Fragmented eigenstate thermalization versus robust integrability in long-range models

This paper demonstrates that in fully connected long-range quantum systems, integrability exhibits a dichotomy of robustness or extreme fragility depending on perturbation type, where only extensive two-body perturbations trigger chaos at infinitesimal strength, leading to a fragmented realization of the eigenstate thermalization hypothesis within symmetry-defined energy bands.

The Gist

Imagine a giant, noisy dance floor where thousands of dancers (quantum particles) are holding hands and moving together. In physics, we want to know: Will this group eventually settle into a predictable, average rhythm (thermalization), or will they get stuck in a weird, repeating pattern (integrability)?

Usually, if you have a small group of dancers who only hold hands with their immediate neighbors (short-range), it's very easy to break their pattern. Just one person tripping or changing the music slightly can make the whole group go chaotic and random.

But what happens if everyone is holding hands with everyone else? This is a "long-range" system, like a massive group hug where every person feels every other person. Scientists have been puzzled by this: Does the chaos spread easily here, or is the group super-stable?

This paper by Soumya Kanti Pal and Lea F. Santos answers that question with a surprising twist: It depends entirely on how you mess with the dance floor.

Here is the breakdown using simple analogies:

1. The Setup: The "Perfectly Connected" Dance Floor

The researchers studied a system where every particle interacts with every other particle equally (like a fully connected network). In this perfect state, the system is "integrable."

• The Analogy: Imagine a choir where every singer is perfectly synchronized. They aren't just singing random notes; they are singing in massive, distinct "blocks" or "bands." Because of their perfect symmetry, they are stuck in these specific patterns and won't naturally turn into a chaotic mess.

2. The Experiment: Three Ways to Break the Silence

The researchers asked: "What happens if we poke this perfect system?" They tried three different types of "pokes" (perturbations):

• Type A: The "Spotlight" (Non-extensive)

  • The Poke: You change the music for just one person, or maybe two people in the middle of the crowd.
  • The Result: Nothing happens. The rest of the choir keeps singing in their perfect blocks. The system remains stable and predictable.
  • Why? The "perfect symmetry" of the group is so strong that ignoring two people out of thousands doesn't break the pattern.

• Type B: The "Whispering Wind" (Extensive One-Body)

  • The Poke: You blow a gentle, slightly different wind on everyone. Maybe the wind is stronger on the left side and weaker on the right, but it touches everyone.
  • The Result: Still nothing happens. Even though everyone feels the wind, the system finds a new way to stay organized. It's like the choir adjusting their pitch slightly to match the wind but still singing in perfect harmony. They remain "integrable."

• Type C: The "Ripple Effect" (Extensive Two-Body)

  • The Poke: You make everyone interact with their immediate neighbor in a new way. You don't just change the music for individuals; you change the rule of how neighbors hold hands.
  • The Result: Total Chaos. Even if you make this change infinitesimally small (like a whisper), the entire system instantly shatters. The perfect blocks break, the dancers start moving randomly, and the system becomes "chaotic."
  • The Twist: This is the opposite of short-range systems, where you usually need a big shove to cause chaos. Here, a tiny nudge to the connections causes a meltdown.

3. The Big Discovery: "Fragmented" Thermalization

When the system goes chaotic (Type C), it doesn't just become one big mess. It breaks into fragments.

• The Analogy: Imagine the dance floor shattering into several separate islands. On each island, the dancers are now chaotic and random. But the islands themselves don't mix.
• The Science: This is called Fragmented Eigenstate Thermalization.
  • In the old view, scientists thought long-range systems might never thermalize (never reach a random state).
  • This paper shows they do thermalize, but only within their own little islands (energy bands).
  • If you look at the whole crowd, it looks weird. But if you zoom in on one specific "island" of energy, the dancers are behaving exactly like a chaotic, thermalized system.

4. Why Does This Happen? (The Secret Sauce)

The reason for this behavior is Symmetry.

• The perfect dance floor has a "super-power" called Permutation Symmetry. It doesn't matter if you swap Dancer A with Dancer B; the dance looks the same. This creates a massive number of "conserved quantities" (rules that never change), keeping the system stable.
• Type A and B pokes respect enough of these rules that the system stays safe.
• Type C pokes break the specific rules that hold the "islands" together, causing the chaos to explode immediately.

Summary for the Everyday Reader

Think of a long-range quantum system as a super-stable fortress.

• If you throw a pebble at the wall (Type A) or blow a breeze at it (Type B), the fortress stands firm. It's incredibly robust.
• But if you slightly loosen the mortar between the bricks (Type C), the whole fortress crumbles into chaotic rubble instantly.
• However, once it crumbles, the rubble doesn't mix into a single pile. It forms separate, chaotic piles. Inside each pile, the chaos is total and predictable (thermalized), but the piles don't talk to each other.

Why does this matter?
This helps scientists understand how to control quantum computers and simulate complex materials. It tells us that in systems where everything talks to everything (like trapped ions or Rydberg atoms), we can't just assume "a little noise causes chaos." We have to be very careful about what kind of noise we introduce. Some noise is harmless; a tiny bit of the wrong kind of noise destroys order completely.

Technical Summary

1. Problem Statement

The stability of integrability in many-body quantum systems is a fundamental issue for understanding thermalization and chaos. While the breakdown of integrability in short-range interacting systems is well-established (where even weak perturbations typically induce chaos), the behavior of long-range interacting systems remains unclear.

• The Gap: Long-range interactions (ubiquitous in trapped ions and Rydberg atoms) violate standard statistical mechanics assumptions (additivity/extensivity). Previous studies offered conflicting results: some suggested that arbitrarily small perturbations induce chaos, while others required finite thresholds.
• The Core Question: How does integrability break down in fully connected (all-to-all) long-range models? Does the Eigenstate Thermalization Hypothesis (ETH) hold in these systems, and if so, in what form?

2. Methodology

The authors investigate a prototypical 1D spin-1/2 model with power-law interactions (realizable in trapped-ion platforms) in the fully connected limit ($\alpha = 0$).

• Model: The Hamiltonian is $\hat{H} = \frac{N_\alpha}{L} \sum_{i>j} \frac{J \hat{\sigma}^x_i \hat{\sigma}^x_j}{|i-j|^\alpha} + h \sum_i \hat{\sigma}^z_i$. In the limit $\alpha \to 0$, this maps to the Lipkin-Meshkov-Glick (LMG) model, which possesses $SU(2)$ symmetry and a highly degenerate, banded energy spectrum.
• Perturbation Classification: The authors systematically classify perturbations into three experimentally relevant categories to test stability:
  1. Non-extensive: Local terms (1-body or 2-body) with support independent of system size $L$ (e.g., a single impurity).
  2. Extensive One-body: Uniform or disordered fields acting on all sites (e.g., $\sum h_i \hat{\sigma}^z_i$).
  3. Extensive Two-body: Local 2-body terms acting on all sites or infinitesimal changes to the interaction exponent $\alpha$ (e.g., $\sum \hat{\sigma}^x_i \hat{\sigma}^x_{i+1}$).
• Analytical & Numerical Tools:
  • Degenerate Perturbation Theory: Applied to the $\alpha=0$ limit to analyze how perturbations lift degeneracies within energy bands.
  • Symmetry Analysis: Utilizing Schur-Weyl duality and permutation symmetry groups ($S_L$) to explain the origin of the banded spectrum and conserved quantities.
  • Chaos Diagnostics: Level spacing ratios ($r$), spectral form factors, entanglement entropy, and ETH checks (diagonal and off-diagonal matrix elements).

3. Key Contributions

The paper establishes a general symmetry-based framework that dictates the stability of integrability in fully connected systems. The central finding is a sharp dichotomy: integrability is either robust or extremely fragile, depending strictly on the structure of the perturbation.

A. The Symmetry Mechanism

The integrability of the $\alpha=0$ model arises from invariance under pairwise permutations of lattice sites ($P_{ij}$). This symmetry generates an extensive set of conserved quantities, resulting in a fragmented spectrum where the Hilbert space decomposes into highly degenerate energy bands.

• Robustness: Perturbations that preserve a subset of these permutation symmetries maintain the extensive conserved quantities, keeping the system integrable.
• Fragility: Perturbations that break these symmetries extensively lift the degeneracies within the bands, inducing chaos even at infinitesimal strength.

B. Classification of Perturbation Outcomes

1. Class (i) Non-extensive & Class (ii) Extensive One-body:

   • These perturbations preserve an extensive number of conserved charges (either by acting on a vanishing fraction of sites or by maintaining block-diagonal structures via residual permutation symmetries).
   • Result: Integrability is robust. The system remains non-chaotic (Poissonian level statistics) even at finite perturbation strength.
   • Note: Disordered transverse fields in Class (ii) map to Richardson-Gaudin integrable models.

2. Class (iii) Extensive Two-body:

   • These perturbations (including nearest-neighbor interactions or infinitesimal $\alpha > 0$) break the pairwise permutation symmetry required for the degenerate bands.
   • Result: Integrability is extremely fragile. Chaos is triggered at infinitesimal strength. The perturbation lifts degeneracies at first order, leading to Wigner-Dyson statistics immediately.

4. Key Results

Fragmented Eigenstate Thermalization (ETH)

Contrary to previous claims that ETH fails in long-range systems, the authors demonstrate that ETH holds, but in a "fragmented" manner.

• The Issue: Analyzing the full spectrum (mixing different symmetry sectors) obscures thermalization and suggests ETH breakdown.
• The Resolution: When the analysis is restricted to individual energy bands (defined by approximate conserved quantities like total spin $S^2$), the system behaves chaotically.
  • Diagonal ETH: Eigenstate expectation values of observables vary smoothly with energy within a band.
  • Off-diagonal ETH: Off-diagonal matrix elements follow a Gaussian distribution.
  • Entanglement: Eigenstates within the band exhibit volume-law entanglement entropy consistent with thermal states.
• Conclusion: Microcanonical shells can be constructed within these symmetry-defined sectors. The "breakdown" of ETH observed in previous works was an artifact of mixing distinct symmetry sectors.

Perturbative Mechanism

Using degenerate perturbation theory, the authors show that for Class (iii) perturbations, the projected perturbation matrix within a degenerate band has distinct eigenvalues with level repulsion (Wigner-Dyson) even at first order. This confirms that chaos emerges within the bands before different bands hybridize.

5. Significance

• Theoretical Framework: Provides a predictive criterion for integrability stability based on symmetry reduction (permutation groups) rather than just interaction range. It explains why some long-range systems thermalize while others do not.
• Resolution of Contradictions: Clarifies conflicting literature regarding ETH in long-range systems by introducing the concept of Fragmented ETH. It reconciles the presence of chaos (Wigner-Dyson statistics) with the existence of conserved quantities.
• Experimental Relevance: The findings are directly applicable to current experiments with trapped ions and Rydberg atoms, where long-range couplings are tunable. It suggests that observing thermalization requires careful isolation of symmetry sectors or the application of specific types of perturbations (Class iii).
• Conceptual Shift: Challenges the notion that "any" perturbation breaks integrability in long-range systems, showing instead that integrability can be remarkably robust against specific classes of noise and disorder.

In summary, the paper demonstrates that in fully connected quantum systems, the fate of integrability is governed by the interplay between permutation symmetries and the spatial structure of perturbations, leading to a novel regime of "fragmented" thermalization where chaos and integrability coexist in different sectors of the Hilbert space.