Locality versus Fock-space structure in East-type models

Imagine a crowded dance floor where everyone is trying to move around. In a normal party (a "ergodic" system), people eventually mix thoroughly; if you start in one corner, you'll likely end up anywhere on the floor after a while. This is how most quantum systems behave: they thermalize, meaning they settle into a uniform, featureless state.

However, physicists are interested in the exceptions: systems where people get stuck in one corner and never mix. This is called localization. Usually, this happens because the floor is covered in random obstacles (disorder). But what if the floor is perfectly smooth, yet people still can't move?

This paper explores a specific type of "smooth floor" system called the East model. In this model, people (spins) can only dance (flip) if their neighbor is already dancing in a specific way. It's like a rule: "You can only spin if the person to your right is already spinning." This simple rule creates a traffic jam, slowing down the system or freezing it completely.

The Big Question

The researchers wanted to know: Is the "jam" caused by the physical distance between people (real-space locality), or is it caused by the specific pattern of who can connect to whom in the abstract "dance map" (Fock space)?

To answer this, they created two versions of the party:

The Original East Model: People are arranged in a line. You can only dance if your immediate neighbor is dancing. The connections are local and orderly.
The "Permuted" East Model: They kept the same rules about how many people can dance together, but they scrambled the connections. Imagine taking the dance floor, cutting out all the people, and randomly shuffling who is standing next to whom. Now, you might be able to dance with someone standing 50 feet away, as long as the "neighbor rule" is mathematically satisfied. The physical distance is gone, but the structure of the connections remains.

The Experiment

They treated these systems like a giant puzzle. They looked at how a single dancer (a quantum state) spreads out over time.

If the system is "delocalized" (ergodic): The dancer spreads out to cover the whole floor.
If the system is "localized": The dancer gets stuck in a small corner and never leaves.

They used two main tools to measure this:

The "Participation Ratio": A way to count how many different spots on the dance floor a dancer visits.
Shannon Entropy: A measure of how "spread out" or "confused" the dancer's position is.

The Surprising Result

The paper found that it didn't matter if the connections were local or scrambled.

Even when they ripped up the "real space" map and randomly shuffled the connections (the Permuted East model), the system behaved almost exactly like the original, orderly model.

At low "hopping" rates, the dancers got stuck (localized) in both models.
At high rates, they spread out (delocalized) in both models.
The point where they switched from stuck to moving was roughly the same for both.

The Takeaway

The authors conclude that for these specific types of constrained systems, the "shape" of the connection map is what matters, not the physical distance.

Think of it like a subway system.

The Old View: You can only get stuck because the stations are far apart and the tracks are broken in specific places.
This Paper's View: You get stuck because of the schedule and the rules of which trains connect to which stations. Even if you teleport the stations to random locations around the world (scrambling the map), as long as the schedule rules (the graph structure) remain the same, the traffic jam persists.

In short: The "traffic jam" in these quantum systems isn't caused by the physical layout of the room; it's caused by the abstract pattern of who is allowed to talk to whom. If you keep that pattern, the jam stays, even if you destroy the room's geometry.

1. Problem Statement

The paper addresses the fundamental mechanisms driving Many-Body Localization (MBL) and ergodicity breaking in quantum systems without real-space disorder.

Context: While MBL is typically associated with disorder, recent research focuses on "disorder-free" localization caused by kinetic constraints (e.g., the Quantum East model).
The Question: In constrained models like the Quantum East model, is the localization transition driven by the geometric locality of spin flips in real space, or is it driven by the topological structure of the connectivity graph in Fock space?
Hypothesis: The authors hypothesize that the specific real-space locality is not the essential ingredient; rather, the hierarchical organization of the Fock space (specifically the connectivity between magnetization sectors) is the critical factor.

2. Methodology

The authors employ a comparative approach using random matrix theory concepts to isolate structural features of the Hamiltonian.

A. Model Construction

Baseline (Ising/QREM): A standard Quantum Random Energy Model (QREM) with Ising-type spin flips ( $\sigma^x_j$ ) and random diagonal disorder. This model has a fully connected Fock space graph (ladder-like structure where every state connects to $L$ neighbors).
Reference Model (Quantum East): A constrained model where a spin can only flip if its right neighbor is up. This introduces kinetic constraints, reducing the connectivity of the Fock space graph. It is known to exhibit a localization transition at a critical hopping rate.
The Novel Model (Permuted East):
- The authors construct a new Hamiltonian by randomizing the off-diagonal connectivity within the Fock space while preserving the coarse block structure.
- Procedure: The Fock basis is ordered by total magnetization $M$ . The Hamiltonian consists of blocks connecting sectors $M$ and $M \pm 1$ . In the Permuted East model, the specific connections within these off-diagonal blocks are randomly shuffled (permuted) using random permutation matrices.
- Result: Real-space locality and translational invariance are destroyed (a spin flip can connect states with Hamming distance $>1$ ), but the "ladder-like" organization of magnetization sectors and the total number of connections between sectors are preserved.

B. Diagnostics

To compare the models, the authors use two primary metrics:

Dynamical Participation Ratio ( $\phi_\infty$ ): Measures the long-time spreading of a wavefunction in Fock space.
- $\phi_\infty \sim O(D^{-1})$ indicates delocalization (ergodic).
- $\phi_\infty \sim O(1)$ indicates localization (non-ergodic).
- Note: Calculated with zero diagonal disorder ( $w=0$ ) to isolate kinetic effects.
Shannon Entropy ( $S$ ) and Spectral Variance: Measures the delocalization of eigenstates.
- The mean entropy $\mu_S$ tracks the degree of spreading.
- The local spectral variance $\sigma^2_S$ (fluctuations of entropy around the microcanonical average) is used as a finite-size estimator for the phase transition. Large fluctuations indicate a transition region.

3. Key Contributions

Decoupling Real-Space Locality from Fock-Space Topology: The paper demonstrates that one can destroy real-space locality entirely while preserving the Fock-space "ladder" structure and still observe the same qualitative physics (localization transition) as the original local model.
Graph-Theoretic Perspective: It reframes the problem of MBL in constrained systems as a problem of graph connectivity in Fock space, rather than spatial proximity.
Complementarity to Existing Work: The work complements recent studies (e.g., Ref [52]) that kept the graph fixed but varied energies. Here, the graph structure (hierarchy) is kept fixed, but the specific edges are randomized.

4. Key Results

Qualitative Equivalence: Both the standard East model and the Permuted East model exhibit a transition from an ergodic (delocalized) phase to a non-ergodic (localized) phase as the constraint parameter $s$ increases.
Dynamical Participation Ratio:
- For $s < 0$ (weak constraints), both models show $\phi_\infty \to 0$ (delocalized) as system size $L$ increases.
- For $s > 0$ (strong constraints), both models show $\phi_\infty \to O(1)$ (localized), whereas the unconstrained Ising model remains delocalized.
Shannon Entropy Analysis:
- The mean scaled entropy $\mu_S / \ln D$ decreases with increasing $s$ for both models, indicating increasing localization.
- Finite-Size Scaling: The local spectral variance $\sigma^2_S$ exhibits a peak that drifts with system size, characteristic of a phase transition.
- Extrapolation: Linear extrapolation of the crossing points and variance maxima to the thermodynamic limit ( $1/L \to 0$ $1/ L \to 0$ ) yields finite critical values ( $s_c$ $s_{c}$ ) for both models:
  - East model: $s_c \approx 1.7 - 1.9$ .
  - Permuted East model: $s_c \approx 2.0 - 2.5$ .
- While the quantitative location of the transition differs slightly, the qualitative behavior (existence of a transition) is identical.

5. Significance and Conclusion

Primary Conclusion: The essential ingredient for slow dynamics and localization in East-type models is not the geometric locality of spin flips, but the hierarchical organization of connectivity in Fock space (specifically, the restriction of transitions to neighboring magnetization sectors).
Implications for Theory:
- This challenges the intuition that real-space locality is required for MBL in clean systems.
- It suggests that analytical approaches relying on real-space geometry (like the Forward Scattering Approximation) may need modification, as the graph distance in Fock space no longer coincides with Hamming distance in the permuted model.
Future Directions: The authors propose a hierarchy of models where further structural features (e.g., connections between non-neighboring magnetization sectors) could be randomized to pinpoint the exact minimal requirements for ergodicity breaking.

In summary, the paper establishes that the topology of the Fock-space graph is the dominant factor governing ergodicity breaking in kinetically constrained systems, rendering the specific real-space locality of interactions secondary.