Floquet-Thermalization via Instantons near Dynamical Freezing

By employing Floquet flow-renormalization, this study reveals that periodically driven quantum many-body systems near dynamical freezing exhibit a universal approach to thermalization characterized by an unstable fixed point with emergent symmetry and a series of instanton events that significantly delay thermalization compared to conventional prethermal regimes.

The Gist

Imagine you have a chaotic, noisy room full of people (a quantum system) who are constantly talking, moving, and interacting. In the world of physics, this chaos usually leads to thermalization: eventually, everyone forgets who they were, where they started, and just becomes a uniform, hot soup of random energy. This is the "default" fate of almost any closed quantum system.

However, scientists have discovered a weird trick called Dynamical Freezing. If you shake this room at just the right rhythm and intensity (a specific frequency and amplitude), the people suddenly stop moving chaotically. They seem to "freeze" in place, remembering their starting positions for a very long time. It's like hitting a perfect note on a guitar string that makes the whole room vibrate in perfect harmony instead of noise.

This paper explores how this freezing happens, why it eventually breaks down, and what happens in the messy middle. Here is the story in simple terms:

1. The Problem: The "Infinite Heat" Trap

In a normal, undriven system, things settle down to a comfortable temperature. But in a Floquet system (one that is being constantly shaken or driven by an external force, like a laser or magnetic field), there is no "comfortable" temperature. The system wants to absorb energy forever until it reaches "infinite temperature"—a state of total, featureless chaos where all memory of the past is erased.

2. The Discovery: The "Magic Shake" (Dynamical Freezing)

The authors found that if you shake the system at a very specific ratio of speed to strength, the chaos is suppressed. The system enters a Prethermal state.

• The Analogy: Imagine a child on a swing. If you push them at random times, they flail. But if you push them at the exact right moment in their swing cycle, they go higher and higher in a smooth, predictable arc.
• In this "frozen" state, the system develops a hidden rule (an emergent conservation law) that keeps it organized. It remembers its initial state for a long time, resisting the urge to turn into hot soup.

3. The New Tool: The "Slow-Motion Filter" (Flow-Renormalization)

Previous studies tried to understand this using two methods:

1. Math approximations (like guessing the answer based on the first few terms of a formula).
2. Computer simulations (trying to calculate every single particle, which is impossible for big systems).

Both methods failed to explain what happens after the initial freezing, or why the freezing eventually stops.

The authors used a new technique called Floquet Flow-Renormalization.

• The Analogy: Imagine you are looking at a blurry, fast-moving video of a car race.
  • Old methods tried to guess the car's path by looking at the first few frames or by freezing the video.
  • The New Method is like a magical filter that slowly zooms out and blurs the details. It gradually removes the "noise" (the fast shaking) from the video, leaving you with a clear, slow-motion view of the car's true path.
• This "flow" allows them to watch the system evolve from its messy, shaking start into a stable, frozen state, and then watch what happens when that stability finally cracks.

4. The Breakdown: "Quantum Tunneling" (Instantons)

The big surprise in this paper is what happens after the system stays frozen for a while. It doesn't just suddenly melt; it breaks down in a very specific, step-by-step way.

The authors discovered that the system escapes the frozen state through events they call Instantons.

• The Analogy: Imagine the system is a ball sitting in a deep valley (the frozen state). To get to the other side (the chaotic, hot state), the ball has to climb a huge mountain.
  • In classical physics, the ball would need infinite energy to climb over.
  • In quantum physics, the ball can tunnel through the mountain.
• The authors found that the system doesn't just tunnel once. It tunnels in a series of small jumps.
  • The system sits in a "valley" (a temporary stable state) for a long time.
  • Then, it suddenly "tunnels" to a slightly different valley.
  • It sits there again, then tunnels again.
  • Each tunneling event is an Instanton.

These tunneling events are the "leaks" in the dam. They are rare and slow, which is why the system stays frozen for so long, but they are inevitable. Eventually, after enough of these jumps, the system loses its memory and turns into the hot, chaotic soup.

5. The Big Picture

• Freezing is an illusion: It's not a permanent stop; it's just a very long pause.
• The "Magic" is approximate: The freezing only works perfectly if you shake the system infinitely fast. In the real world, it's just very good for a while.
• The Path to Chaos: The journey from "frozen" to "chaotic" isn't a smooth slide. It's a staircase. The system hops down one step (an instanton), waits, hops down another, and so on.

Why Does This Matter?

Understanding this "slow leak" is crucial for building quantum computers.

• Quantum computers are very fragile; they lose their information (decohere) too quickly.
• If we can use "Dynamical Freezing" to keep the system in a stable, memory-retaining state for longer, we can perform more complex calculations.
• This paper tells us exactly how long we can expect that stability to last and how it will eventually fail, giving engineers a roadmap to design better, more stable quantum machines.

In summary: The paper uses a new "slow-motion filter" to watch a quantum system get frozen by a rhythmic shake. It discovers that this freeze isn't permanent; the system slowly escapes through a series of quantum "tunneling" jumps (instantons) until it finally succumbs to chaos.

Technical Summary

Here is a detailed technical summary of the paper "Floquet-Thermalization via Instantons near Dynamical Freezing" by Mukherjee, Guo, and Chowdhury.

1. Problem Statement

Periodically driven (Floquet) quantum many-body systems generally thermalize to an infinite-temperature state due to the lack of energy conservation, erasing memory of the initial state. However, recent studies have identified a phenomenon called Dynamical Freezing, where specific ratios of drive amplitude ($A$) and frequency ($\Omega$) lead to emergent conservation laws and slow entanglement growth, effectively "freezing" the system's dynamics.

Key Challenges Addressed:

• Limitations of Existing Methods: Previous studies relied heavily on the Floquet-Magnus expansion (perturbative, often divergent for generic interacting systems) and Exact Diagonalization (ED) (limited to small system sizes and finite times). Neither method systematically captures the slow approach to thermalization or the non-perturbative mechanisms governing the breakdown of freezing at asymptotically late times.
• Open Questions:
  • What determines the precise location of freezing points in generic non-integrable systems?
  • Is dynamical freezing exact in the thermodynamic limit, or does it eventually decay?
  • What is the mechanism driving the eventual thermalization away from the frozen state?

2. Methodology: Floquet Flow-Renormalization (fRG)

The authors employ a Floquet flow-renormalization group (fRG) framework. This approach generates a continuous sequence of infinitesimal unitary transformations to decouple the time-dependent driving component of the Hamiltonian from an effective static Hamiltonian.

• Flow Equations: The system evolves in a fictitious "fRG time" ($\lambda$) governed by coupled differential equations for the static part $H_0(\lambda)$ and the oscillatory part $H_1(\lambda)$:
  $$\partial_\lambda H_0(\lambda) = 2[H_1(\lambda), H_1^\dagger(\lambda)]$$
  $$\partial_\lambda H_1(\lambda) = -\Omega H_1(\lambda) - [H_0(\lambda), H_1(\lambda)]$$
  Here, $H(t) = H_0 + H_1 e^{i\Omega t} + H_1^\dagger e^{-i\Omega t}$. The flow preserves the spectrum of the stroboscopic unitary evolution operator.

• Numerical Implementation:

  • Exact Diagonalization (ED): Used for small system sizes ($L \le 10$) to solve the flow equations exactly using Runge-Kutta methods.
  • Matrix Product Operators (MPO): Used for larger systems ($L \le 40$) to handle the operator growth, utilizing the ITensor library.
  • Analytical Treatment: Perturbative expansions for early times (ramp-up) and non-perturbative analytical solutions for late-time instanton events.

3. Key Contributions

1. Universal Description of Freezing: The paper provides a universal framework describing dynamical freezing not as a static property, but as a trajectory in the space of Hamiltonians flowing through a sequence of unstable fixed points.
2. Identification of Instanton Events: The authors identify that the breakdown of the prethermal regime (and the eventual thermalization) occurs via a series of instanton events. These are non-perturbative "tunneling" events in fRG time where the system rapidly transitions between intermediate fixed points.
3. Analytical vs. Perturbative Distinction: The work demonstrates that while the early-time flow matches the Floquet-Magnus expansion, the flow formalism yields a gauge-invariant effective Hamiltonian. Crucially, it reveals that the leading-order correction to the Magnus expansion is of order $O(1/\Omega^2)$, differing from the standard $O(1/\Omega)$ gauge-dependent terms.
4. Thermalization Mechanism: The paper establishes that thermalization is driven by the reorganization of the energy spectrum (band folding) via these instanton events, which occur when matrix elements of $H_1(\lambda)$ grow large enough to couple eigenstates separated by energy differences $\sim \Omega$.

4. Key Results

A. The Flow Trajectory and Fixed Points

The flow trajectory exhibits three distinct regimes (see Fig. 1 in the paper):

1. **Ramp-up Regime ($0 \le \lambda \lesssim \Omega^{-1}$):** Rapid renormalization of$H_0$. The flow can be described perturbatively.
2. Prethermal Regime ($\Omega^{-1} \lesssim \lambda \lesssim \lambda_{min}$): $H_1(\lambda)$ decays exponentially ($\sim e^{-\Omega \lambda}$). The system approaches an unstable prethermal fixed point ($P$) characterized by an emergent approximate symmetry.
   • At freezing points, the commutator $[H_0(\lambda), Q]$ (where $Q$ is the emergent charge) becomes very small, scaling as $O(1/\Omega^2)$.
   • This leads to a plateau in the commutator norm $P(\lambda)$, indicating slow dynamics.
3. Thermalization Regime ($\lambda \gtrsim \lambda_{min}$): The system flows away from the prethermal fixed point.

B. Instanton Events

• Mechanism: As $\lambda$ increases beyond $\lambda_{min}$, $H_1(\lambda)$ stops decaying and begins to grow. This growth is associated with instanton events.
• Signature: In the fRG time evolution, these appear as sharp peaks in the Frobenius norm $\|H_1(\lambda)\|$ and step-like features in the static Hamiltonian $H_0(\lambda)$.
• Physics: An instanton corresponds to a rapid hybridization of two eigenstates of $H_0(\lambda)$ whose energy difference is close to $\Omega$. This "folds" the energy spectrum, reducing the bandwidth until it fits within an interval of width $\Omega$.
• Delay at Freezing: At dynamical freezing points, the onset of these instanton events is significantly delayed compared to non-frozen points, extending the prethermal lifetime.

C. Numerical Diagnostics

• Operator Entanglement Entropy (OPEE):
  • For low frequencies ($\Omega < \Omega^*(L)$), OPEE saturates to a volume-law value consistent with Random Matrix Theory (RMT), indicating full thermalization.
  • For high frequencies, OPEE at freezing points saturates to a lower, sub-volume-law value, preserving memory of the initial state.
  • The threshold frequency $\Omega^*(L)$ weakly increases with system size $L$.
• Diagonal Ensemble (DE) Average:
  • At freezing points, the system retains memory of the initial state (expectation values remain near initial values) for longer times.
  • However, this memory is not exact; it decays eventually, confirming that freezing is an approximate phenomenon at any finite $\Omega$.

D. Analytical Insights

• Scaling: The violation of the emergent symmetry scales as $1/\Omega^2$.
• Timescales: The prethermal lifetime scales exponentially with frequency: $t_{pre} \sim \exp(\Omega/J)$. The instanton events occur at fRG times $\lambda \sim \lambda_{min} \sim \frac{1}{J} \ln(\Omega/J)$.
• Gauge Invariance: The flow formalism naturally averages over the gauge dependence of the initial time $t_0$, providing a more robust description of the effective Hamiltonian than the standard Magnus expansion.

5. Significance and Outlook

• Beyond Perturbation Theory: This work moves beyond the limitations of the Floquet-Magnus expansion, offering a non-perturbative understanding of how generic driven systems thermalize.
• Universal Mechanism: The identification of instanton-mediated thermalization provides a new paradigm for understanding the breakdown of prethermalization in driven systems, applicable beyond the specific spin chain model studied.
• Experimental Relevance: The results suggest that while dynamical freezing can protect quantum states for exponentially long times, it is not a permanent solution to heating in the thermodynamic limit unless the drive frequency is infinitely high or the system size is finite. The "step-like" evolution of the effective Hamiltonian suggests a discrete, quantized absorption of energy quanta ($\Omega$) in real-time dynamics.
• Future Directions: The authors propose that the intermediate fixed points between instantons correspond to new "prethermal" regimes. Verifying this requires simulating real-time dynamics over exponentially long times, a challenge for current numerical methods but a promising avenue for future research.

In summary, the paper establishes that dynamical freezing is an asymptotic phenomenon characterized by a sequence of unstable fixed points connected by non-perturbative instanton events, which ultimately drive the system toward thermalization.