🔬 condensed matter

Thermal avalanches in a quasiperiodic XXZ model

This study reveals that while particle-number entropy effectively detects thermal avalanches in a quasiperiodic XXZ chain, two-point correlations fail to do so, indicating that standard avalanche theory requires revision to account for the unique short-range correlations inherent in quasiperiodic potentials.

Original authors: Paolo Molignini, Antonio Štrkalj

Published 2026-07-27

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CC BY 4.0

Original authors: Paolo Molignini, Antonio Štrkalj

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 world made of tiny, spinning tops, all jostling together in a quantum dance. In most chaotic systems, if you poke one part, the energy spreads out until everything settles into a warm, uniform temperature, like a cup of coffee cooling down to room temperature. This is called "thermalization," and it's the rule for almost everything we see. But there's a weird exception called "Many-Body Localization" (MBL). Think of it as a quantum traffic jam where the spinning tops get stuck in place, refusing to share their energy or forget their initial state, even after a very long time. They stay cold and frozen in their own little pockets.

Now, scientists have been wondering: Is this traffic jam truly unbreakable? What if you placed a "hot zone"—a group of tops that can move and share energy—right next to the frozen ones? Would the heat slowly creep into the frozen zone, melting the jam and turning the whole system into a warm, chaotic soup? This creeping heat is called a "thermal avalanche." It's a bit like a single warm spot in a block of ice that starts to melt, and if the ice is weak enough, that melting spot grows and eventually destroys the whole ice block. Understanding this is crucial because it tells us if quantum computers can stay stable or if they will eventually melt down into chaos.

In this paper, the authors set up a digital experiment to test exactly this scenario. They built a model of a chain of quantum spins, but instead of using random disorder (like a messy pile of rocks), they used a "quasiperiodic" pattern. Imagine a pattern that repeats but never quite lines up perfectly, like a wallpaper design that shifts slightly every time you look at it. They took a chunk of this chain and made it "hot" (ergodic) while keeping the rest "cold" (localized), essentially creating a thermal bath to see if it could trigger an avalanche.

Here is where the story gets surprising. The researchers used two different ways to watch the heat spread. The first method was like looking at a map of how much the spinning tops were "talking" to each other (correlations). When they looked at this map, it told a boring story: the heat barely moved. The map suggested the avalanche died out quickly, and the frozen zone remained safe. The "talking" between the tops grew very slowly, logarithmically, which usually means the system is still stuck in its frozen state. Based on this map alone, one would conclude that the avalanche failed.

However, the second method told a completely different story. The authors looked at the "particle-number entropy," which is a fancy way of measuring how much the particles are actually moving and swapping places across the boundary. This measurement was like a motion sensor that detected the heat spreading much further than the map suggested. It showed that for certain conditions, the "hot" bath was successfully melting the frozen zone, sending an avalanche front deep into the system.

The paper finds a major disagreement between these two views. The "map" (correlations) said the avalanche stopped, but the "motion sensor" (entropy) said it was still marching forward. The authors suggest that in these quasiperiodic systems, the map is actually lying to us. The unique, structured pattern of the potential creates short-range "resonances"—little shortcuts that allow energy to jump further than the simple correlation map can see. Because of these hidden shortcuts, the avalanche can grow even when the correlation length looks too short to support it.

The authors are careful to note that while their simulations clearly show this discrepancy, they haven't proven a new mathematical law yet. They suggest that the standard theories used to predict avalanches need a serious update to account for these specific, structured shortcuts found in quasiperiodic systems. They also point out that if you make the hot bath too small, the avalanche stops, which fits the general idea that you need a big enough "spark" to start a fire. But the key takeaway is that in these specific quantum systems, you can't just look at how things are connected to know if they are melting; you have to watch the particles actually move. This means that for future experiments with cold atoms, scientists might need to use different tools than they thought to tell if a quantum system is truly stable or if it's slowly falling apart.

Technical Summary: Thermal Avalanches in a Quasiperiodic XXZ Model

Problem Statement
The stability of the Many-Body Localized (MBL) phase in the thermodynamic limit is a central open question in non-equilibrium quantum statistical mechanics. While MBL systems evade thermalization via the Eigenstate Thermalization Hypothesis (ETH), their stability is challenged by "quantum avalanches." In randomly disordered systems, rare regions of weak disorder (Griffiths regions) can act as thermal inclusions. If sufficiently large, these inclusions can hybridize with neighboring localized degrees of freedom, growing and eventually destroying localization throughout the system.

While this mechanism is well-studied in random disorder models, its applicability to quasiperiodic systems remains less understood. Quasiperiodic systems, often realized in ultracold atomic platforms, lack the rare Griffiths regions inherent to random disorder but can host MBL. The authors investigate whether thermal avalanches can be induced in a quasiperiodic XXZ spin chain by artificially engineering a thermal inclusion (a "bath") and how such avalanches manifest in different observables. A key motivation is to determine if standard avalanche diagnostics, developed for random systems, remain valid for quasiperiodic potentials.

Methodology
The authors study an interacting spin-1/2 XXZ chain of length $L$ partitioned into a localized subsystem $A$ and a thermal bath $B$ .

Model: The system is governed by an XXZ Hamiltonian with a quasiperiodic magnetic field. The field strength in subsystem $A$ ( $W_A$ ) is varied to tune between ergodic and localized regimes, while the bath $B$ is set to a weak field strength ( $W_B = 0.5$ ) to ensure it acts as an ergodic thermal reservoir. The incommensurate frequency is fixed to the inverse golden ratio.
Numerical Approach: The authors employ Exact Diagonalization (ED) on chains up to $L=18$ sites. Time evolution is calculated exactly from an initial Néel state. To mimic disorder averaging in a deterministic system, they average over an ensemble of realizations generated by randomizing the phase offsets of the quasiperiodic potential.
Observables: Two complementary observables are used to detect thermalization and avalanche propagation:
1. Two-point connected correlation function ( $C_{ij}^{zz}$ ): Specifically, a bath-averaged correlation profile $g^{(2)}(i,t)$ is analyzed to extract a time-dependent correlation length $\xi_d(t)$ . In standard avalanche theory, $\xi_d(t)$ is expected to grow logarithmically ( $\sim \log t$ ) until saturation, or faster if an avalanche proliferates.
2. Particle-number entropy ( $S_N$ ): This observable quantifies fluctuations in conserved quantities (particle number/magnetization) between subsystems. It is sensitive to transport across the boundary and is expected to show a secondary growth phase when an avalanche front reaches a specific partition.

Key Results
The study reveals a striking discrepancy between the two observables regarding the presence of thermal avalanches:

Correlation Function Analysis:
- The bath-averaged correlations exhibit an exponential decay with distance from the bath.
- The extracted correlation length $\xi_d(t)$ grows logarithmically in time for all potential strengths $W_A$ where the isolated system is expected to be localized.
- Crucially, the extracted Local Integrals of Motion (LIOM) localization length $\xi$ remains consistently well below the critical threshold ( $\xi_{crit} \approx 2.9$ ) predicted by standard avalanche theory.
- Based on correlations alone, the data suggests that avalanches terminate and localization is preserved, even for large baths ( $L_B=8$ ).
Particle-Number Entropy Analysis:
- In contrast, $S_N$ displays clear signatures of avalanche proliferation for $W_A \leq 4$ .
- For intermediate potential strengths (e.g., $W_A=3$ ), $S_N$ shows a distinct secondary growth phase after an initial saturation. This growth corresponds to the arrival of the thermalization front at partitions deep within the localized subsystem.
- The propagation of this front depends on bath size and potential strength: larger baths sustain avalanches further into the system, while stronger potentials ( $W_A=5$ ) cause the avalanche to die out before reaching distant partitions.
- This behavior confirms that the bath successfully thermalizes the localized region, contradicting the conclusion drawn from the correlation function.
Discrepancy and Interpretation:
- At $W_A=3$ , $S_N$ indicates a full avalanche, while the correlation analysis yields $\xi \approx 0.4$ , far below the theoretical critical value.
- The authors argue that two-point correlations are insufficient to identify thermal avalanches in quasiperiodic systems, unlike in random systems where they serve as a reliable diagnostic.
- The quantitative deviations from standard avalanche theory (e.g., the mismatch between the observed avalanche and the predicted critical $\xi$ ) are attributed to short-range resonances and structured correlations generated by the quasiperiodic potential. These features likely renormalize the effective coupling between the LIOMs and the bath, allowing avalanches to occur even when the standard exponential decay criteria suggest they should not.

Significance and Claims
The paper claims that the standard avalanche framework, which relies on the assumption of featureless random disorder, requires revision to account for the specific properties of quasiperiodic potentials.

Diagnostic Limitations: The primary finding is that two-point correlations can be misleading in quasiperiodic systems, potentially failing to detect active avalanches that are clearly visible via transport-sensitive observables like particle-number entropy.
Theoretical Implication: The results suggest that short-range resonances in quasiperiodic systems can lower the effective critical localization length required for avalanche proliferation.
Experimental Relevance: The authors note that their findings have direct implications for ultracold atomic experiments, where quasiperiodic potentials are readily engineered. They suggest that experimentalists should complement correlation measurements with transport-sensitive observables to correctly diagnose the stability of the MBL phase against thermal inclusions.

The work does not claim to resolve the thermodynamic stability of MBL definitively but highlights the necessity of refining avalanche theories to include the structured nature of quasiperiodic disorder. Future extensions proposed by the authors include investigating long-range interactions and higher-dimensional systems.

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