Fragmentation, Zero Modes, and Collective Bound States in Constrained Models

This paper investigates quantum kinetically constrained models with $U(1)$ symmetry to demonstrate how the interplay of constraints and chiral symmetry induces Hilbert space fragmentation and a parametric increase in zero modes, leading to the emergence of robust collective bound states that challenge ergodicity.

The Gist

Imagine you are hosting a massive, chaotic dance party in a giant hall. Usually, in physics, we expect that if you wait long enough, everyone will mix, mingle, and the energy of the party will spread evenly until it looks like a uniform, boring soup. This is called "thermalization."

But in this paper, the authors discover something strange happening in a specific type of quantum dance party. Instead of mixing, the dancers get stuck in little groups, refusing to move, or they form tight-knit circles that stay exactly the same no matter how much you expand the dance floor.

Here is a simple breakdown of what they found, using everyday analogies.

1. The "Traffic Jam" Rules (Kinetic Constraints)

Imagine a dance floor where people can only move if certain conditions are met. For example, in the "East Model," a dancer can only move to the right if there is already someone standing to their left. If you are the leftmost dancer, you are frozen in place because there is no one to your left to "unlock" your movement.

This creates a traffic jam. The rules of the game (the constraints) stop the dancers from spreading out freely. This is what physicists call "kinetic constraints."

2. The "Ghost Dancers" (Zero Modes)

In these constrained systems, the authors found a special group of dancers called Zero Modes. Think of them as "ghosts" at the party.

• They exist at a special energy level (zero energy).
• Because of the specific rules of the dance (chiral symmetry), these ghosts come in huge numbers.
• Usually, you'd expect a few ghosts, but because the dance floor is broken into disconnected rooms (a concept called Hilbert Space Fragmentation), the number of these ghosts explodes exponentially. It's like finding that every time you add a new room to the house, the number of ghosts doubles.

3. The "Self-Contained Bubbles" (Collective Bound States)

This is the paper's biggest discovery. The authors found that some of these ghost dancers form Collective Bound States.

The Analogy:
Imagine a group of friends holding hands in a tight circle.

• In a normal party: If you add more people to the room, the circle might break, or the friends might get distracted and wander off.
• In this quantum party: These friends form a "bubble." No matter how big you make the dance floor (adding more empty space around them), this specific group of friends stays exactly the same. They are robust. They don't care if the room gets bigger; they just stay in their little pocket of the universe.

The authors call these "Collective Bound States." They are like a self-contained ecosystem that refuses to interact with the rest of the system, even as the system grows.

4. The "Lego Bricks" (Factorizable States)

Because these bubbles are so stable, you can build bigger structures out of them.
Imagine you have a small, perfect Lego castle that doesn't fall apart. You can build a huge wall by just placing these small castles next to each other, separated by a few empty bricks.

• The authors showed that you can take these small "bound state" bubbles and string them together with empty space to create massive, complex states that still don't mix with the rest of the party.
• These are called Factorizable Eigenstates. They are essentially "Lego castles" made of smaller "Lego castles."

5. Why Does This Matter?

Why should you care about quantum ghosts and Lego castles?

• Breaking the Rules of Thermodynamics: Normally, quantum systems are supposed to forget their past and become random (thermalize). These bound states are "memory keepers." If you start the party with a specific pattern, these bound states ensure that part of that pattern survives forever, defying the usual rules of chaos.
• Quantum Computing: If you want to store information in a quantum computer, you need things that don't get messed up by noise or by the system getting bigger. These "bound states" are naturally protected. They are like a vault that stays locked even if you build a bigger building around it.
• New Physics: The authors showed this isn't just a fluke of one specific model. They found these "bubbles" in 1D lines, 2D grids, and even in systems that don't conserve particle numbers. It suggests that nature might be full of these "islands of stability" in a sea of chaos.

Summary

The paper is about discovering stable islands in a quantum ocean.

1. The Rules: Special movement rules create traffic jams.
2. The Result: These jams create a massive number of "ghost" states (Zero Modes).
3. The Discovery: Some of these ghosts form tight, unbreakable bubbles (Bound States) that stay the same even if you make the universe bigger.
4. The Application: You can stack these bubbles to build complex, stable structures that remember their past, offering a new way to think about quantum memory and stability.

It's like realizing that in a chaotic crowd, there are secret clubs that can form, stay together, and ignore the rest of the party forever, no matter how much the party grows.

Technical Summary

Here is a detailed technical summary of the paper "Fragmentation, Zero Modes, and Collective Bound States in Constrained Models" by Nicolau, Ljubotina, and Serbyn.

1. Problem Statement

The paper addresses the structure and implications of zero-energy modes (ZMs) in quantum kinetically constrained models (KCMs) that possess both chiral symmetry and U(1) particle-number conservation.

• Context: KCMs are known for slow relaxation and Hilbert space fragmentation. Their quantum counterparts often exhibit exponentially degenerate zero-energy subspaces due to chiral symmetry.
• Gap: While the existence of ZMs is known, their internal structure, the mechanism behind their exponential degeneracy in the presence of U(1) symmetry, and their relationship to ergodicity breaking (thermalization) are poorly understood. Specifically, it is unclear how particle conservation interacts with kinetic constraints to enhance the ZM subspace and whether these subspaces contain special non-ergodic eigenstates.

2. Methodology

The authors employ a combination of analytical graph theory, operator construction, and numerical exact diagonalization.

• Model Selection: They focus on two primary families of 1D models:
  • U(1) East: A model with kinetic constraints allowing hopping only if particles exist to the left (breaks inversion symmetry).
  • U(1) East-West: A model allowing hopping if particles exist to the left or right (restores inversion symmetry).
  • They also generalize to 2D (North-East model) and non-particle-conserving systems (Pair-flip model).
• Graph Representation: The Hamiltonian is treated as an adjacency matrix of a graph where vertices represent product states (Fock states) and edges represent allowed transitions.
  • Chiral Symmetry: Implies the graph is bipartite. The mismatch in the number of vertices between the two sublattices ($M = |N_{even} - N_{odd}|$) provides a lower bound on the number of ZMs.
• Fragmentation Analysis: They analyze how kinetic constraints disconnect the Hilbert space into dynamically isolated sectors, calculating the mismatch $M$ within each sector to determine the total ZM count.
• State Construction:
  • They define Collective Bound States (generalizing single-particle Compact Localized States) as many-body eigenstates robust to system size expansion.
  • They utilize Matrix Product Operators (MPOs) and unitary rotations within the degenerate ZM subspace to construct these states numerically.
  • They define Factorizable Zero Modes as tensor products of bound states separated by "decoupling" strings of empty sites.

3. Key Contributions

A. Parametric Enhancement of Zero Modes via Fragmentation

The authors demonstrate that Hilbert space fragmentation significantly increases the number of zero modes beyond the standard chiral symmetry lower bound.

• Mechanism: In the U(1) East model, kinetic constraints freeze particles, creating dynamically disconnected sectors.
• Result: The lower bound on ZMs is the sum of mismatches ($M_{frag}$) across all fragmented sectors. Because fragmentation creates sectors where the parity imbalance is reversed or amplified, the total number of ZMs grows exponentially faster with system size than in the unconstrained case.
• Comparison: The U(1) East-West model (inversion symmetric) has fewer frozen states and saturates the standard mismatch bound, whereas the U(1) East model (inversion breaking) exhibits a much larger ZM subspace due to extensive fragmentation.

B. Definition and Existence of Collective Bound States

The paper introduces the concept of Collective Bound States:

• Definition: Many-body eigenstates that remain eigenstates when the system size is increased by adding sites (padding) to the boundaries. They are localized in Fock space (low entanglement) and robust to system enlargement.
• Sufficient Conditions: The authors formulate three conditions for their existence:
  1. Recursivity: The graph of the larger system contains the smaller system's graph as an induced subgraph.
  2. Sparse Connectivity: The new vertices in the larger graph are not adjacent to the support of the bound state.
  3. Compactness: The wavefunction has zero amplitude on the vertices adjacent to the new sites.
• Construction: In the U(1) East model, these states are found by rotating the degenerate ZM subspace to find states with zero amplitude on the "active" boundary vertices.

C. Factorizable Zero Modes and Non-Ergodicity

• Factorizable ZMs: The authors show that bound states can be combined with "decoupling" strings (e.g., $r+1$ empty sites) to form factorizable eigenstates. These states have zero entanglement across specific cuts.
• Spectral Impact:
  • In the U(1) East model, factorizable states can exist at non-zero energies, "poisoning" the spectrum with non-thermal states away from $E=0$.
  • In the U(1) East-West model, factorizable states are confined strictly to the ZM subspace due to inversion symmetry.

D. Generalization to Other Models

• North-East (2D): Demonstrates that 2D bound states exist, forming tree-like structures in the many-body graph that host ZMs.
• Pair-Flip Model: Shows that bound states can exist even without U(1) symmetry and at non-zero energies, proving the generality of the mechanism.

4. Key Results

1. Exponential Growth of ZMs: In the U(1) East model, the number of ZMs scales exponentially with the number of particles ($N$), significantly exceeding the mismatch bound of the unfragmented system.
2. High Fraction of Bound States: Numerical results show that a large fraction (up to ~70-90% depending on range $r$) of the ZM subspace consists of factorizable states built from collective bound states.
3. Dynamical Signatures:
   • Fidelity Revivals: Initial states with overlap with bound states exhibit long-lived fidelity revivals, indicating a lack of thermalization (memory of initial conditions).
   • Robustness: These revivals persist even under perturbations that break kinetic constraints (uncorrelated hopping), provided the perturbation strength is not too large. The damping of revivals correlates with particle leakage through the "padding" sites.
4. Classical vs. Quantum Fragmentation:
   • Classical: Arises when bound states admit a product state basis (e.g., frozen particles).
   • Quantum: Arises when bound states are entangled superpositions (e.g., in the U(1) East model's ZM subspace), leading to fragmentation visible only in an entangled basis.

5. Significance

• Theoretical Framework: The work provides a rigorous definition and construction method for collective bound states in interacting many-body systems, extending the concept of Compact Localized States (CLS) from single-particle flat bands to constrained many-body physics.
• Ergodicity Breaking: It establishes a direct link between kinetic constraints, Hilbert space fragmentation, and the existence of non-ergodic eigenstates (ZMs and bound states) that violate the Eigenstate Thermalization Hypothesis (ETH).
• Experimental Relevance: The models discussed (East, East-West, North-East) are realizable on digital quantum simulators (e.g., Rydberg atom arrays, superconducting qubits) using controlled swap gates (Fredkin gates). The dynamical signatures (fidelity revivals) offer a clear experimental probe for these phenomena.
• New Perspective on Fragmentation: The paper suggests that studying bound states offers a new approach to distinguishing between classical and quantum Hilbert space fragmentation, potentially applicable to other models like the PXP model and dipole-conserving systems.

In summary, the paper reveals that the interplay between chiral symmetry, particle conservation, and kinetic constraints creates a rich landscape of zero-energy states. These states are not merely accidental degeneracies but are structured, robust collective bound states that prevent thermalization and can be systematically constructed and identified.