Dressed Floquet scars from protected zero modes in a Rydberg chain

This paper presents an approximate analytic construction and numerical verification of two robust quantum many-body scars in a periodically driven Rydberg chain, which are protected by an index theorem and can be understood as dressed versions of the Rydberg vacuum and a volume-law scar.

The Gist

Imagine a crowded dance floor where everyone is trying to move randomly. In a typical quantum system (like the one described in this paper), if you start with a specific pattern of dancers, they will quickly lose that pattern, mix up completely, and eventually look like a random, chaotic mess. This is the "thermalization" the paper talks about—everything eventually forgets its starting point and becomes a hot, disordered soup.

However, this paper discovers two special "ghosts" on the dance floor that refuse to forget their starting moves, even as the music (the driving force) changes. These are called Quantum Many-Body Scars.

Here is a simple breakdown of what the researchers found, using everyday analogies:

1. The Setting: A Rigid Dance Floor

The scientists are studying a chain of atoms (like a line of dancers) that have a strict rule: No two neighbors can be "up" (excited) at the same time. This is called the "Rydberg blockade." It's like a dance floor where if one person jumps up, their immediate neighbors must stay seated.

They are also "driving" this system by changing the music rhythmically (periodically). Usually, this kind of rhythmic pushing makes the system heat up and forget its initial state very quickly.

2. The Discovery: Two Special "Dressed" Ghosts

The researchers found that despite the chaotic music, two specific starting patterns survive. They call these "Dressed Floquet Scars."

Think of these scars as two distinct dancers who manage to keep their original formation, but they get "dressed up" in a costume that changes slightly depending on how fast the music is playing.

• Dancer A: The "Empty Room" (Rydberg Vacuum)

  • The Start: Imagine a dance floor where everyone is sitting down (all "down" spins). This is a very simple, unentangled state.
  • The Magic: Even though the music is loud and chaotic, this "all-sitting" pattern doesn't dissolve. It survives, but it gets slightly "dressed" (modified) by the rhythm. The researchers found that even when the music is very slow or the volume is low (where the math usually breaks down), this pattern stubbornly refuses to thermalize. It's like a dancer who keeps sitting perfectly still even when the DJ is playing the wildest, most chaotic remix.

• Dancer B: The "Perfectly Correlated" Partner (Ivanov-Motrunich Scar)

  • The Start: Imagine a complex pattern where every dancer on the left side of the room is perfectly mirrored by a partner on the right side. This is a highly entangled, complex state.
  • The Magic: This pattern also survives, but it needs a specific "outfit change" (a mathematical rotation) to survive the driving force. The researchers found that if you rotate the dancers' positions by a specific angle based on the music's speed, this complex pattern becomes a "zero-energy" state that the system loves to stay in.
  • The Limit: This dancer is more fragile. If the music gets too slow, the "costume" falls apart, and the dancer eventually joins the chaotic crowd. The paper shows that this happens when the "real" part of the music's rhythm stops dominating the "imaginary" part (a technical way of saying the system becomes too random).

3. Why This Matters (The "Zero Mode" Concept)

In physics, there is a mathematical rule (an index theorem) that guarantees a huge number of "zero-energy" states exist in this system. Usually, these states are boring, featureless, and look like random noise (thermal).

The paper's big claim is that two of these zero-energy states are special. They aren't random noise; they are "dressed versions" of the two specific starting patterns mentioned above.

• They act like anchors. Even though the system is being pushed and pulled, these two states remember where they started.
• They are robust. They survive across a wide range of music speeds and volumes, not just at one perfect setting.

4. The "Dressing" Analogy

The term "Dressed" is key. Imagine you have a plain white t-shirt (the parent state).

• If you put it in a washing machine with a specific setting (the drive parameters), it comes out with a specific pattern of dye on it.
• The "Dressed Scar" is that t-shirt with the dye. It's still the same shirt (the memory of the parent state is there), but it looks different because of the environment.
• The researchers showed that they can predict what the "dye pattern" looks like using math, and they confirmed with computer simulations that these "dressed shirts" really do exist and stay intact for a long time.

Summary

The paper shows that in a quantum system of atoms that can't be neighbors when excited, there are two special "memory states."

1. One is a simple "all-sitting" state that is surprisingly tough and survives even when the math gets messy.
2. The other is a complex "mirror-image" state that survives as long as the rhythm isn't too slow.

These states are "protected" by the rules of the system, allowing them to resist the natural tendency of quantum systems to turn into random, thermal chaos. They are the exceptions to the rule that "everything eventually forgets its past."

Technical Summary

Technical Summary: Dressed Floquet scars from protected zero modes in a Rydberg chain

Problem Statement
The Eigenstate Thermalization Hypothesis (ETH) posits that generic eigenstates of interacting many-body quantum systems appear thermal for local observables. Quantum Many-Body Scars (QMBSs) represent a violation of ETH, manifesting as athermal mid-spectrum eigenstates that prevent thermalization for specific initial states. While QMBSs have been identified in static models (e.g., the PXP chain), their existence and structure in periodically driven (Floquet) systems remain less understood. A specific challenge in Floquet systems is that the effective Floquet Hamiltonian ($H_F$) is inherently nonlocal, even if the time-dependent Hamiltonian $H(t)$ is local. This nonlocality complicates the analytic construction of scars. Furthermore, while index theorems guarantee an exponentially large subspace of exact zero-energy modes (zero modes) in certain Floquet settings due to symmetry constraints, it is unclear whether these modes retain memory of specific initial states or simply act as infinite-temperature thermal states. This work addresses the construction and characterization of "dressed" QMBSs within the exponentially large nullspace of a Floquet Hamiltonian for a driven Rydberg chain.

Methodology
The authors investigate a periodically driven PXP model on a ring of $L$ Rydberg atoms with periodic boundary conditions. The system is subject to a strong Rydberg blockade, restricting the Hilbert space to states with no two adjacent Rydberg excitations. The drive is a piecewise-constant protocol where the detuning $\lambda(t)$ alternates between $-\lambda$ and $+\lambda$ with period $T$.

The study employs a dual approach:

1. Floquet Perturbation Theory (FPT): For high drive frequencies ($\omega_D = 2\pi/T$) and amplitudes, the authors construct an approximate analytic form of the Floquet Hamiltonian $H_F$ as a series expansion ($H_F = H_F^{(1)} + H_F^{(3)} + \dots$). They utilize a local unitary rotation $U(x)$, dependent on the drive parameter ratio $x = \lambda/(2\omega_D)$, to transform the leading-order term $H_F^{(1)}$ into a real symmetric operator. This allows them to identify "parent states" that are exact zero modes of the truncated Hamiltonian.
2. Exact Diagonalization (ED): To validate the perturbative findings and explore the non-perturbative regime, the authors perform ED on finite chains ($L \le 30$). They numerically compute the exact Floquet Hamiltonian via the matrix logarithm of the time-evolution operator $U(T,0)$. They calculate the overlap $W_0(\psi)$ between specific parent states and the exact zero-mode subspace of $H_F$ to quantify the "dressing" effect. They also analyze the Frobenius norm ratio of the real and imaginary parts of $H_F$ and time-averaged spin correlations to assess the stability and memory retention of these states.

Key Contributions and Results
The paper identifies two distinct "dressed" zero modes that persist as QMBSs over a range of drive parameters:

1. Dressed Ivanov-Motrunich (IM) Scar:

   • Parent State: A highly entangled zero mode of the undriven PXP model, characterized by perfect antipodal spin correlations.
   • Mechanism: The authors show that applying a local unitary rotation $U(x) = \exp(-i\pi x \sum_j \sigma_j^z / 2)$ to the IM scar creates a state $|\Lambda_r\rangle$ that is an exact zero mode of the leading-order perturbative Hamiltonian $H_F^{(1)}$.
   • Results: Numerical analysis reveals that $|\Lambda_r\rangle$ maintains a high overlap ($W_0 \approx 1$) with the exact zero-mode subspace at high frequencies. As $\omega_D$ decreases, the overlap drops in a "knee" regime where higher-order nonlocal terms ($H_F^{(3)}$) become significant. The survival of this scar is linked to the condition that the real part of $H_F$ dominates its imaginary part (measured by the Frobenius norm ratio $R$). When $R \approx 1$, the scar melts, and the system behaves like a generic complex Hermitian random matrix.

2. Dressed Vacuum Scar:

   • Parent State: The Rydberg vacuum state $|vac\rangle = |\downarrow \dots \downarrow\rangle$, which is completely unentangled and thermalizes rapidly in the static PXP model.
   • Mechanism: The vacuum state acts as a parent for a zero mode when the effective single-spin flip coefficient $w_{eff}$ (generated by $H_F$ acting on $|vac\rangle$) is minimized. This occurs at specific drive frequencies $\omega_D^f(\lambda)$.
   • Results: The authors find that $|vac\rangle$ retains a large overlap with the zero-mode subspace ($W_0 \sim O(1)$) even in the non-perturbative regime where $\lambda \sim O(1)$. Unlike the IM scar, the vacuum scar exhibits robust non-perturbative features. Even when $H_F$ becomes highly nonlocal, the dressed vacuum scar remains an outlier in terms of low entanglement entropy and low magnetization $\langle S^z \rangle$, distinct from the infinite-temperature ensemble. The overlap $W_0$ remains finite for system sizes up to $L=30$, suggesting potential stability in the thermodynamic limit for certain parameters.

Significance
The paper claims to provide a distinct mechanism for constructing QMBSs in interacting Floquet systems. By demonstrating that zero modes in the exponentially large nullspace of $H_F$ can be viewed as "dressed" versions of parent states with varying degrees of complexity (from unentangled to highly entangled), the work bridges the gap between protected zero modes and observable non-thermal dynamics.

Key implications highlighted by the authors include:

• Adiabatic Continuation: The dressed scars can be interpreted as an adiabatic continuation of parent states, where the "dressing" is a function of drive parameters.
• Robustness: The existence of these scars is not limited to the perturbative regime; the dressed vacuum scar, in particular, survives when the Floquet Hamiltonian loses any local representation.
• Hierarchy of Complexity: The work suggests the existence of a hierarchy of anomalous zero modes with intermediate complexities between the two extremes studied, pointing to a broader structure of protected states in Floquet systems.

The study concludes that understanding the structure of these protected zero modes offers insight into weak ergodicity breaking in driven quantum matter, though the non-perturbative behavior of the dressed vacuum scar remains an open issue for future investigation.