Quantum many-body scars leading to time-translation symmetry breaking in kicked interacting spin models

This paper demonstrates that a periodically kicked long-range interacting Ising model exhibits robust time-translation symmetry breaking and persistent period doubling for various initial states, driven by a minority of exponentially numerous Floquet eigenstates that display quantum many-body scars, $\pi$-spectral pairing, and long-range order.

The Gist

The Big Idea: A Quantum Clock That Ticks Twice as Slowly

Imagine you have a giant, complex machine made of billions of tiny spinning tops (these are the spins in the quantum model). Normally, if you shake this machine up and down in a regular rhythm (a "kick"), the tops eventually get so chaotic and hot that they stop caring about your rhythm. They just spin randomly, like a pot of boiling water. This is called thermalization.

However, this paper discovers a special trick. Under certain conditions, even though the machine is huge and complex, a specific group of tops refuses to get chaotic. Instead, they lock into a rhythm where they flip back and forth once every two kicks, instead of every single kick.

This is called Time-Translation Symmetry Breaking (or a "Time Crystal"). It's like a clock that ticks once every two seconds, even though you are pushing the button once every second. It breaks the rule that "what you put in is what you get out" in terms of timing.

The Cast of Characters

To understand how this works, we need to meet the "characters" in this quantum story:

1. The Crowd (Thermal States): Most of the spinning tops in the machine. If you look at them, they are chaotic, hot, and random. They represent the "normal" behavior of the universe.
2. The Scars (Quantum Scars): A small, special group of tops that refuse to join the chaos. They are like "scars" on the surface of the machine—remnants of a simpler, more ordered past that survived the chaos. They are the heroes of this story.
3. The Kicker: The external force that pushes the system periodically.
4. The Long-Range Connection: In this specific machine, every top can "feel" every other top, even if they are on opposite sides of the room. This is like a telepathic link connecting everyone instantly.

The Plot: How the Magic Happens

The scientists set up an experiment with two main ingredients:

1. Long-Range Telepathy: The spins talk to each other across the whole system.
2. The Kick: They push the system periodically.

They found that if they start the system in a very specific way (like arranging the tops in a specific pattern, such as a wall of "up" spins next to a wall of "down" spins, or tilting them all slightly), the system wakes up the Scars.

The "Doublet" Dance

Inside the machine, the special "Scar" states come in pairs, called doublets. Think of them as two dancers holding hands.

• The Rhythm: These two dancers are perfectly synchronized but slightly out of step with the Kicker. They swap places every two kicks.
• The Gap: There is a tiny, invisible gap between their energy levels. If this gap is perfect, they dance forever. If the gap is messy, they eventually get out of sync and stop dancing.

The paper shows that for these special starting patterns, the "Scar" dancers find a partner with a perfect gap, allowing them to dance in period doubling (2 kicks = 1 dance cycle) for a very, very long time.

The Surprise: It's Not Just One or Two

Usually, in quantum physics, if you have a huge system, the "special" states are so rare that they are like finding a single needle in a haystack. You would need a microscope to see them, and they wouldn't affect the whole machine.

But here is the twist:
The scientists found that while the "Scars" are still a minority compared to the chaotic crowd, their number is exponential.

• Imagine a room with 10 people. Maybe 1 is a Scar.
• Now imagine a room with 100 people. Maybe 10 are Scars.
• Now imagine a room with 1,000 people. Maybe 100 are Scars.

Even though they are still the minority, there are so many of them that they can't be ignored. If you start the machine with a random pattern, there is a very high chance you accidentally hit one of these Scar groups, and the whole system starts dancing in that special 2-kick rhythm.

The "Finite-Size" Secret

The researchers also looked at what happens as the machine gets bigger (more spins).

• They found that as the machine grows, the "gap" between the Scar dancers gets smaller and smaller.
• The Analogy: Imagine a tightrope walker. If the rope is short, a tiny wobble makes them fall. But if the rope is infinitely long and the wobble is microscopic, they can walk for a lifetime.
• Because the gap gets smaller as the system gets bigger, the "dance" (the period doubling) lasts longer and longer. In fact, the time it lasts grows exponentially with the size of the system. This means for a large enough system, the rhythm could last longer than the age of the universe.

Why This Matters

1. Defying Chaos: It shows that even in systems that should be chaotic and hot, order can persist if you know how to start them.
2. New Kind of Time Crystal: Previous time crystals needed "disorder" (like a messy room) to work. This paper shows you can get time crystals just by having long-range connections and specific starting patterns, without needing a messy room.
3. Quantum Memory: Because these rhythms are so stable and long-lasting, they could be useful for storing information in future quantum computers. If you can make a quantum system "remember" a rhythm for a long time, you can use it to store data.

Summary in One Sentence

The paper discovers that in a giant, telepathically connected quantum machine, a surprisingly large group of "special" particles can ignore the chaos and keep a perfect, slow rhythm forever, as long as you start the machine in the right way.

Technical Summary

1. Problem Statement

The paper investigates the emergence of Floquet time crystals (phases exhibiting persistent period-doubling oscillations) in a periodically driven, long-range interacting Ising spin chain.

• Context: Traditional Floquet time crystals often rely on disorder and Many-Body Localization (MBL) to prevent thermalization to an infinite-temperature state. However, disorder-free systems with long-range interactions are of significant interest.
• Challenge: In generic driven systems, the Eigenstate Thermalization Hypothesis (ETH) suggests that all eigenstates thermalize, destroying time-translation symmetry breaking (TTSB). The authors aim to demonstrate that TTSB can persist in a disorder-free, chaotic system if a specific subset of eigenstates—known as Quantum Scars—exhibits non-thermal properties.
• Specific Question: Can a kicked Ising model with power-law interactions support a time-crystalline phase driven by a minority of "scarred" Floquet states, even when the majority of the spectrum is thermal?

2. Methodology

The authors employ a combination of analytical framework and exact numerical diagonalization.

• The Model:

  • A 1D Ising model with long-range interactions decaying as $1/|i-j|^\alpha$.
  • Hamiltonian: $\hat{H} = -\frac{J}{K(N,\alpha)} \sum_{i,j} \frac{\hat{\sigma}^z_i \hat{\sigma}^z_j}{|i-j|^\alpha} + h_z \sum_i \hat{\sigma}^z_i + h_x \sum_i \hat{\sigma}^x_i$.
  • Driving: A periodic "kick" protocol consisting of unitary evolution under $\hat{H}$ for time $\tau$, followed by a global spin-flip kick $\hat{U}_K = e^{i(\pi/2 + \epsilon)\sum \hat{\sigma}^x_i}$.
  • Symmetries: The system utilizes periodic boundary conditions and exploits dihedral group symmetries (inversion and rotation) to reduce the Hilbert space dimension for numerical simulation.

• Diagnostic Tools for TTSB:
  To identify time-crystalline behavior, the authors analyze the properties of Floquet eigenstates $|\phi_p\rangle$ with quasienergies $\alpha_p$:

  1. $\pi$-Spectral Pairing: They identify doublets of states $|\phi_p\rangle, |\phi_q\rangle$ where the quasienergy difference is close to $\pi/\tau$. The "gap" is defined as $\Delta^\pi_p = \min_q ||\alpha_p - \alpha_q| - \pi/\tau|$. Small gaps imply coherent Rabi oscillations with period $2\tau$.
  2. Long-Range Order (Cat States): Within each doublet, they diagonalize the total magnetization operator $\hat{J}_z$. If the resulting eigenvalues $\Lambda_{p,\pm}$ are extensive (scaling with system size $N$), the doublet represents a superposition of macroscopically distinct states (a "cat state"), a prerequisite for TTSB.
  3. Overlap Analysis: They calculate the overlap between specific initial states $|\Psi(0)\rangle$ and the Floquet doublets. They define weighted distributions for the $\pi$-shifted gap and the normalized magnetization eigenvalues ($\lambda = \Lambda/N$).

• Initial States:

  • Domain Wall States: Classical configurations with specific arrangements of up/down spins (e.g., $|\uparrow\downarrow\dots\downarrow\uparrow\dots\rangle$).
  • Tilted States: Uniformly rotated spins $|\psi_t(\theta)\rangle = \bigotimes_i (\cos\theta|\uparrow\rangle + \sin\theta|\downarrow\rangle)$, allowing for finite-size scaling analysis.

3. Key Contributions

• Identification of Scars in Driven Systems: The paper demonstrates that in a kicked long-range Ising model, TTSB arises not from a fully non-thermal spectrum (as in MBL), but from a weak ergodicity breaking scenario where a minority of Floquet states are "scars" (non-thermal, low entanglement) embedded in a thermal bulk.
• Quantitative Link between Scars and Dynamics: The authors establish a rigorous connection between the initial state's overlap with specific scarred doublets and the persistence of period-doubling oscillations.
• Exponential Scaling of Scar Count: They show that while scars are a vanishing fraction of the Hilbert space in the thermodynamic limit, their absolute number grows exponentially with system size $N$. This explains why period doubling is observed for a wide variety of initial states.
• Robustness beyond Mean-Field: The phenomenon persists for power-law exponents $\alpha$ well beyond the mean-field regime ($\alpha > 1$), up to $\alpha \approx 3$, suggesting the mechanism is robust against interaction range variations.

4. Results

• Spectral Properties:

  • The average level spacing ratio $\langle r \rangle$ indicates that the bulk of the spectrum is chaotic (thermal, $\langle r \rangle \approx 0.52$, consistent with Circular Orthogonal Ensemble), not integrable.
  • However, a subset of doublets exhibits $\pi$-spectral pairing (very small $\Delta^\pi_p$) and extensive magnetization eigenvalues ($|\lambda| \sim O(1)$).
  • These "scarred" doublets have significantly lower half-chain entanglement entropy compared to the Page value (typical of thermal states), confirming their non-thermal nature.

• Dynamics of Initial States:

  • Domain Wall States: Initial state $|\psi_1\rangle$ (specific domain wall configuration) has a large overlap with the scarred doublets. It exhibits persistent period-doubling oscillations in magnetization. In contrast, state $|\psi_2\rangle$ overlaps mostly with thermal states and shows no period doubling.
  • Tilted States: For small tilt angles ($\theta = \pi/20, \pi/10$), the system shows period doubling. For larger angles ($\theta = \pi/5$), the overlap with scars diminishes, and oscillations vanish.

• Finite-Size Scaling:

  • For tilted states, the average logarithmic gap $\langle \log_{10} \Delta^\pi \rangle$ decreases linearly with system size $N$.
  • This implies the dephasing time scale $t^* \sim \exp(-\langle \log_{10} \Delta^\pi \rangle)$ grows exponentially with $N$.
  • The mode of the magnetization distribution ($\lambda_{max}$) remains non-zero and constant with $N$ for the oscillating cases, confirming the stability of the time-crystalline phase.

• Count of Scars:

  • The number of scarred states (defined by $|\lambda| > 0.2$) scales as $\log_{10}(\text{count}) \propto N$. While the fraction of scars vanishes as $N \to \infty$, the absolute number is exponential, ensuring that "typical" initial states will likely have non-zero overlap with them.

5. Significance

• New Mechanism for Time Crystals: The work provides a concrete example of a disorder-free Floquet time crystal driven by quantum scarring rather than disorder-induced localization. This expands the known universality classes of time-crystalline phases.
• Weak Ergodicity Breaking: It highlights a specific form of weak ergodicity breaking where the system is globally chaotic (thermal bulk) but retains memory of initial conditions due to a sparse set of non-thermal eigenstates.
• Experimental Relevance: Since the mechanism relies on long-range interactions (which can be engineered in trapped ions or Rydberg atom arrays) rather than disorder, it offers a viable pathway for observing time-crystalline behavior in current quantum simulators.
• Theoretical Insight: The paper clarifies that the existence of a time crystal does not require the entire spectrum to be non-thermal; a sufficiently large (exponentially growing) subset of scars is sufficient to sustain oscillations for experimentally relevant timescales.