🔬 condensed matter

Anomalous suppression of quantum chaos between two integrable limits

This paper demonstrates that the interacting Su-Schrieffer-Heeger model exhibits an anomalous suppression of quantum chaos, characterized by a reduced mean level-spacing ratio deep within the non-integrable regime, due to the incomplete hybridization of many-body band states inherited from the non-interacting structure.

Original authors: Roopayan Ghosh, Henry Davenport, Frank Schindler

Published 2026-07-08

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CC BY 4.0

Original authors: Roopayan Ghosh, Henry Davenport, Frank Schindler

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

Imagine a crowded dance floor where everyone is trying to move around. In most chaotic, energetic systems (like a mosh pit), everyone eventually mixes with everyone else. In the world of quantum physics, this "mixing" is called ergodicity, and when it happens perfectly, the system is considered "chaotic" in a very specific mathematical way. Scientists usually expect that if you take a system that isn't perfectly ordered (non-integrable) and add some interactions, it will immediately become this chaotic, well-mixed dance floor.

This paper, however, discovers a strange exception to that rule. The researchers found a specific scenario where the dance floor should be chaotic, but instead, the dancers stay in their own little groups, refusing to mix fully.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: The Two-Lane Highway

The scientists studied a specific model called the SSH model (Su-Schrieffer-Heeger). Think of this as a one-dimensional highway with two lanes (let's call them Lane A and Lane B).

The Non-Interacting Case: If cars (electrons) don't talk to each other, they just drive in their lanes. The highway has a very clear structure: sometimes the lanes are wide apart, sometimes they are close.
The "Integrable" Limits: There are two special settings where the system is perfectly predictable (like a train on a track).
- Setting 1: The cars don't interact at all.
- Setting 2: The highway is broken into isolated pairs of spots (dimers), so cars can only move within their own pair.

2. The Surprise: The "Ghost" of the Old Structure

Usually, when you turn on the "interaction" switch (making the cars bump into or influence each other), you expect the system to lose its old structure and become a chaotic mess where every car eventually meets every other car.

The researchers found that even when they turned up the interactions to make the system "non-integrable" (supposedly chaotic), a strange line of order appeared deep inside the chaos.

The Metric: They measured how "close" the energy levels of the system were to each other. In a chaotic system, these levels repel each other (like magnets with the same pole), keeping a healthy distance. In an ordered system, they can clump together.
The Anomaly: They found a specific path where the levels didn't repel each other as much as they should. The "chaos" was suppressed. The system was behaving as if it were still partially ordered, even though it shouldn't be.

3. The Cause: Incomplete Blending (The "Hybridization" Problem)

Why did this happen? The authors explain it using the idea of hybridization (mixing).

Imagine you have two different types of dough: one is blue, one is yellow.

Normal Chaos: If you mix them well, you get a uniform green dough. You can't tell where the blue or yellow started.
The Anomaly: In this specific quantum system, even though the "mixing" (interaction) started, the blue and yellow doughs didn't blend completely. They remained as distinct "remnants" of their original shapes.

The researchers traced this back to the band structure (the layout of the highway lanes).

In the "ordered" limits, the system has distinct "minibands" (groups of energy states).
When interactions are added, these minibands start to overlap.
The Catch: The states coming from the top of one miniband and the bottom of the next look very different from each other. To mix them, the system has to perform a very complex, multi-step "dance move" (a high-order quantum process).
Because this dance move is so difficult and rare, the states fail to mix. They sit right next to each other in energy but remain "strangers." This lack of mixing weakens the "repulsion" between energy levels, creating the anomaly.

4. The Mechanism: A High-Order Detour

The paper uses perturbation theory (a way of calculating small effects) to show why they don't mix.

To connect a state from the "blue" group to a state in the "yellow" group, the system has to go through a series of virtual, temporary states that cost a lot of energy.
It's like trying to walk from one side of a city to the other, but the direct bridge is out. You have to take a detour through three different neighborhoods, paying a toll at each stop.
Because the "tolls" (energy costs) are so high, the traffic (quantum probability) barely flows between the two groups. The groups remain distinct, suppressing the chaos.

5. Robustness: It's Not Just a Fluke

The researchers checked if this was just a trick of the math or a small system size.

They tested different system sizes and found the effect persisted.
They broke the "symmetries" (like flipping the highway or changing the rules of the lanes) and found the anomaly still existed.
This proves the effect isn't protected by a special symmetry; it is a fundamental result of the geometry of the energy bands themselves.

Summary

In short, the paper shows that in certain clean, interacting quantum systems, the "memory" of the original energy bands is so strong that it prevents the system from becoming fully chaotic, even when it shouldn't be. The system gets stuck in a middle ground where different groups of states overlap in energy but refuse to mix, creating a "ghost" of order inside the chaos. This challenges the standard idea that non-integrable systems always become fully chaotic.

Problem Statement
The emergence of ergodicity in isolated quantum many-body systems is a central problem in condensed matter physics. Generic interacting systems are expected to obey the Eigenstate Thermalization Hypothesis (ETH), leading to spectral correlations described by the Gaussian Orthogonal Ensemble (GOE), a hallmark of quantum chaos. Deviations from this behavior, typically signaled by Poisson level statistics, are usually attributed to identifiable mechanisms such as disorder-induced many-body localization (MBL), quantum many-body scars, or Hilbert-space fragmentation due to kinetic constraints. A critical open question is whether non-ergodic spectral signatures can arise in clean, translationally invariant systems lacking these specific ingredients, and if so, what mechanism governs their emergence.

Methodology
The authors investigate the interacting Su-Schrieffer-Heeger (SSH) model, a simple 1D spinless fermion chain with nearest-neighbor density-density interactions. The study employs:

Exact Diagonalization (ED): Numerical computation of the full many-body eigenspectrum for system sizes up to $L=24$ (Hilbert space dimension $D \approx 5.7 \times 10^4$ ) under periodic boundary conditions.
Level Statistics: Analysis of the adjacent level-spacing ratio $\langle r \rangle$ as a primary diagnostic for ergodicity, distinguishing between GOE ( $\langle r \rangle \approx 0.535$ ) and Poisson ( $\langle r \rangle \approx 0.386$ ) statistics.
Eigenstate Probes: Calculation of the Inverse Participation Ratio (IPR) and projection of eigenstates onto many-body (MB) minibands of the non-interacting limit to assess hybridization.
Perturbative Analysis: Semi-analytical arguments using perturbation theory to explain the suppression of hybridization near integrable limits (the non-interacting limit $V_{int}=0$ and the fully dimerized limit $\delta=1$ ).
Symmetry Breaking: Extensions of the model to break chiral and inversion symmetries to test the robustness of the observed phenomena.

Key Results

Anomalous Suppression of Chaos: The interacting SSH model exhibits a pronounced line of suppressed mean level-spacing ratio $\langle r \rangle$ deep within the non-integrable regime. Along this line, $\langle r \rangle$ remains systematically reduced (saturating near $\approx 0.508$ ) compared to the GOE value, despite the system being far from integrable limits.
Origin in Incomplete Hybridization: The suppression is not caused by integrability or localization. Instead, it arises from the incomplete hybridization of many-body states inherited from different MB minibands of the non-interacting band structure.
- In the dimerized limit ( $\delta \to 1$ ), states at the edges of adjacent MB bands possess qualitatively different structures (e.g., differing numbers of doubly occupied dimers). Hybridization between these states requires high-order perturbation theory, leading to exponentially weak effective couplings ( $t_{eff} \sim \epsilon^\ell / V_{int}^{\ell-1}$ ).
- In the non-interacting limit ( $V_{int} \to 0$ ), states at the edges of adjacent MB bands correspond to particle-hole excitations localized at different momenta ( $k=0$ vs. $k=\pi$ ). Hybridizing these states requires high-order scattering processes ( $\sim V_{int}^n$ ) due to momentum conservation constraints, suppressing the coupling.
Spectral Restructuring: As parameters move away from the integrable limits, the MB bands broaden and eventually overlap. However, near-degenerate states originating from different minibands remain weakly hybridized for a significant parameter range. This weak hybridization reduces level repulsion without restoring integrability or inducing global localization.
Robustness: The phenomenon persists in extensions of the model that break chiral and inversion symmetries, indicating the mechanism is driven by single-particle band geometry rather than symmetry protection. The anomaly is observed up to the largest accessible system sizes ( $L=24$ ), though its fate in the thermodynamic limit remains an open question.

Significance and Claims
The paper identifies a distinct mechanism for suppressing quantum chaos in clean systems: the persistence of weakly hybridized states due to the underlying single-particle band structure. This challenges the conventional association between non-integrability and fully chaotic spectral statistics. The authors demonstrate that "intermediate" statistics can emerge not from disorder or dynamical constraints, but from the specific geometric constraints of the band structure that inhibit the efficient mixing of many-body states.

The work suggests that the crossover between band-structured and fully ergodic spectra involves a regime where near-degenerate states suppress level repulsion without producing global localization. The authors posit that these atypical states may leave observable dynamical signatures, such as anomalously slow relaxation or long-lived prethermal behavior. The study raises the broader question of whether similar intermediate-statistics regimes emerge generically in interacting multiband systems and how band-geometry-induced constraints should be incorporated into generalized theories of quantum chaos and ergodicity.