Finite-time Scaling with Arbitrary Driving Rates: Bridging the Kibble-Zurek and De Grandi-Gritsev-Polkovnikov Limits

This paper establishes a generalized finite-time scaling framework that unifies the Kibble-Zurek and De Grandi-Gritsev-Polkovnikov limits, providing a universal description of driven critical dynamics across the full range of driving rates in quantum many-body systems.

The Gist

Imagine you are trying to cross a busy, chaotic intersection (the "critical point") where traffic rules change instantly. How you cross depends entirely on how fast you drive your car (the "driving rate").

For decades, physicists have had two different rulebooks for this intersection, but they only worked in extreme situations:

1. The Slow Driver (Kibble-Zurek): If you drive very slowly, you have time to react to every change in the road. You can smoothly navigate the chaos, and the number of "accidents" (defects) you cause follows a predictable pattern based on your speed.
2. The Instant Jumper (De Grandi-Gritsev-Polkovnikov): If you teleport instantly from one side of the intersection to the other, you don't react to the road at all. You just land where you are, and the number of accidents depends entirely on where you started and where you landed, ignoring the speed of the jump.

The Problem:
What happens if you drive at a medium speed? Or if you start your journey right in the middle of the chaos, rather than far away from it? The old rulebooks said, "We don't know," or "This only works if you start far away and drive slowly." They hit a wall: if you drove too fast, the "Slow Driver" math broke down.

The New Discovery:
This paper introduces a Universal GPS (a new mathematical framework called "Generalized Finite-Time Scaling") that works for any speed, from a slow crawl to a lightning-fast jump, as long as you are driving within the chaotic intersection itself.

Here is how the authors explain it using simple concepts:

1. The "Freeze" vs. The "Memory"

• The Old View: The authors explain that in the past, if you drove too fast, the system would "freeze" before it even reached the chaotic center. It was like trying to take a photo of a speeding car with a slow camera; the image would be blurry and useless. The old math required the "freeze" to happen inside the chaotic zone, which limited how fast you could go.
• The New View: The authors realized that if you start your journey inside the chaotic zone, the system never really "freezes" in a way that breaks the rules. Instead, the system keeps a memory of where it started.
  • Slow Speed: The system forgets where it started and just follows the traffic rules (the critical point).
  • Fast Speed: The system remembers its starting point vividly. It's like a runner who starts in the middle of a storm; if they sprint, they carry the memory of the wind's direction with them.

2. The Unified Equation

The paper proposes a single, master equation (Equation 3 in the text) that acts like a Swiss Army Knife for physics.

• If you plug in a slow speed, the equation automatically simplifies to the old "Slow Driver" rule.
• If you plug in a fast speed, it automatically simplifies to the "Instant Jumper" rule.
• If you plug in any speed in between, it gives you the correct answer, blending the two behaviors seamlessly.

3. The Proof (The Simulation)

To prove this wasn't just a pretty theory, the authors ran computer simulations (like a video game) using two different "worlds":

• World 1: A standard magnetic chain (the Quantum Ising model).
• World 2: A more complex, exotic magnetic system (the Tricritical point).

In both worlds, they tested driving speeds ranging from very slow to extremely fast.

• The Result: When they used the old math, the data points scattered like confetti and didn't line up. But when they used their new "Universal GPS," all the data points from slow, medium, and fast speeds collapsed perfectly onto a single, smooth line.

The Takeaway

The paper claims to have found a single, universal language to describe how quantum systems behave when pushed through a phase transition, regardless of how fast you push them.

It bridges the gap between the "slow and steady" world and the "fast and furious" world. It tells us that as long as we are inside the critical region, the system's behavior is always predictable and follows a specific scaling law, provided we account for the system's memory of where it started. This unifies two previously separate theories into one complete picture.

Technical Summary

Technical Summary: Finite-time Scaling with Arbitrary Driving Rates

Problem Statement
The paper addresses a fundamental gap in the theoretical description of nonequilibrium critical dynamics in quantum many-body systems. While two distinct regimes are well-understood, a unified framework for arbitrary driving rates within the critical region has been absent.

1. The Kibble-Zurek (KZ) Limit: For slow, near-adiabatic ramps starting far from the critical point, the KZ mechanism predicts a specific scaling of topological defect density ($n \propto R^{d/r}$). However, this theory relies on an "adiabatic-impulse-adiabatic" scenario where the system freezes only within the critical region. This imposes a strict upper bound on the driving rate $R$; if $R$ is too high, the impulse regime extends beyond the critical region, breaking the universality of the scaling.
2. The De Grandi-Gritsev-Polkovnikov (DGP) Limit: For sudden quenches starting exactly at the critical point, the defect density follows a different scaling ($n \propto |g_f|^{d\nu}$), determined by projecting the initial state onto the final Hamiltonian's eigenstates.
3. The Gap: Existing theories treat these as separate limiting cases. It was unclear how to describe driven dynamics restricted entirely within the critical region for arbitrary driving rates, particularly when the initial state is close to (but not exactly at) the critical point.

Methodology
The authors develop a Generalized Finite-Time Scaling (FTS) framework to bridge these limits.

• Theoretical Formulation: They propose a generalized scaling form for a macroscopic quantity $P$ (such as defect density or order parameter) driven by a rate $R$ within the critical region:
  $$P[\lambda_i, \lambda(t), R] = R^{\kappa/r} f[g_i R^{-1/\nu r}, g(t) R^{-1/\nu r}, \{X R^{-x/r}\}]$$
  Here, $g_i = \lambda_i - \lambda_c$ and $g(t) = \lambda(t) - \lambda_c$ represent the distance from the critical point. Crucially, unlike previous FTS forms, this equation explicitly incorporates the initial parameter $g_i$ as a scaling variable, removing the requirement for an upper bound on $R$.
• Numerical Verification: The theory is tested using two distinct models:
  1. 1D Quantum Ising Model: A standard critical point with known exponents ($\beta=1/8, \nu=1, z=1$). The system is quenched across the critical point with initial and final parameters close to $\lambda_c$.
  2. Tricritical Point Model: A more complex model involving coupled spin chains (relevant to Rydberg atom systems) exhibiting a tricritical point ($\beta=1/4, \nu=5/4, z=1$).
• Simulation Technique: The authors employ the infinite time-evolving block decimation (iTEBD) algorithm to simulate the time evolution of matrix product states, allowing for precise calculation of the order parameter dynamics across a wide range of driving rates.

Key Results

• Unified Scaling: The numerical data confirms that the generalized FTS form successfully collapses the dynamic evolution of the order parameter $M$ for arbitrary driving rates $R$, spanning from the slow-driving regime to the sudden-quench limit.
• Role of Initial Conditions:
  • In the slow-driving regime ($R < |g_i|^{\nu r}$), the initial state is effectively "frozen" at infinite fixed points, and the standard KZ/FTS behavior is recovered (the $g_i$ term becomes irrelevant).
  • In the fast-driving regime ($R > |g_i|^{\nu r}$), the system retains memory of its initial state within the critical region. The generalized scaling form correctly accounts for this by treating $g_i$ as a relevant scaling variable.
• Robustness: The scaling collapse holds for both increasing and decreasing driving directions and is verified for both the standard Ising critical point and the tricritical point, demonstrating the theory's generality beyond simple critical points.
• Transition to DGP: The results show that as $R$ increases, the system transitions smoothly from KZ-like behavior to the DGP scaling limit, with the generalized FTS form providing the continuous mathematical description of this crossover.

Significance and Claims
The paper claims to establish a universal theory for nonequilibrium critical dynamics that spans the full range of driving rates within the critical region.

• Unification: It provides the first unified framework that bridges the KZ scaling (slow driving) and the DGP scaling (sudden quench), resolving the issue of the "upper bound" on driving rates found in original KZ/FTS theories.
• Generality: The theory accommodates initial states displaced from the critical point, offering a complete description of driven dynamics where the entire process occurs inside the critical region.
• Experimental Relevance: The authors note that this framework is immediately relevant to state-of-the-art quantum devices (such as those using Rydberg atoms or superconducting qubits) where high controllability allows for quenches starting inside the critical region. It enables the reliable characterization of universal nonequilibrium behaviors in these experiments without the constraints of the conventional KZ framework.
• Future Scope: The authors suggest the theory is extendable to other critical points (including those beyond the Landau paradigm) and alternative driving protocols involving varying symmetry-breaking fields or temperature changes near quantum critical points.