Exponentially slow thermalization and the robustness of Hilbert space fragmentation

This paper demonstrates that even when exact dynamical constraints are perturbed by a boundary thermal bath, Hilbert space fragmentation can induce exponentially slow thermalization in both Hamiltonian and random unitary circuit models due to strong bottlenecks that severely restrict the system's exploration of configuration space.

The Gist

Imagine you are trying to get a crowd of people from one side of a giant, complex maze to the other side. Usually, if you open a door on one side (like a "thermal bath" or a heat source), the people would quickly wander through the maze, mix up, and eventually spread out evenly everywhere. This is what physicists call thermalization—the system reaching a state of equilibrium where everything looks the same.

However, this paper discovers a special kind of maze where the people get stuck for an incredibly long time, even with the door open. In fact, the time it takes for them to mix isn't just "long"; it's exponentially long. If the maze is size 10, it might take 10 minutes. If the maze is size 20, it might take 100 minutes. But if the maze is size 30, it might take millions of years.

Here is the breakdown of this discovery using simple analogies:

1. The "Frozen" Maze (Hilbert Space Fragmentation)

In the quantum world, the "maze" is called Hilbert Space. It's the map of all possible arrangements a system of particles can be in.

Usually, this map is one big, open room where you can walk from any spot to any other spot. But in this specific model (called the Pair-Flip model), the rules of the game are very strict. Imagine that in this maze, you can only move if you are standing next to someone wearing the exact same color shirt as you. If you are next to someone with a different shirt, you are frozen in place.

Because of these strict rules, the maze gets chopped up into millions of tiny, isolated islands (or "fragments"). If you start on one island, you can never leave it. This is called Hilbert Space Fragmentation. The system is "frozen" and never thermalizes.

2. Breaking the Rules (The Impurity)

The researchers asked: "What happens if we break the rules just a little bit?"

They took one end of the chain and connected it to a "chaotic" system (a thermal bath) that doesn't follow the strict rules. Imagine opening a gate at the edge of the maze and letting a group of people who can move freely run in. You would expect them to quickly run through the maze, break the ice, and get everyone else moving.

The Surprise: Even with this "free-moving" gate, the people in the deep parts of the maze stay frozen for an incredibly long time. The system does eventually mix, but it takes an exponentially long time.

3. The Tree Analogy (Why is it so slow?)

To understand why, the authors used a clever geometric trick. They mapped every possible arrangement of the particles to a walk on a giant tree.

• The Tree: Imagine a tree where the trunk is the start, and it branches out into $N$ directions, and then those branches split again, and again.
• The Walk: Every time the particles arrange themselves, it's like taking a step on this tree.
• The Constraint: The strict "same shirt" rule means that if you take two steps in the same direction, you have to immediately step back (backtrack). This keeps you from wandering too far out.
• The "Frozen" States: The states that are completely frozen are the ones at the very tips of the tree branches.

The Bottleneck:
Now, imagine you are at the very tip of a branch (a frozen state). You want to get to the center of the tree (the mixed, thermal state).

• To get out of the tip, you have to walk back toward the center.
• But the tree is shaped such that there is only one path back toward the center, but many paths leading further out to the tips.
• It's like being in a funnel that is wider at the top than the bottom. Even if you try to walk back, the sheer number of paths leading away from the center makes it statistically very hard to find the single path back in.

The "thermal bath" at the edge of the chain acts like a person trying to push you from the tip toward the center. But because the tree structure is so biased against moving inward, you get stuck in a "traffic jam" (a bottleneck) for a very long time.

4. The Result: Exponential Slowness

The paper proves mathematically that because of this tree structure, the time it takes for the "chaotic" energy from the edge to reach the center grows exponentially with the size of the system.

• Small System: The jam clears quickly.
• Large System: The jam becomes a traffic nightmare that lasts forever (in human terms).

Why Does This Matter?

This is a big deal for physics and technology:

1. Robustness: It shows that even if you don't perfectly tune your system (if you break the rules slightly), the "frozen" behavior is surprisingly robust. It doesn't just disappear; it just gets slower.
2. Quantum Memory: Because the system stays in its initial state for so long, it could be used as a quantum memory. You could store information in these frozen states, and it would stay there for a very long time without being scrambled by the environment.
3. New Physics: It challenges the idea that "if you add enough chaos, everything eventually mixes." This paper shows there are special geometric structures (like this tree) that can resist mixing for an incredibly long time, even when chaos is present.

In a nutshell: The researchers found a quantum maze that is so strangely shaped that even if you open the gate to let chaos in, the particles get stuck in a traffic jam for an exponentially long time, preserving their initial order against all odds.

Technical Summary

1. Problem Statement

The central theme of the paper addresses the mechanisms that impede or arrest thermalization in isolated quantum systems. While Many-Body Localization (MBL) relies on spatial disorder, a more recent mechanism known as Hilbert Space Fragmentation (HSF) arises from dynamical constraints that split the Hilbert space into exponentially many disconnected sectors (Krylov sectors), preventing ergodicity.

The Core Question: HSF typically relies on exact dynamical constraints. In realistic experimental settings, these constraints are often only approximately satisfied due to perturbations or coupling to an environment. The authors investigate:

• Does HSF remain robust when constraints are weakly broken?
• Can a system with fragmented dynamics thermalize, and if so, on what timescale?
• Specifically, they study a 1D spin chain with "pair-flip" constraints coupled to an ergodic thermal bath at one boundary. Does the bath thermalize the bulk, and how does the timescale ($t_{th}$) scale with system size ($L$)?

2. Methodology

The authors employ a combination of numerical simulations and rigorous analytical proofs using two distinct dynamical frameworks:

A. Hamiltonian Dynamics (Numerical)

• Model: A 1D spin chain of size $L$ with $N$ states per site.
  • Constrained Region ($1 \le i \le L_{cons}$): Governed by a Hamiltonian $H_0$ that only flips neighboring spins if they have identical values ($|aa\rangle \leftrightarrow |bb\rangle$). This preserves "pair-flip" constraints.
  • Impurity/Bath Region ($L_{cons} < i \le L$): Governed by $H_{imp}$, which breaks the constraints and restores ergodicity.
• Simulation: They perform quantum quenches starting from "frozen" states (product states annihilated by $H_0$) and use the Time-Evolving Block Decimation (TEBD) algorithm to simulate dynamics.
• Observables:
  • Bipartite entanglement entropy ($S_A$).
  • Expectation values of local charges ($Q_a$).
  • Krylov entropy (measuring spreading across sectors).

B. Random Unitary (RU) Circuit Dynamics (Analytical)

To obtain rigorous bounds, the authors map the problem to a stochastic process:

• Setup: The constrained region undergoes constrained random unitary dynamics (Haar-random gates preserving pair-flip constraints), while the boundary site undergoes depolarizing noise (simulating an infinite thermal bath).
• Mapping: The circuit-averaged evolution is mapped to a Markov process on the space of computational basis product states.
• Graph Theory:
  • Product states are mapped to random walks on an $N$-valent tree ($T_N$).
  • Krylov Sectors: States with the same "irreducible string" (walk endpoint) form a sector.
  • Krylov Graph ($G_K$): The graph where nodes are Krylov sectors and edges represent transitions induced by the bath.
• Analysis: They analyze the spectral gap ($\Delta_M$) of the Markov generator and the graph expansion ($\Phi$) using Cheeger's inequality to bound relaxation times.

3. Key Contributions and Results

A. Exponentially Slow Thermalization

The primary result is that even with a constraint-breaking bath at the boundary, the thermalization time $t_{th}$ scales exponentially with the size of the constrained region ($L_{cons}$):
$$t_{th} \sim e^{c L_{cons}}$$
This contrasts with generic diffusive ($L^2$) or ballistic ($L$) thermalization.

B. The Mechanism: Bottlenecks in Configuration Space

The slow dynamics arises from a strong bottleneck in the Krylov graph:

1. Tree Structure: The Krylov sectors form a tree structure. The "frozen" states lie at the leaves (edge) of the tree, while the largest sectors (highest entropy) are near the center.
2. Biased Random Walk: The bath induces transitions between sectors. However, the structure of the tree creates a competition:
   • Outward Bias: Moving from the center toward the edge (leaves) is statistically favored because there are $N-1$ ways to go "out" vs. 1 way to go "in."
   • Inward Bias: The number of states (volume of sectors) increases exponentially toward the center.
3. Dominance of Outward Bias: The authors prove that the "outward" bias dominates the dynamics for typical states. A system initialized at the edge (frozen state) moves inward quickly but gets "stuck" near the center (depth $d^* \approx v_N L$) because the probability of escaping the massive central sectors back to the edge is exponentially small.
4. Spectral Gap: The spectral gap of the Markov process is exponentially small:
   $$\Delta_M \sim L^{-3/2} \rho^L$$
   where $\rho = \frac{2\sqrt{N}-1}{N} < 1$ for $N > 2$.

C. Analytical Proofs (Theorems)

The paper provides rigorous bounds for the RU dynamics:

• Theorem 1: The spectral gap $\Delta_M$ is exponentially small in $L$.
• Theorem 2 (Entanglement): The time to reach a fraction $\gamma$ of maximal entanglement entropy scales as $t_S \sim \sqrt{L} e^{L \lambda_\gamma}$. The system rapidly reaches a sub-maximal entropy but takes exponentially long to saturate.
• Theorem 3 (Charge Relaxation): The relaxation time for local charge operators $Q_a$ also scales exponentially. Crucially, local observables can diagnose this slow thermalization even though they cannot distinguish between different Krylov sectors directly.

D. Robustness and $N=2$ vs. $N>2$

• Robustness: The slow thermalization persists even when constraints are broken only at the boundary. The "memory" of the initial fragmented state is retained for exponentially long times.
• Dimensionality Dependence:
  • $N > 2$: Strong fragmentation; exponential thermalization time.
  • $N = 2$: Weak fragmentation (sectors form a line, not a branching tree). The bottleneck is weak, leading to polynomial (diffusive) thermalization times ($t \sim L^2$).

E. Temperley-Lieb Models

In the supplementary material, the authors prove that for specific $SU(N)$ symmetric Temperley-Lieb models, even a single-site impurity fails to thermalize the system in the thermodynamic limit. The system retains memory of initial conditions indefinitely due to exponentially large degeneracies in the spectrum.

4. Significance

• New Mechanism for Slow Dynamics: The paper identifies a new route to anomalously slow thermalization that does not require disorder (unlike MBL) but relies on the topological structure of the Krylov space (bottlenecks).
• Experimental Relevance: It suggests that systems with approximate dynamical constraints (e.g., cold atoms with kinetic constraints, dipole-conserving systems) may exhibit glassy dynamics and slow relaxation even in the presence of weak perturbations or coupling to environments.
• Universality: The authors conjecture that this "bottleneck" mechanism is universal to all strongly fragmented systems, generalizing beyond pair-flip models to dipole-conserving and semigroup-based dynamics.
• Diagnostic Tools: It demonstrates that simple local observables (like charge relaxation) can serve as robust diagnostics for Hilbert space fragmentation, even when the constraints are not perfectly exact.

In summary, the paper establishes that Hilbert Space Fragmentation is robust against local perturbations, leading to exponentially slow thermalization driven by geometric bottlenecks in the configuration space, fundamentally altering the understanding of ergodicity breaking in clean quantum systems.