamic limit because it systematically misses crucial information about the actual eigenstates.

Lev: What this means practically is that if we are trying to model complex non-Hermitian physics, we have to use a method that looks at states, not just eigenvalues, because the eigenvalue distance can be unstable for these systems.

Kai: So the authors are pushing for a shift in how we analyze these systems where state-centric analysis becomes indispensable for understanding non-Hermitian topology and many-body physics.

Mira: They’re suggesting that this is a necessary adjustment because the eigenvalue spectrum alone isn't sufficient to capture the full picture of what’s happening in these physical systems when you scale up.

Lev: For us building things, it means any method we develop for characterizing non-Hermitian topology needs to account for these hidden macroscopic features that eigenvalues just aren't showing us.

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