🔬 mesoscale physics

Universal aspects of bulk density of states in non-Hermitian lattices

This paper demonstrates that despite apparent boundary sensitivity, non-Hermitian lattice systems possess a universal bulk density of states characterized by identical multipole moments and finite-time dynamics across different boundary conditions, with the Brown measure serving as a canonical representative and point-gap topology providing a systematic criterion for when boundary-dependent eigenvalue accumulation retains physical significance.

Original authors: Mykhailo Pavliuk, Askar Iliasov, Emil J. Bergholtz, Tomáš Bzdušek

Published 2026-08-17

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CC BY 4.0

Original authors: Mykhailo Pavliuk, Askar Iliasov, Emil J. Bergholtz, Tomáš Bzdušek

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Invisible Map of Quantum Chaos

Imagine you are trying to map the landscape of a strange, new world. In the familiar world of standard physics, this landscape is predictable: if you build a giant crystal, the rules inside it are the same whether you look at the middle or the edges. But there is a newer, weirder branch of physics called "non-Hermitian" physics. Here, the rules are different. In this world, energy isn't just a number; it's a complex number with a real part and an imaginary part, like a coordinate on a map with two directions. Because of this, the edges of the system act like a giant magnet, pulling all the particles to the boundaries and leaving the middle empty. This is called the "non-Hermitian skin effect."

For a long time, scientists were confused about how to describe the "bulk" (the middle) of these systems. They had two main ways to calculate the density of states (a fancy way of saying "how many energy levels exist in a specific area"). One way assumed the system was a perfect loop (like a video game world that wraps around), and the other assumed it had hard walls (like a real box). These two methods gave completely different maps. It seemed like the middle of the system didn't have a single, true identity; it depended entirely on how you looked at it. This paper asks a simple but profound question: Is the middle of the system actually ambiguous, or is our confusion just a matter of perspective?

The Great Illusion of the Middle

The authors of this paper, a team of physicists from Switzerland and Sweden, have discovered that the confusion is largely an illusion. They show that while the "maps" drawn by different boundary conditions look different on the surface, they are actually describing the exact same underlying reality when you zoom out to look at the big picture.

To understand this, imagine the energy levels of a system as a cloud of charged dust floating in a room. In the old view, if you put a wall on the left, the dust piles up on the left. If you put a wall on the right, it piles up on the right. It seemed like the dust had no fixed shape. The authors, however, point out that if you stand far enough away from the room (a concept they call the "far-field"), the electric field created by that dust cloud looks exactly the same, no matter how the dust is piled up inside.

In the language of physics, the "Green's function" is like the electric field generated by this dust. The paper proves that for any system with a limited range of interactions (like atoms only talking to their nearest neighbors), the Green's functions calculated from different boundary conditions become identical outside a certain boundary. This means that the "multipole moments"—which are like the overall shape, weight, and balance of the cloud—are identical for every definition. Just as a sphere and a cube of the same mass create the same gravitational pull if you stand far enough away, different boundary conditions create the same bulk physics if you look at the system from the right distance.

The "Brown Measure": The True Map

So, if the maps are different but the physics is the same, which map should we trust? The authors propose a "canonical" map called the Brown measure. Think of this as the "Goldilocks" definition of the system's energy. It is constructed using a clever mathematical trick called "Hermitization," which turns the messy, non-Hermitian problem into a clean, standard one.

The Brown measure is special because it is defined directly from the infinite system itself, without needing to guess how the system was cut or bounded. It turns out that for systems with periodic boundaries (the loop world), the Brown measure matches the standard calculation perfectly. But for systems with open boundaries (the box world), the Brown measure reveals a crucial distinction regarding the "skin effect." While the skin effect often involves a massive pile-up of particles at the edge (an extensive effect), the paper clarifies that this is not always the case. Specifically, only boundary states arising from strong higher-dimensional topological invariants are "sub-extensive." This means that in those specific high-dimensional cases, the particles pile up at the edge but not enough to change the overall "density" of the middle in the infinite limit. They are like a thin layer of paint on a massive wall; the paint is there, but it doesn't change the volume of the wall. In contrast, the usual skin effect in one dimension is extensive and does contribute to the density.

When the Rules Break: The Topology Trap

The paper also explains when and why the different maps might actually disagree. It turns out that the "topology" of the system acts like a set of invisible guardrails. If the system has a "point gap" (a hole in the energy map) that is topologically trivial (boring), then all definitions of the density of states must agree. The Green's functions are forced to be the same, and the bulk is truly universal.

However, if the system has a "non-trivial" point gap (a hole with a twist, like a knot), the rules change. In these specific regions, the boundary conditions can force the eigenvalues (the energy levels) to accumulate in different ways. The authors show that this accumulation is tied to the existence of "zero-energy modes" in a related mathematical structure. But here is the catch: even in these tricky regions, the paper suggests that the accumulation of energy levels is often "sub-extensive." This means that while you might see a line of energy levels appearing in a specific spot on the map, they are so sparse that they don't contribute to the overall density. They are like a few scattered pebbles on a beach; they are visible, but they don't change the fact that the beach is mostly sand.

The Verdict

The paper concludes that the "ambiguity" of the bulk density of states in non-Hermitian systems is not a fundamental feature of nature, but a limitation of how we choose to look at it. The bulk dynamics—the way waves move and evolve in the middle of the system—are insensitive to the boundaries. Whether you calculate the density of states using periodic boundaries, open boundaries, or any other method, the "far-field" information (the multipole moments) remains identical.

The authors argue that the Brown measure is the best representative of this universal bulk physics because it is intrinsic to the infinite system and independent of how we approximate it. While different boundary conditions can produce different-looking spectra, they are all just different projections of the same underlying reality. The "skin effect" is real, but it doesn't rewrite the rules of the bulk; it just adds a thin, often invisible, layer of complexity to the edges. For a curious teenager, this means that even in a chaotic, edge-hugging quantum world, there is still a stable, universal core that doesn't care how you build the box.

Technical Summary: Universal Aspects of Bulk Density of States in Non-Hermitian Lattices

Problem Statement
Non-Hermitian lattice Hamiltonians exhibit a fundamental departure from Hermitian physics: their eigenvalues are complex, and their spectra are highly sensitive to boundary conditions. In the thermodynamic limit, periodic boundary conditions (PBC) and open boundary conditions (OBC) often yield strikingly different spectra, a phenomenon known as the non-Hermitian skin effect. Consequently, it has been widely accepted that non-Hermitian systems lack a unique bulk density of states (DOS), with different definitions (e.g., PBC DOS vs. OBC DOS) representing inequivalent bulk physics. This creates an apparent ambiguity: if the bulk DOS is not unique, how can one define intrinsic bulk dynamics, such as wave packet evolution far from boundaries, which physically should be insensitive to boundary conditions?

Methodology
The authors address this ambiguity by analyzing finite-range tight-binding Hamiltonians through the lens of thermodynamic limits and Green's functions (resolvents). Their methodology relies on three core theoretical tools:

Electrostatic Analogy: They establish a direct mapping between the Green's function $G(z)$ and the spectral density $\rho(z)$ in the complex plane to a two-dimensional electrostatic problem. Here, the spectral density acts as a charge distribution, and the Green's function corresponds to the electric field. This analogy allows the authors to utilize Gauss's law and multipole expansions to relate different limiting distributions.
Strong Resolvent Convergence (SRC): They examine the convergence of approximating finite-size Hamiltonians $\hat{H}_{\Omega N}$ to an infinite-volume operator $\hat{H}$ . They demonstrate that for sufficiently large complex frequencies ( $|z| > M$ , where $M$ is related to the operator norm), the resolvents of any boundary-condition-dependent sequence converge strongly to the resolvent of the infinite operator.
Hermitization and Brown Measure: To define a canonical bulk DOS, the authors employ the "Hermitization" procedure, constructing a doubled Hermitian Hamiltonian $\hat{\mathcal{H}}(z)$ . They utilize the Brown measure, a spectral measure defined for non-normal operators in a tracial von Neumann algebra, which is derived from the resolvent of this Hermitized system.

Key Contributions and Results

Universality of Bulk Dynamics: The paper proves that while different boundary conditions (PBC, OBC, etc.) produce distinct finite-size spectra and potentially distinct limiting spectral densities, they all share identical multipole moments. In the electrostatic analogy, this means all valid bulk DOS definitions generate the same "far-field" configuration of the Green's function.
Equivalence of Green's Functions: The authors establish that for any finite-range Hamiltonian, the Green's functions of different thermodynamic limits coincide for all complex frequencies $z$ outside a bounded region (specifically, where $|z| > \limsup \|\hat{H}_{\Omega N}\|$ ). Since the time evolution operator is related to the resolvent via a contour integral, this implies that bulk dynamics (e.g., the evolution of wave packets far from boundaries) are insensitive to the choice of boundary conditions in the thermodynamic limit.
The Brown Measure as a Canonical Representative: The authors identify the Brown measure, obtained via Hermitization, as the canonical and convenient representative of the bulk DOS. Unlike PBC or OBC DOS, the Brown measure is:
- Intrinsic to the infinite-volume Hamiltonian operator.
- Independent of the specific limiting procedure used to define the infinite system.
- Supported strictly on the spectrum of the infinite operator.
- Operationally accessible via finite flakes with arbitrary boundary conditions (provided a regulator is handled correctly).
Constraints on Spectral Accumulation via Topology: The paper clarifies the role of point-gap topology in determining where eigenvalues can accumulate.
- Trivial Point Gaps: In regions of the complex plane corresponding to topologically trivial point gaps of the Hermitized system, the limiting Green's functions for any boundary condition must coincide with the Brown Green's function. Consequently, the limiting spectral density must vanish in these regions.
- Nontrivial Point Gaps: Differences in limiting spectral densities can only occur within the spectrum of the infinite operator or inside nontrivial point gaps.
- Extensivity vs. Subextensivity: The authors distinguish between extensive spectral features (contributing to the normalized DOS) and subextensive ones. They argue that only point gaps carrying nontrivial weak one-dimensional winding invariants support an extensive number of boundary states (the standard skin effect). Stronger, higher-dimensional point-gap invariants or "pump" topologies may enforce boundary states, but these are typically subextensive (scaling with boundary area rather than volume) and thus carry zero weight in the normalized thermodynamic DOS, even though they appear in the spectrum as a set.
Symmetry Considerations: The framework is extended to systems with inherent non-Hermitian symmetries. The authors show that lifted symmetries act on the spectral parameter $z$ (e.g., mapping $z \to -\bar{z}$ ), refining the symmetry class of the Hermitized double on specific loci (real axis, imaginary axis, origin). This allows for a systematic determination of where spectral collapse occurs based on the refined symmetry classes.

Significance and Claims
The paper claims to resolve the apparent ambiguity regarding the non-uniqueness of the bulk DOS in non-Hermitian systems. It argues that this ambiguity is largely illusory from the perspective of bulk physics. While different limiting procedures yield different spectral distributions, they are physically equivalent in the bulk because they share identical multipole moments and generate identical bulk dynamics at finite times.

The significance of the work lies in:

Restoring a Unique Bulk Concept: It provides a rigorous justification for defining a unique bulk DOS (the Brown measure) that is independent of boundary conditions, aligning with the physical expectation that bulk properties should not depend on boundaries.
Systematic Prediction of Spectral Collapse: It offers a systematic criterion, valid in arbitrary dimensions, to determine where and how eigenvalues of different boundary truncations can accumulate. It delineates the regime where DOS ambiguity retains physical significance (nontrivial point gaps) versus where it does not (trivial point gaps).
Generalization: The approach applies to non-Hermitian lattice Hamiltonians in arbitrary dimensions and extends to systems with inherent symmetries, providing a unified language (Green's functions and resolvent analysis) to discuss phenomena previously treated as distinct, such as the skin effect and spectral collapse.

The authors maintain a modest stance, noting that while Hermitization identifies where boundary states may accumulate, it does not automatically guarantee that these states are extensive (i.e., that they contribute to the normalized DOS). They emphasize that the distinction between the spectrum as a set and the normalized spectral density is crucial, particularly for subextensive features arising from higher-dimensional topological invariants.