⚛️ high-energy theory

Krylov-Space Memory Cores

This paper introduces "Krylov-space memory cores" as stationary, depth-resolved structures within the Krylov space of thermalizing nonintegrable systems that reveal how anomalous initial states retain structured quantum memory through compact regions characterized by residual fluctuations, Gibbs mismatch, and persistent current activity, distinguishing them from generic reference states and integrable dynamics.

Original authors: Mohsen Alishahiha, Mohammad Javad Vasli

Published 2026-07-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohsen Alishahiha, Mohammad Javad Vasli

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 Hidden Library of Quantum Chaos

Imagine a crowded, chaotic dance floor where everyone is spinning, jumping, and colliding. In the world of quantum physics, this dance floor is a system of many interacting particles. Usually, when you start a dance with a specific pattern, the chaos of the crowd eventually washes it away. The dancers forget the original steps, and the room settles into a generic, "thermal" state where everything looks the same on average. This is called thermalization, and it's why your hot coffee eventually cools down to room temperature; the specific energy you put in gets scrambled until it's indistinguishable from the background noise.

But sometimes, the dance floor has a secret. Even in a chaotic room, a few dancers might remember the original steps perfectly, or at least keep a weird, persistent rhythm that refuses to fade. Scientists have been trying to figure out where this memory hides. Is it scattered everywhere? Is it locked in a specific corner? To find out, researchers use a mathematical tool called "Krylov space." Think of this not as a physical room, but as a giant, one-dimensional library. The first shelf holds the starting dance move. The second shelf holds what happens if you apply the rules of the dance once more, the third shelf twice more, and so on. As the system evolves, the "probability" of finding the system in a certain state moves from shelf to shelf, like a wave traveling down the library. The big question is: when the system stops changing on average, does that wave just sit there randomly, or does it form a specific, organized shape that holds onto the memory of the start?

The Paper's Discovery: The "Memory Core"

In this paper, Mohsen Alishahiha and Mohammad Javad Vasli introduce a new way to look at this library. They propose the existence of "Krylov-space memory cores." These are compact, deep pockets of activity where the system's memory of its starting state stays alive, even after the rest of the system has forgotten everything.

The authors didn't just look at how much probability is on each shelf (the "occupation profile"). Instead, they built a three-part detective kit to see what that probability is doing.

The Wiggle Factor: They checked if the probability on a shelf is just sitting still or if it's still vibrating and fluctuating wildly.
The Mismatch Meter: They compared the shelf's contents to what a "normal" thermal system should look like. If it's different, that's a sign of memory.
The Flow Tracker: They measured if probability is still bouncing back and forth between neighboring shelves, like a pendulum that won't stop swinging.

When they applied this kit to three different types of "weird" quantum systems—ones that are supposed to be chaotic but act strangely—they found a recurring pattern. In these systems, the anomalous starting states (the ones that don't forget) developed a compact, low-depth memory core.

Imagine the library again. For a normal, chaotic system, the probability wave spreads out all the way to the back of the building, filling thousands of shelves, but it's weak and quiet everywhere. For the "anomalous" systems, the wave still spreads out to fill a huge "halo" of shelves (the stationary occupation), but right at the front of the library, within the first few dozen shelves, there is a tiny, dense core. Inside this core, the probability is not just sitting there; it is vigorously fluctuating, it looks very different from the thermal average, and it is constantly exchanging energy back and forth.

The paper shows this happens in three very different scenarios:

Weak Thermalization: In a chaotic Ising chain (a model of magnetic spins), a specific starting state called $|X^+\rangle$ kept its memory in a core of just 32 shelves, even though the probability cloud stretched out to 2,053 shelves.
Confinement: In a different setup where particles are trapped by a potential, a "Néel" state (alternating spins) formed a core of 41 shelves, while the cloud reached 228.
Quantum Scars: In the famous PXP model (used to study Rydberg atoms), the "scarred" Néel state kept its memory in a core of 109 shelves, while the probability cloud extended to 9,818 shelves.

In all these cases, the "memory core" was a tiny fraction of the total space. For the PXP model, the core was only about 0.42% of the total cyclic dimension, yet it held all the interesting, non-thermal activity.

What This Paper Rules Out

It is crucial to understand what this paper says is not the answer. The authors explicitly argue against the idea that this memory core is just a result of the system spreading out quickly or having high "complexity."

They tested this by comparing their results to "escaping" geometries—mathematical models where the probability wave runs away to infinity, never stopping. In these models, the wave spreads fast and the "complexity" grows, but because the wave never settles down, there is no stationary memory core. The probability just leaks out forever. This proves that fast spreading alone does not create a memory core. You need a system that stops and settles, but settles in a very specific, organized way.

They also checked if this was just a feature of "integrable" systems (systems that are perfectly predictable and never chaotic). By comparing their chaotic systems to an integrable Ising chain, they found that while integrable systems can have structured states, the specific combination of a compact core with strong fluctuations and Gibbs mismatch is a feature of state-selective anomalies in non-integrable systems. It's not just about the rules of the game; it's about the specific starting move you choose.

The Bottom Line

The paper doesn't claim to have solved the mystery of quantum memory forever, nor does it claim this happens in every single weird system. Instead, through detailed numerical simulations, it suggests a powerful new way to visualize these phenomena. It proposes that when a quantum system refuses to forget, it doesn't just "hold on" vaguely; it builds a tiny, hyper-active fortress near the beginning of its mathematical journey. This fortress, the Krylov-space memory core, is where the system keeps its secrets, vibrating and exchanging energy, while the rest of the library just sits there, quietly thermalized.

The authors emphasize that this is a "stationary framework," meaning it describes how the system looks after it has settled down, not how it got there. The discovery is that this "core-halo" structure—where a tiny, active core is embedded in a broad, quiet cloud—appears to be a common language spoken by very different types of quantum anomalies. It's a map for finding where the quantum memory lives, showing us that even in a chaotic universe, the past can be kept safe in a very small, very busy room.

Technical Summary: Krylov-Space Memory Cores

Problem Statement
The equilibration and thermalization of isolated quantum many-body systems remain central problems in nonequilibrium quantum physics. While the Eigenstate Thermalization Hypothesis (ETH) explains how generic non-integrable systems approach thermal equilibrium, specific initial states in non-integrable systems exhibit "anomalous" relaxation. Examples include weak thermalization, confinement-induced dynamics, and quantum many-body scars. Current diagnostics (local observables, entanglement, spectral statistics, complexity) identify that these dynamics are atypical but fail to reveal whether the surviving late-time information acquires a common spatial organization within the state-dependent dynamical subspace. Specifically, it is unclear if structured quantum memory resides in a specific region of the Krylov space and how it remains dynamically encoded.

Methodology
The authors introduce a stationary, depth-resolved framework within the Krylov space to analyze anomalous initial-state dynamics.

Krylov Space Construction: For a Hamiltonian $H$ and initial state $|\psi_0\rangle$ , the Lanczos algorithm generates an orthonormal basis $\{|n\rangle\}$ spanning the cyclic subspace $\mathcal{K}$ . The Schrödinger equation maps to a one-dimensional hopping problem on a chain where the index $n$ represents a state-dependent "depth."
Stationary Occupation Profile ( $\Pi_n$ ): Defined as the infinite-time average of the Krylov probability distribution $P_n(t) = |\langle n|\psi(t)\rangle|^2$ . This profile describes the background probability distribution along the Krylov chain after dephasing but does not inherently distinguish between equilibrated and non-equilibrated regions.
Three Complementary Diagnostics: To characterize the physical content of the stationary background, the authors define three depth-resolved quantities:
- Residual Equilibration Fluctuations ( $\chi_n$ ): The long-time variance of the Krylov occupation, quantifying persistent temporal motion.
- Gibbs-Deviation Profile ( $d_n$ ): The signed difference between the stationary profile $\Pi_n$ and a reference Gibbs state $\rho_{\text{Gibbs}}$ constructed on the cyclic subspace with matching energy. This identifies regions where the stationary state mismatches the equilibrium benchmark.
- Krylov-Current Fluctuation Activity ( $\Gamma_n$ ): The long-time variance of the probability current $J_n(t)$ between neighboring Krylov sites. This quantifies persistent back-and-forth probability exchange, distinguishing dynamic activity from static occupation.
Cumulative Analysis: The authors define a "memory core" not by the occupation profile alone, but by the spatial concentration of the total weights of $\chi_n$ , $d_n^2$ , and $\Gamma_n$ . The core depth $n_{mc}$ is defined as the maximum depth required to contain a fraction $q$ (typically 0.95) of each of these three residual weights.

Key Contributions and Results
The paper applies this framework to three distinct non-integrable models: a chaotic mixed-field Ising chain, a confinement regime of the same chain, and the periodic PXP model (known for many-body scars).

Discovery of Memory Cores: In all three models, specific anomalous initial states (e.g., weakly thermalizing states, domain-wall states, and scarred N'eel states) develop a compact, low-depth memory core. Within this region (typically extending only a few tens of Krylov steps, $n_{mc} \ll D_0$ ), the three diagnostics ( $\chi_n, d_n, \Gamma_n$ ) are simultaneously appreciable.
Core-Halo Structure: These compact cores are embedded within substantially broader stationary occupation "halos." The stationary occupation profile $\Pi_n$ often extends to depths orders of magnitude larger than the memory core (e.g., $n^{(0.95)}_\Pi \gg n_{mc}$ ). The "halo" contains probability but lacks the strong residual fluctuations, Gibbs mismatch, and current activity found in the core.
State Selectivity: The phenomenon is state-selective. Generic reference states (random product states or efficiently thermalizing states) evolving under the same Hamiltonians do not exhibit this combination. Their residual weights are either weak, broadly distributed across the Krylov chain, or lack one of the three components.
Distinction from Integrability: An auxiliary comparison with an integrable Ising model shows that compact residual profiles can arise in integrable systems due to conserved quantities, but the specific "memory core" structure identified here is defined by the contrast between anomalous and typical states within non-integrable systems.
Distinction from Krylov Growth: The authors compare their finite-system results with exactly solvable infinite Krylov chains (exhibiting ballistic, accelerating, or linear spreading). In these "escaping" geometries, probability flows to arbitrarily large depths, and no normalizable stationary profile exists ( $\Pi_n = 0$ ). This demonstrates that strong Krylov spreading and large transient complexity do not generate a memory core; the core is a feature of stationary organization in finite-dimensional spaces.

Significance and Claims
The paper claims that Krylov-space memory cores provide a stationary framework for identifying where structured quantum memory resides in otherwise thermalizing systems.

Unified Phenomenology: The repeated appearance of this core-halo structure across weak thermalization, confinement, and many-body scarring suggests a common stationary organization for anomalous dynamics, despite their different microscopic mechanisms.
Operational Definition: The memory core is operationally defined by the co-localization of residual fluctuation, equilibrium mismatch, and current activity. It is not merely a region of high occupation probability.
Limitations: The authors explicitly state that the memory core is not proposed as a criterion for distinguishing integrable from non-integrable Hamiltonians, nor is it claimed to be a unique or exhaustive measure of information storage capacity. The results are based on finite-size systems, and while the data suggests the active depth scales linearly with system size ( $n_{mc} \sim O(L)$ ) while the Krylov dimension grows exponentially, asymptotic scaling requires larger systems to confirm.

In summary, the work establishes that anomalous initial states in non-integrable systems retain non-thermal information in a compact, dynamically active region of the Krylov space, distinct from the broader stationary probability background.

The Hidden Library of Quantum Chaos

The Paper's Discovery: The "Memory Core"

What This Paper Rules Out

The Bottom Line

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