⚛️ quantum physics

Local integrals of motion encoded in a few eigenstates

This paper demonstrates that local integrals of motion in the XXZ model can be accurately estimated from a vanishingly small fraction of eigenstates in the thermodynamic limit, a property that fundamentally distinguishes true integrability from Hilbert space fragmentation, which requires the majority of eigenstates for similar reconstruction.

Original authors: J. Pawłowski, P. Łydżba, M. Mierzejewski

Published 2026-03-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: J. Pawłowski, P. Łydżba, M. Mierzejewski

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 you have a massive, incredibly complex lock with billions of tumblers. This lock represents a quantum system (like a chain of tiny magnets). Usually, to understand how this lock works, you'd need to see every single tumbler in every possible position. That's like calculating the behavior of every single energy state (eigenstate) of the system. For a large system, that number is so huge it's practically impossible to compute.

However, this paper discovers a clever shortcut. It turns out that for certain types of locks (called integrable systems), you don't need to see the whole picture. You only need to peek at a tiny handful of tumblers—maybe just a few dozen out of billions—to figure out the lock's secret rules.

Here is the breakdown of their discovery using simple analogies:

1. The "Secret Rules" (Integrals of Motion)

In physics, some systems follow strict, unbreakable rules. Imagine a game of billiards where, no matter how the balls hit each other, the total "spin" and "energy flow" never change. These unchanging quantities are called Local Integrals of Motion (LIOMs).

The Old Way: To find these rules, physicists thought they needed to analyze the entire game (all possible ball positions) to see the pattern.
The New Discovery: The authors found that if you look at just a tiny, random sample of the game's states (like watching 100 specific moments in a 10-billion-year game), you can mathematically reconstruct the secret rules.

2. The "Compression" Trick

How do they do it? They use a technique called compression, which is like taking a high-resolution photo of a crowd and shrinking it down to a tiny thumbnail.

They take a list of local operators (think of these as "questions" they ask the system, like "What is the spin at this spot?").
They look at the answers for a small number of energy states.
They use a mathematical "squeezer" (Singular Value Decomposition) to compress this data.
The Magic: If the system is "integrable," the compression doesn't lose the important secrets. The "noise" disappears, and the "signal" (the secret rules) pops out clearly, even with very little data.

3. The "Two Types of Locks" Comparison

This is the most exciting part of the paper. The authors compared two types of quantum systems to see if this shortcut works for both.

Type A: The "Integrable" Lock (The XXZ Model)

Think of this as a well-organized library where every book has a perfect, logical place.

The Result: The shortcut works perfectly. Whether the library has 100 books or 100 billion books, you only need to look at a fixed, tiny number of books to find the cataloging system.
The Analogy: It's like finding the rule of a song. If you listen to just the first 10 seconds of a repeating melody, you can predict the whole song. As the song gets longer, you don't need to listen to more seconds to understand the pattern; the pattern is already fully encoded in those first few seconds.

Type B: The "Fragmented" Lock (The Folded XXZ Model)

Now, imagine a library where the books are scattered into millions of separate, locked rooms. You can't walk from one room to another; the library is "fragmented."

The Result: The shortcut fails. Even if you look at a few random books, you can't figure out the rules. You have to visit almost every single room to understand how the whole library is organized.
The Analogy: This is like trying to guess the rules of a game by looking at a single player in a massive, disconnected crowd. Because the crowd is broken into isolated groups, a small sample tells you nothing about the whole. You need to see the entire crowd to see the big picture.

4. Why This Matters

This paper highlights a fundamental difference between two ways nature can "break" the usual flow of energy (thermalization):

Integrability: The system is so orderly that a tiny sample reveals the whole truth.
Fragmentation: The system is so broken up that you need the whole dataset to understand it.

The Takeaway:
In the world of quantum physics, we used to think that to understand a complex system, we needed a supercomputer to calculate everything. This paper says: "Not always!" If the system is integrable, a tiny peek is enough. But if the system is fragmented, you really do need to see the whole mess. This helps scientists distinguish between these two mysterious states of matter and understand how information is stored in the universe.

1. Problem Statement

The paper addresses a fundamental question in non-equilibrium quantum statistical mechanics: How much information about a quantum system's conserved quantities (integrals of motion) is encoded in its energy eigenstates?

Context: In ergodic (chaotic) systems, the Eigenstate Thermalization Hypothesis (ETH) implies that a single eigenstate contains all thermodynamic information. In integrable systems, the existence of an extensive number of Local Integrals of Motion (LIOMs) prevents thermalization, requiring a Generalized Gibbs Ensemble (GGE) for description.
The Gap: While it is known that single eigenstates can identify ergodicity, it is unclear if a small subset of eigenstates is sufficient to reconstruct the LIOMs in integrable systems. Furthermore, it is unknown how this compares to systems exhibiting Hilbert Space Fragmentation (HSF), another mechanism for ergodicity breaking where the Hilbert space decomposes into exponentially many disconnected sectors.
Goal: To determine the minimum number of eigenstates required to accurately estimate LIOMs in the XXZ model (integrable) versus the folded XXZ model (fragmented), and to identify fundamental differences between integrability and HSF.

2. Methodology

The authors employ a compression-based numerical approach to extract LIOMs from an incomplete spectrum.

The Algorithm:
1. Operator Basis: A set of $D_O$ orthonormal local operators $\{A_s\}$ is defined (e.g., Pauli strings acting on $M$ consecutive sites).
2. Matrix Construction: For a selected subset of $N_S$ eigenstates $\{|n\rangle\}$ , a matrix $R$ is constructed containing the diagonal matrix elements $A^s_{nn} = \langle n | \hat{A}_s | n \rangle$ .
3. Singular Value Decomposition (SVD): The matrix $R$ is decomposed ( $R = U \Lambda V^T$ ).
4. Compression: According to the Eckart–Young–Mirsky theorem, the "true" LIOMs correspond to the largest singular values ( $\lambda_\alpha$ ). The LIOMs are reconstructed as linear combinations of the local operators: $\hat{Q}_\alpha = \sum_s V_{s\alpha} \hat{A}_s$ .
5. Sampling: The authors test the accuracy of this reconstruction by varying $N_S$ (the number of eigenstates used) from a tiny fraction to the full Hilbert space dimension ( $Z$ ). They use the shift-invert method (or exact diagonalization for degenerate cases) to target specific energy sectors.
Models Studied:
- XXZ Spin Chain: A paradigmatic integrable model.
- Folded XXZ Model: A model derived from the XXZ chain in the limit of infinite anisotropy, known to exhibit Hilbert Space Fragmentation. A perturbation term ( $g$ ) is added to break Bethe-ansatz integrability while preserving fragmentation.

3. Key Contributions and Results

A. Integrable Systems (XXZ Model)

Few Eigenstates Suffice: The study demonstrates that LIOMs in the XXZ chain can be accurately reconstructed using a vanishingly small fraction of the total eigenstates.
- For a system size $L=24$ , only $N_S \approx 100$ eigenstates (approx. $10^{-5}$ of the Hilbert space) are sufficient.
- Crucially, the required number of eigenstates ( $N^*$ ) does not increase with system size; in fact, it slightly decreases or remains constant as $L$ grows.
Scaling: The accuracy of the reconstruction depends on the ratio $N_S / \text{rank}(R)$ . Once $N_S$ exceeds the rank of the operator matrix (roughly $N_S \gtrsim D_O$ ), the compression effectively isolates the LIOMs.
Quasilocal Integrals of Motion (QLIOMs): The method also successfully detects quasilocal operators (which have exponentially decaying tails), confirming that even these complex conserved quantities are encoded in a few eigenstates.
Symmetry Sectors: These results hold across different symmetry sectors (real/imaginary operators, spin-flip parity) and for operators with varying support sizes ( $M$ ).

B. Fragmented Systems (Folded XXZ Model)

Failure of Sparse Sampling: In the folded XXZ model, the behavior is fundamentally different.
- LIOMs arising specifically from Hilbert Space Fragmentation (e.g., domain wall conservation) cannot be reconstructed from a small subset of eigenstates.
- To accurately estimate these fragmentation-induced LIOMs, one requires almost the entire set of eigenstates ( $N_S \approx Z$ ).
Robustness to Perturbation: The authors introduce a perturbation ( $g=2.0$ $g = 2.0$ ) that breaks Bethe-ansatz integrability but preserves fragmentation.
- The "Bethe-ansatz" LIOMs (like the energy current) disappear or become approximate, and they can still be estimated from a few eigenstates (if they exist).
- However, the fragmentation-induced LIOMs persist under this perturbation but still require the full spectrum to be reconstructed.
Degeneracy Handling: The model possesses massive degeneracies. The authors show that the inability to reconstruct LIOMs from few states is not merely an artifact of degeneracy (where one might need to sample the whole subspace), but a fundamental property of how information is distributed in fragmented systems.

4. Significance and Conclusions

Fundamental Distinction: The paper establishes one of the few known fundamental differences between Integrability and Hilbert Space Fragmentation.
- Integrability: LIOMs are "local" in the sense that their information is redundantly encoded in every typical eigenstate. A few samples suffice to decode them.
- Fragmentation: LIOMs arising from fragmentation are "global" in the sense that the information is distributed across the disconnected Krylov subspaces. Reconstructing them requires accessing the vast majority of the Hilbert space.
Implications for Hydrodynamics: Since LIOMs are the starting point for constructing Generalized Hydrodynamics (GHD), this result suggests that GHD for integrable systems can be derived from limited spectral data, whereas systems with fragmentation may require full spectral knowledge to characterize their stationary states.
Methodological Advance: The compression-based SVD method is validated as a powerful tool for identifying conserved quantities without full diagonalization of the Hamiltonian, provided the system is integrable.

In summary, the work reveals that while integrable systems are "efficiently compressible" regarding their conservation laws (requiring few eigenstates), fragmented systems are not, highlighting a deep structural difference in how quantum information is stored in eigenstates for these two classes of ergodicity-breaking phenomena.