⚛️ quantum physics

Fock-space fragmentation in quenches of disordered interacting fermions

This paper demonstrates that Fock-space fragmentation, typically associated with translation-invariant systems, also emerges in disordered interacting fermions like the XXZ model with random fields, where increasing disorder fragments the accessible potential-energy shell in Fock space, offering a geometric mechanism to explain finite-time relaxation slowing and providing a heuristic estimate for the strong-fragmentation scale.

Original authors: Ishita Modak, Rajesh Narayanan, Ferdinand Evers, Soumya Bera

Published 2026-10-14

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CC BY 4.0

Original authors: Ishita Modak, Rajesh Narayanan, Ferdinand Evers, Soumya Bera

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

In the quiet, isolated world of a quantum computer or a super-cooled cloud of atoms, particles do not behave like the billiard balls of our everyday experience. Instead, they exist in a state of constant potential, where their positions and energies are linked in a vast, invisible web. When scientists disturb such a system—by suddenly changing its conditions—they expect the particles to scramble, sharing their energy until everything settles into a uniform, thermal state. This process, known as thermalization, is the rule for most physical systems; it is why a hot cup of coffee eventually cools to room temperature. However, in systems filled with disorder and strong interactions, this rule sometimes breaks down. Instead of mixing, the particles can get stuck, refusing to forget their initial arrangement. This phenomenon, called many-body localization, has puzzled physicists for years because it challenges our fundamental understanding of how matter reaches equilibrium. The key to understanding why this happens lies not just in the particles themselves, but in the vast landscape of all possible arrangements they could occupy, a mathematical space that dictates how easily they can move from one state to another.

A team of researchers has now mapped this landscape with a new level of detail, revealing that the reason particles get stuck is often due to the very shape of the space they inhabit. By simulating a one-dimensional chain of interacting fermions—particles like electrons that cannot occupy the same space—with random, disordered forces, the scientists tracked how the system evolved after a sudden change. They discovered that the system's behavior is governed by a specific region within the vast space of all possible configurations, which they call a "potential-energy shell." Think of this shell as a specific floor in a massive, multi-story building where the particles are allowed to roam; the height of the floor is determined by the energy the system started with. In a perfectly ordered world, this floor would be a single, open room where particles could wander freely. But in the disordered systems studied here, the floor is often broken up.

The researchers found that as the disorder in the system increases, this accessible floor does not just shrink; it shatters into isolated islands. When the disorder is weak, the floor is mostly connected, allowing the particles to explore the entire space and eventually thermalize. However, once the disorder crosses a certain threshold, the floor fragments. The particles, starting on one specific island, find themselves cut off from the rest of the accessible space. They can move around on their small island, but they cannot reach the other islands without a rare, unlikely event that bridges the gap. This geometric fragmentation explains why the system slows down dramatically. The particles are not trapped by a lack of energy, but by the broken geometry of the space they are trying to navigate.

This discovery helps explain a long-standing mystery in the field: why some simulations show that systems thermalize quickly while others seem to freeze, even when the conditions look similar. The researchers observed that in the regime where disorder is moderate, the size of the accessible floor varies wildly from one sample to another. In some cases, the floor is large and connected; in others, it is tiny and broken. This variation accounts for the huge differences seen in experiments and simulations, where some samples appear to thermalize while others remain stuck. As the disorder grows stronger, the fragmentation becomes more pronounced, and the floor breaks into many small, disconnected pieces. The initial state of the system ends up on a small fragment, isolated from the rest of the potential energy landscape.

The study also sheds light on the nature of the transition between a flowing, thermal state and a frozen, localized one. The researchers noticed that as they increased the size of the system in their simulations, the point at which the floor seemed to break apart shifted. At first, the fragmentation appeared to be a permanent feature of the system, suggesting a sharp transition to a localized phase. However, as they looked at larger and larger systems, they saw a "turn-around" effect. The flow of the system seemed to reverse, suggesting that the fragmented islands might eventually reconnect if the system were large enough. This implies that the transition to a fully localized state might not happen at a fixed level of disorder, but rather requires extremely strong disorder to become permanent. The fragmentation they observed is a geometric property of the landscape, a structural barrier that slows down the system, but it is not the final word on localization. Quantum tunneling and interference effects, which allow particles to jump between these isolated islands, remain outside the scope of this geometric picture.

By focusing on the geometry of the potential-energy shell, the researchers have provided a fresh perspective on why disordered systems behave the way they do. They have shown that the "freezing" of particles is often a result of the landscape breaking apart, creating a maze of isolated rooms that the particles cannot easily escape. While this geometric fragmentation is a crucial piece of the puzzle, the full story of many-body localization involves a competition between this structural barrier and the quantum ability of particles to tunnel through walls. The work suggests that the path to understanding these complex systems lies in understanding the shape of the space they occupy, revealing that sometimes, the most difficult obstacle is not a wall, but a broken floor.

Problem Statement

The paper addresses the ongoing debate regarding the existence and nature of the Many-Body Localization (MBL) phase in generic models of disordered interacting fermions. While numerical and experimental studies utilize charge imbalance decay as a primary indicator of thermalization, there is no consensus on whether MBL truly exists in the thermodynamic limit. A major challenge is the difficulty of accessing the double limit of long observation times and large system sizes.

Previous studies have identified intriguing transient phenomena in the weak-to-moderate disorder regime ( $W \lesssim W_c$ ), including:

A qualitative change in relaxation behavior from accelerated to decelerated as disorder increases.
Drastic sample-to-sample fluctuations in imbalance relaxation, where ensembles contain both perfectly thermalizing and highly insulating samples.

The authors aim to identify a geometric mechanism within Fock space that underlies these finite-size and finite-time crossover phenomena, specifically focusing on the regime of moderate to strong disorder ( $W \gtrsim t_{hop}$ ).

Methodology

The authors investigate a paradigmatic model of disordered, spinless fermions on a 1D lattice described by the Hamiltonian $\hat{H} = \hat{T} + \hat{V} + \hat{U}$ , where $\hat{T}$ is the kinetic hopping term, $\hat{V}$ is the nearest-neighbor interaction, and $\hat{U}$ represents random on-site potentials drawn from a uniform distribution $[-W, W]$ . The interaction strength is fixed at $V = t_{hop}$ .

Key Methodological Steps:

Fock Space Perspective: The analysis adopts a Fock-space representation where basis states $|b\rangle$ are eigenstates of local occupation numbers. In the regime where potential energy dominates, the site-occupation basis is the natural choice.
Quench Protocol: The system is initialized in a Néel state ( $|1010\dots\rangle$ ) and evolved under the Hamiltonian. The time evolution conserves the initial energy $E_{in}$ and all moments of the Hamiltonian.
Potential-Energy Shell: The authors define a "potential-energy shell" as the subset of Fock-space basis states $|b\rangle$ satisfying $|E_{in} - E_b| \lesssim \alpha \Delta_{in}$ , where $E_b$ is the potential energy of the basis state, $\Delta_{in}$ is the energy variance of the initial state, and $\alpha$ is a phenomenological parameter (set to $1/2$ ). This shell represents the energetically accessible region of the configuration space.
Cluster Statistics: The connectivity of states within the shell is analyzed. States are considered connected if linked by the hopping term $\hat{T}$ . The authors study the size of the connected cluster containing the initial state ( $N_{csh}$ ) relative to the total mass of the shell ( $N_{sh}$ ).
Numerical Tools: Time propagation is performed using the Chebyshev expansion technique. The primary observable is the sublattice imbalance $I(t)$ . System sizes range up to $L=28$ , with extensive analysis of sample-to-sample statistics.

Key Contributions and Results

1. Geometric Interpretation of Sample-to-Sample Fluctuations
The paper demonstrates that the large fluctuations in imbalance relaxation observed at moderate disorder are a direct manifestation of fluctuations in the "mass" (number of states, $N_{sh}$ ) of the potential-energy shell.

At moderate disorder ( $W \approx 1.5$ ), the initial state energy $E_{in}$ fluctuates significantly relative to the distribution of shell energies.
Consequently, the fraction of the total Fock space occupied by the shell ( $N_{sh}/N_{FS}$ ) varies wildly between samples. Some samples have a large shell mass (leading to thermalization), while others have a small mass (leading to insulating behavior).

2. Identification of Fock Space Fragmentation
The authors identify a phenomenon termed "shell fragmentation." As disorder increases beyond a threshold ( $W \approx 3.5 - 4$ ), the potential-energy shell breaks into disconnected clusters (fragments).

Weak/Moderate Disorder: The shell is typically singly connected ( $N_{csh}/N_{sh} \approx 1$ ).
Strong Disorder: The initial state becomes confined to a small fragment ( $N_{csh}/N_{sh} \ll 1$ ), disconnected from the majority of the shell.
This fragmentation restricts the directly connected region available to the initial state. Relaxation beyond this fragment requires coupling different fragments, which the authors argue leads to the observed "creep" or slow dynamics.

3. Turn-Around Flow and Finite-Size Scaling
A critical finding is the behavior of the fragmentation transition with system size.

The authors observe a "turn-around flow" in the renormalization group (RG) sense. For a given disorder $W$ , as system size $L$ increases, the ratio $N_{csh}/N_{sh}$ initially decreases (suggesting fragmentation) but eventually turns around and increases (suggesting re-connection/ergodicity) at very large $L$ .
This implies that the "fragmented" phase observed in finite systems is transient. The flow towards a fragmented fixed point is interrupted by the growth of many-body resonances (increasing connectivity) at larger scales.
By extrapolating the position of the variance peak of $N_{csh}/N_{sh}$ , the authors provide a heuristic estimate for the strong-fragmentation scale $W_{frag} \approx 5$ . However, they emphasize that the critical disorder for the true MBL transition may differ due to quantum tunneling effects not captured by this geometric analysis.

4. Distinction from Percolation and MBL
The paper clarifies that shell fragmentation is a geometric property of the Fock-space graph and is distinct from the MBL transition.

Unlike percolation on random regular graphs (RRGs) where connectivity is fixed, the Fock-space graph here has connectivity that grows with system size, and energies are strongly correlated.
Fragmentation restricts dynamics but does not preclude it; quantum tunneling between fragments and interference effects (outside the scope of this geometric construction) are required to determine the true localization properties.

Significance and Claims

The authors claim that their work offers a fresh geometric perspective on the dynamics of disordered interacting systems:

Mechanism for Slow Dynamics: They propose that shell fragmentation provides a geometric mechanism contributing to the finite-time slowing down of relaxation (creep) observed in numerical and experimental studies.
Explanation of Fluctuations: The concept explains why sample-to-sample fluctuations are large at moderate disorder (fluctuations in shell mass) and why they appear suppressed at stronger disorder (the shell fragments, and the distribution of fragment sizes hosting the initial state becomes narrow).
Heuristic Estimates: While not providing a rigorous proof of MBL, the analysis offers a heuristic estimate for the disorder scale where fragmentation becomes dominant ( $W_{frag} \approx 5$ ) and highlights the existence of a turn-around flow regime.
Caveat: The authors explicitly state that fragmentation is not a diagnostic of MBL. The full MBL problem involves a competition between localization on a fragmented backbone and delocalization via tunneling. The geometric fragmentation scale $W_{frag}$ should not be conflated with the MBL critical disorder $W_{MBL}$ .

In summary, the paper argues that the complex phenomenology of relaxation in disordered systems, particularly the transient slowing and fluctuations, can be understood through the lens of the geometry and connectivity of the potential-energy shell in Fock space, independent of the ultimate fate of the system in the thermodynamic limit.

Problem Statement

Methodology

Key Contributions and Results

Significance and Claims

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