Long-range resonances in quasiperiodic many-body localization

This study reveals that quasiperiodic many-body localized systems harbor a broad unconventional regime at strong potentials, characterized by atypical eigenstates with long-range correlations and fat-tailed distributions that challenge the expectation of a sharp ergodic-MBL transition.

The Gist

The Big Picture: The "Frozen" Party

Imagine a crowded dance floor (a quantum system). Usually, if you start a dance, everyone eventually mixes, swaps partners, and forgets who they started with. This is called thermalization or being "ergodic." The energy spreads out, and the system reaches a comfortable, random equilibrium.

However, in a special type of party called Many-Body Localization (MBL), something weird happens. The dancers get stuck. They remember exactly where they started and who they were dancing with, even after an infinite amount of time. The party is "frozen."

For a long time, physicists believed this "frozen" state was very stable, especially in systems with quasiperiodic disorder (a specific, non-random pattern of obstacles, like a perfectly repeating but non-repeating wallpaper). They thought that because the obstacles were perfectly arranged, there were no "weak spots" (called Griffiths regions) where the party could accidentally start dancing again. They thought the transition from "frozen" to "dancing" would be a sharp, clean line.

This paper says: "Not so fast."

The researchers found that even in these perfectly ordered, frozen systems, there are hidden "ghosts" in the machine. These are rare, strange states that allow the system to "leak" energy over long distances, threatening the stability of the frozen state.

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The Analogy: The Perfectly Arranged Bookshelf

To understand the discovery, let's use an analogy of a giant bookshelf (the quantum system) filled with books (particles/spins).

1. The Setup: The books are arranged in a very specific, mathematical pattern (the quasiperiodic potential). It's not random; it's like a Fibonacci sequence of book sizes.
2. The Expectation: If you push a book on the left, the force shouldn't travel to the right. The shelf is "localized." The books are stuck in their spots.
3. The Standard Check: Physicists usually check if the shelf is stable by looking at the average behavior. "On average, do the books move?" The answer was usually "No." The shelf looked perfectly frozen.
4. The New Discovery: The authors decided to look at the extremes. They asked: "Is there any pair of books, no matter how far apart, that suddenly start vibrating together?"

They found that yes, there are.

The "Cat" in the Room: The Resonant Cat States

The paper identifies these rare, vibrating pairs as "Resonant Cat States."

• What is a Cat State? In quantum physics, a "Schrödinger's Cat" is a state where a cat is both alive and dead at the same time. A "Cat State" here means two completely different arrangements of the whole bookshelf are happening simultaneously.
• The Resonance: Imagine two books on opposite ends of the shelf (one on the far left, one on the far right). Normally, they don't know about each other. But in these rare "Cat States," the universe finds a way to make them vibrate in perfect sync, as if they are holding hands across the entire room.
• The "Fat Tail": When the researchers plotted the data, they saw a "fat tail." Imagine a graph where most data points are clustered in the middle (normal behavior). A "fat tail" means there are a few extreme outliers that are way off the chart. These outliers are the "Cat States." They are rare, but when they happen, they are huge.

Why This Matters: The "Achilles' Heel"

The authors found that these "Cat States" appear in a region where everyone thought the system was perfectly safe and frozen.

• The Old View: "The transition from frozen to dancing is a sharp cliff. Once you are on the frozen side, you are safe."
• The New View: "The frozen side isn't a solid cliff; it's more like a cliff with hidden cracks."

Even though the system looks frozen on average, these rare "Cat States" act like long-range bridges. They allow information to travel from one side of the system to the other. If you have enough of these bridges, the whole "frozen" state could eventually collapse into a "dancing" state.

The "Creep" Metaphor

Think of the frozen state as a block of ice.

• Standard diagnostics tell you the ice is solid.
• This paper says, "Look closely, and you'll see tiny, invisible cracks forming deep inside the ice."
• These cracks are the long-range resonances. They don't melt the ice immediately, but they suggest the ice is less stable than we thought. Over time, or as the system gets bigger, these cracks could grow, causing the ice to shatter (the system to thermalize).

The "Detective Work"

How did they find this?

1. They stopped looking at the average. Instead of asking "What is the average distance a book moves?", they asked, "What is the longest distance any two books ever vibrate together?"
2. They looked at the "tails" of the data. They found that while 99% of the time, the books stay put, 1% of the time, two books on opposite ends of the shelf vibrate in unison.
3. They proved it's not random. In random systems, these cracks are caused by "weak spots" in the disorder. In this system, the disorder is perfectly ordered (quasiperiodic). The fact that these cracks exist without random weak spots suggests this is a universal rule of quantum physics, not just a fluke of messy randomness.

The Takeaway for the Future

This discovery is a big deal for two reasons:

1. Theory: It challenges the idea that quasiperiodic systems are "safer" or more stable than random ones. It suggests that the "frozen" phase might be unstable in a way we didn't expect, even without random chaos.
2. Experiment: The authors suggest that scientists can actually see this in the lab using ultracold atoms (atoms cooled to near absolute zero). By measuring how density correlations behave (how the "books" shake together), experimentalists can spot these "Cat States" and test if the ice is really as solid as we thought.

In short: The universe is full of hidden connections. Even in a system designed to be frozen and isolated, there are rare, ghostly bridges connecting the farthest corners, waiting to shake the whole thing apart.

Technical Summary

1. Problem Statement

The paper addresses the stability of the Many-Body Localization (MBL) phase in quasiperiodic (QP) systems, specifically the Heisenberg spin chain with an Aubry-André potential.

• Context: While MBL is well-studied in random disorder systems, its stability in the thermodynamic limit remains debated. In random systems, instabilities arise from "Griffiths regions" (rare, weak-disorder inclusions) and "thermal avalanches."
• The Gap: Quasiperiodic systems lack random disorder and thus lack Griffiths regions. It was previously hypothesized that this absence would make the MBL phase more robust and the transition to the ergodic phase sharper. However, recent studies suggest instabilities might still exist.
• Specific Question: Do rare, system-spanning resonances exist in deterministic QP systems that could destabilize the MBL phase, even when standard diagnostics suggest the system is localized?

2. Methodology

The authors investigated the spin-1/2 XXZ chain in a deterministic Aubry-André potential using exact diagonalization techniques.

• Hamiltonian:
  $$H = \sum_{i=1}^L \left( S_i^x S_{i+1}^x + S_i^y S_{i+1}^y + \Delta S_i^z S_{i+1}^z + h_i S_i^z \right)$$
  where $\Delta = 1$ (Heisenberg case) and the field is $h_i = h \cos(2\pi\beta i + \phi)$ with $\beta = (\sqrt{5}-1)/2$ (inverse golden ratio).
• Numerical Approach:
  • System Sizes: $L = 10$ to $22$ spins.
  • Technique: Shift-invert exact diagonalization to access mid-spectrum eigenstates ($\epsilon = 0.5$).
  • Samples: Thousands of disorder samples (varying phase $\phi$) and eigenstates per sample.
• Observables:
  1. Standard Diagnostics: Gap ratio ($r$), Participation Entropy (PE), Entanglement Entropy (EE), and Extreme Magnetization (EM). These are used to locate the standard ergodic-MBL transition.
  2. Novel Probe: Long-distance pairwise spin correlations, specifically the longitudinal correlation at half-chain distance:
     $$C_{L/2}^{zz} = \langle S_i^z S_{i+L/2}^z \rangle - \langle S_i^z \rangle \langle S_{i+L/2}^z \rangle$$
     The authors analyzed the distribution of the rescaled variable $\ell_z = -\ln |4 C_{L/2}^{zz}|$.

3. Key Contributions

• Beyond Standard Diagnostics: The paper demonstrates that standard observables (like EE and gap ratio) can fail to detect subtle instabilities in QP systems. While these indicators suggest a stable MBL phase at intermediate disorder strengths ($h \approx 3.6$), long-range correlations reveal a different reality.
• Discovery of Fat-Tailed Distributions: The authors identified a broad, unconventional regime characterized by fat-tailed distributions of long-distance correlations. This indicates the presence of rare eigenstates with anomalously strong long-range correlations, even in a regime where the system appears localized by conventional metrics.
• Identification of "Cat States": The anomalous eigenstates are identified as quasi-degenerate pairs of resonant cat states. These are coherent superpositions of macroscopically distinct spin configurations (e.g., $|\uparrow \downarrow \dots \rangle \pm |\downarrow \uparrow \dots \rangle$) involving distant sites.
• Universal Instability Mechanism: The work suggests that the instability of the MBL phase is not unique to random disorder (Griffiths regions) but is a universal feature driven by rare many-body resonances, even in deterministic potentials.

4. Key Results

• Phase Diagram:
  • Ergodic ($h < 3$): Standard diagnostics show volume-law entanglement and GOE statistics. Long-range correlations decay as a power law.
  • Fat-Tail Regime ($3 \lesssim h \lesssim 5$): Standard diagnostics (EE, gap ratio) indicate MBL (area law, Poisson statistics). However, the distribution of long-range correlations $P(\ell_z)$ develops fat tails that do not decay with system size. The typical longitudinal correlation length $\xi_z^{typ}$ grows with $L$, signaling an incipient instability.
  • Fully Localized ($h \gtrsim 5$): Both standard diagnostics and correlation distributions show exponential decay, consistent with a stable MBL phase.
• Nature of Anomalous States:
  • At $h=3.6$, the authors found nearly degenerate eigenstate pairs with energy splitting $\Delta E \sim 10^{-5}$ (comparable to level spacing).
  • These states exhibit strong $O(1)$ correlations between distant sites (e.g., sites 5 and 16 in a chain of 22) while local magnetization profiles remain similar.
  • These states fit an ansatz of resonant cat states: $|\Phi_{\pm}\rangle \sim (|\uparrow_i \downarrow_j \downarrow_k \uparrow_l\rangle \pm |\downarrow_i \uparrow_j \uparrow_k \downarrow_l\rangle) \otimes |\psi_{rest}\rangle$.
• Non-Interacting Limit: In the non-interacting limit ($\Delta = 0$), the fat tails disappear, confirming that the long-range resonances are an emergent phenomenon driven by interactions.

5. Significance

• Theoretical Impact: The findings challenge the view that the MBL transition in QP systems is "sharper" or more robust than in random systems. Instead, they suggest that rare long-range resonances destabilize the MBL phase in a similar manner to avalanches in random systems, implying a universal mechanism for MBL breakdown.
• Experimental Relevance: The paper proposes that density-correlation measurements in ultracold atomic systems (which can directly measure $C_{L/2}^{zz}$) are a powerful probe for detecting these rare resonances. This offers a concrete experimental signature to test the stability of MBL in deterministic potentials.
• Methodological Advance: It highlights the necessity of analyzing the full distribution of observables (specifically long-range correlations) rather than just their averages, to capture the physics of rare events in finite-size systems.

In summary, the paper uncovers a hidden "fat-tail" regime in quasiperiodic MBL systems, driven by resonant cat states, which suggests that the MBL phase is generically unstable against rare long-range resonances even in the absence of random disorder.