Floquet scars and prethermal fragmentation in a driven spin-one chain

Imagine you have a long row of spinning tops (a "spin chain"). Usually, if you poke them or shake them, they eventually get chaotic, mix up their energy, and settle into a messy, random state. This is called thermalization—like stirring sugar into hot coffee until it's all sweet and uniform.

But this paper explores what happens when you shake these tops in a very specific, rhythmic pattern (a "square-pulse drive"). The researchers found that depending on how fast you shake them, the tops can do three very different, surprising things:

The "Ghost" Dance (Quantum Scars): They keep dancing in a perfect, repeating loop, refusing to get messy.
The "Traffic Jam" (Hilbert Space Fragmentation): The row of tops gets stuck in different lanes that never mix with each other.
The "Freeze" (Prethermalization): They stay frozen in place for a very long time before finally giving in to chaos.

Here is a breakdown of the paper's discoveries using simple analogies.

1. The Setup: A Row of Spinning Tops

The scientists are studying a chain of particles that can spin in three directions (x, y, or z).

The Rule: There is a hidden rule (a "conserved quantity") for every pair of neighbors. Think of it like a traffic light between every two cars. Some lights are always green ( $W=1$ ), and some follow a pattern like Green-Green-Red ( $W=\{1, 1, -1\}$ ).
The Drive: They shake the whole chain with a strong, rhythmic pulse. Imagine a giant hand slapping the tops back and forth.

2. The Fast Shake: "Quantum Scars" (The Ghosts)

The Scenario: You shake the tops very, very fast (high frequency).
What Happens: Instead of getting chaotic, the system remembers its starting position. If you start with a specific pattern, the tops will oscillate back and forth, returning to that exact pattern over and over again.
The Analogy: Imagine a group of people in a crowded room. Usually, if you tell them to move, they bump into each other and scatter randomly. But in this "Scar" state, it's like they are all wearing invisible ghost costumes. They pass right through each other and return to their original spots perfectly, ignoring the chaos that should be happening.
Why it matters: This violates the normal laws of physics (thermalization) and suggests a special, protected order in the quantum world.

3. The "Just Right" Shake: Hilbert Space Fragmentation (The Traffic Jam)

The researchers found two special "sweet spot" frequencies where the shaking creates a massive traffic jam. They call this Hilbert Space Fragmentation (HSF).

Think of the "Hilbert Space" as a giant parking lot where every possible arrangement of the spinning tops is a parking spot.

Normal Physics: Cars can drive from any spot to any other spot.
Fragmentation: The parking lot suddenly gets divided into isolated islands. Once a car (the system) is on one island, it can never cross to another.

The paper found two types of islands:

A. Strong Fragmentation (The Big Island)

When: At specific frequencies like $\omega = Q_0 / (2n)$ .
What Happens: The parking lot breaks into exponentially many tiny islands. Most of the time, the system gets stuck in one tiny island and can't escape.
The Difference:
- In the All-Green sector (all $W=1$ ), the biggest island is still chaotic (ergodic). The tops move around wildly within that island, but they can never leave it.
- In the Patterned sector (Green-Green-Red), the biggest island is Integrable. This is like a perfectly choreographed dance. The tops move in a simple, predictable rhythm forever. They don't get chaotic at all.

B. Weak Fragmentation (The Small Islands)

When: At slightly different frequencies like $\omega = Q_0 / (2n+1)$ .
What Happens: This only happens in the "All-Green" sector. The parking lot breaks into islands, but there aren't that many of them (only a few more as the chain gets longer). It's a mild traffic jam, not a total gridlock. The "Patterned" sector doesn't get jammed at these frequencies; it just flows normally.

4. The "Frozen" State

When the system is in these fragmented islands, it behaves like it's frozen in time.

The Analogy: Imagine you are in a room with a locked door. You can run around inside the room (dynamics), but you can never get out. If you start in a corner, you might stay in that corner for a very long time, or bounce around in a small circle, but you will never reach the other side of the room.
The Result: The system retains "memory" of where it started for a very long time (prethermalization). It only eventually breaks free and becomes chaotic after a time that grows exponentially with how hard you shake it.

Summary of the "Phase Diagram"

The paper draws a map (Figure 11) showing what happens based on how fast you shake the tops:

Very Fast: The tops dance in a loop (Scars).
Medium/Slow: The tops get chaotic and mix (Thermalization).
Special Frequencies: The tops get stuck in isolated lanes (Fragmentation).
- Some lanes are chaotic but trapped.
- Some lanes are perfectly ordered and never chaotic.

Why Should We Care?

This isn't just about spinning tops.

Memory: If we can keep quantum systems in these "fragmented" or "scarred" states, they won't lose their information to chaos. This is crucial for building quantum computers that don't crash.
Control: It shows us how to use rhythm and timing to force nature to behave in ways it normally wouldn't, creating new states of matter that are stable and predictable.

In short, the paper shows that by tapping a quantum system at the exact right rhythm, you can turn a chaotic mess into a perfectly ordered dance, or trap it in a tiny, isolated room where it can never get lost.

Here is a detailed technical summary of the paper "Floquet scars and prethermal fragmentation in a driven spin-one chain" by Krishanu Ghosh, Diptiman Sen, and K. Sengupta.

1. Problem Statement

The paper investigates the non-equilibrium dynamics of a periodically driven (Floquet) spin-1 chain, specifically a variant of the Kitaev model. The system is driven by a square-pulse protocol with amplitude $Q_0$ and frequency $\omega_D$ . The central challenge addressed is understanding how such driven many-body systems thermalize or fail to thermalize.

Key phenomena of interest include:

Quantum Many-Body Scars (QMBS): Special eigenstates that violate the Eigenstate Thermalization Hypothesis (ETH), leading to persistent oscillations and fidelity revivals.
Hilbert Space Fragmentation (HSF): The breaking of the Hilbert space into dynamically disconnected sectors (fragments), preventing thermalization.
Prethermalization: A long-lived regime where the system is governed by an effective Hamiltonian before eventually heating up to infinite temperature.

The authors specifically focus on a spin-1 chain with $Z_2$ -valued conserved quantities ( $W_\ell$ ) on the links, which naturally fragments the Hilbert space. They aim to map out the phase diagram of this driven system as a function of drive frequency and amplitude, identifying regimes of scars, ergodicity, and strong/weak fragmentation.

2. Methodology

The study employs a combination of analytical perturbative techniques and exact numerical simulations:

Model Hamiltonian: The system is defined by $H = H_0 + H_1$ , where $H_0 = J \sum S^x_j S^y_{j+1}$ and $H_1 = Q(t) \sum (S^x_j S^y_{j+1})^2$ . The drive $Q(t)$ is a square pulse of amplitude $Q_0$ and period $T = 2\pi/\omega_D$ .
Conserved Quantities: The model possesses local conserved quantities $W_\ell = \Sigma^y_j \Sigma^x_{j+1}$ $W_{ℓ} = Σ_{j}^{y} Σ_{j + 1}^{x}$ (where $\Sigma^\alpha = e^{i\pi S^\alpha}$ $Σ^{α} = e^{iπ S^{α}}$ ), which are $Z_2$ $Z_{2}$ valued ( $\pm 1$ $\pm 1$ ). This allows the study of dynamics within specific sectors defined by fixed patterns of $W_\ell$ $W_{ℓ}$ . Two sectors are analyzed:
1. Sector A: All $W_\ell = 1$ .
2. Sector B: Periodic pattern $W_\ell = \{\dots, 1, 1, -1, 1, 1, -1, \dots\}$ .
Floquet Perturbation Theory (FPT): In the large drive amplitude regime ( $Q_0 \gg J$ ), the authors derive the first-order Floquet Hamiltonian ( $H_F^{(1)}$ ) using perturbation theory. This effective Hamiltonian governs the prethermal dynamics.
Transfer Matrix Methods: To analytically count the number of fragments and the size of the largest fragment (Hilbert Space Dimension, HSD) at special drive frequencies, the authors construct transfer matrices based on the constraints imposed by $H_F^{(1)}$ .
Exact Diagonalization (ED): Numerical simulations are performed on finite chains of length $L \le 24$ $L \leq 24$ . The authors compute:
- Entanglement Entropy ( $S$ ): To detect thermalization vs. fragmentation.
- Fidelity ( $F$ ): To detect quantum scars and revivals.
- Correlation Functions ( $C_0$ ): To track dynamical freezing.

3. Key Contributions and Results

A. High-Frequency Regime ( $\hbar\omega_D \gg Q_0$ )

Quantum Scars: In this regime, $H_F^{(1)} \approx H_0$ . The system exhibits quantum many-body scars.
Dynamics: For specific initial states (e.g., Néel-like states), the system shows oscillatory dynamics and periodic fidelity revivals, violating ETH.
Transition: As $\omega_D$ is lowered, the system crosses over to fast thermalization consistent with the Floquet ETH.

B. Special Drive Frequencies and Hilbert Space Fragmentation

The authors identify two classes of special drive frequencies where the system exhibits distinct prethermal fragmentation behaviors:

1. Strong HSF at $\omega^*_n = Q_0 / (2n\hbar)$
At these frequencies, the first-order Floquet Hamiltonian simplifies to $H_F^{(1)} \propto H_a$ (or $H'_a$ in Sector B), which contains projection operators that annihilate a vast number of states.

Sector A ( $W_\ell=1$ ): The Hilbert space undergoes strong fragmentation. The number of fragments grows exponentially with system size $L$ $L$ . The largest fragment is ergodic (thermalizes within itself).
- Signature: Entanglement entropy saturates to different values depending on the initial state (violating ETH).
Sector B ( $W_\ell=\{1,1,-1...\}$ ): Also exhibits strong fragmentation. However, the largest fragment is integrable.
- Signature: The largest fragment maps to a set of decoupled spin-1/2 systems ( $H_{eff} \propto \sum \tau^x_k$ ). This leads to perfect periodic revivals in fidelity and oscillatory entanglement entropy between $S=0$ and $S=\ln 2$ .

2. Weak HSF at $\omega'_n = Q_0 / [\hbar(2n+1)]$

Sector A: Exhibits weak fragmentation. The number of fragments grows only linearly with $L$ $L$ , while the largest fragment size grows exponentially.
- Signature: Entanglement entropy saturates to a few distinct values (not a continuum), indicating a small number of disconnected sectors.
Sector B: No fragmentation occurs at these frequencies; the system thermalizes.

C. Prethermal Timescales

The prethermal regime, governed by $H_F^{(1)}$ , persists for a timescale $t_p$ that grows exponentially with the drive amplitude $Q_0$ .
Deviations from the prethermal behavior (due to higher-order terms in the Floquet Hamiltonian) eventually lead to thermalization, but this occurs on extremely long timescales for large $Q_0$ .

D. Numerical Verification

Entanglement: Numerical results confirm that at $\omega^*_1$ , $S$ saturates to values $< S_{Page}$ (the Page value for the full sector), with the specific value depending on the initial state.
Fidelity: Shows perfect revivals for integrable fragments (Sector B) and strong finite-size fluctuations for ergodic fragments (Sector A).
Correlations: Correlation functions remain pinned to initial values for long times at $\omega^*_n$ , indicating dynamical freezing.

4. Significance

Unified Framework: The paper provides a comprehensive phase diagram for a driven spin-1 chain, unifying the concepts of Floquet scars, ETH, and both strong and weak Hilbert space fragmentation.
Integrability vs. Ergodicity in Fragments: A crucial finding is that strong HSF does not imply integrability for all fragments. The authors demonstrate that the largest fragment can be ergodic (Sector A) or integrable (Sector B) depending on the specific sector constraints, offering a nuanced view of fragmentation.
Experimental Relevance: The model is related to experimentally realizable Rydberg atom chains (PXP model) and spin-1 systems in optical lattices. The predicted signatures (oscillatory entanglement, fidelity revivals, and distinct saturation values) provide clear targets for experimental verification of prethermal HSF.
Control of Heating: The identification of regimes where heating is suppressed for exponentially long times (prethermal HSF) is significant for the stability of driven quantum systems.

In summary, the work establishes that by tuning the drive frequency of a spin-1 chain, one can access distinct dynamical phases ranging from scar-induced oscillations to various forms of Hilbert space fragmentation, with the nature of the fragmentation (strong/weak, ergodic/integrable) being highly sensitive to the specific symmetry sector of the system.