Universal Symmetry-Breaking Dynamics at Continuous Phase Transitions: Evidence for a New Dynamical Critical Exponent

This paper identifies a new form of universal far-from-equilibrium dynamics in Ising models following a symmetry-breaking quench, characterized by a previously unknown dynamical critical exponent and a lower critical effective dimension that distinguishes observable scaling in higher-dimensional systems from lower-dimensional ones.

The Gist

Imagine you have a giant, crowded dance floor. Everyone is dancing in a chaotic, perfectly balanced way right at the edge of a "phase transition"—a moment where the crowd is about to decide whether to all dance in a synchronized line (ordered) or stay completely random (disordered).

In physics, this is called a critical point. Usually, scientists know how to predict what happens if you gently nudge this crowd. But what happens if you suddenly shout a command that forces everyone to break that balance? That's what this paper investigates.

Here is the story of their discovery, broken down into simple concepts:

1. The Experiment: The "Sudden Shout"

The researchers set up a simulation of magnetic spins (think of them as tiny compass needles) on a grid.

• The Setup: They start the system in a state of perfect, critical chaos.
• The Action: At a specific moment ($t=0$), they suddenly switch on a magnetic field. This is like a conductor suddenly shouting, "Everyone face North!"
• The Result: This isn't a gentle nudge; it's a massive shock that throws the system far away from equilibrium. The energy fluctuations become huge, and the system enters a wild, unpredictable state.

2. The Mystery: The "Magic Collapse"

When the scientists watched how the "order" (the compass needles aligning) fluctuated over time, they saw something strange.

• They tried this with different sized dance floors (system sizes) and different volumes of the shout (field strengths).
• The Expectation: Usually, a small dance floor behaves differently than a huge one. A quiet shout behaves differently than a loud one. You would expect a messy tangle of different curves.
• The Surprise: When they plotted the data correctly, all the different curves collapsed into a single, perfect line.

The Analogy: Imagine you have a recipe for baking a cake. Usually, if you double the size of the pan, you have to change the baking time and temperature in complex ways. But here, the researchers found that if you mix the "pan size" and the "oven temperature" in a very specific, secret way, every single cake, regardless of size or heat, bakes at the exact same rate.

3. The New Rule: A "Secret Ingredient"

To explain why all these different scenarios fit onto one single line, the scientists realized they were missing a piece of the puzzle.

• In physics, we use "exponents" (mathematical numbers) to describe how things scale.
• They found that the existing rules weren't enough. They had to invent a new, previously unknown number (which they call the exponent $w$) to make the math work.
• This new number acts like a "universal dial" that explains how the system reacts to the sudden shock, regardless of how big the system is.

4. The "Goldilocks" Zone: Where It Works (and Where It Doesn't)

The most fascinating part of their discovery is that this "magic collapse" doesn't happen everywhere. It only works in specific dimensions (sizes of the universe they simulated):

• It Works:
  • In 2D Quantum systems (like a flat sheet of quantum spins).
  • In 3D and 4D Classical systems (like a cube or hyper-cube of magnetic spins).
• It Fails:
  • In 1D Quantum systems (a single line of spins).
  • In 2D Classical systems (a flat sheet of classical spins).

The Analogy: Think of this like a specific type of music that only sounds good in a concert hall with a certain shape. If the room is too small (1D) or shaped differently (2D classical), the music sounds muddy and doesn't harmonize. But in the "Goldilocks" zones (2D quantum, 3D/4D classical), the music is perfect, and everyone sings in tune.

This suggests there is a "lower limit" to the complexity of the universe required for this specific type of universal behavior to emerge.

5. How They Did It (The Quantum Challenge)

Simulating a 2D quantum system is incredibly hard because the math gets exponentially complicated as you add more particles. It's like trying to predict the movement of every single water molecule in a swimming pool simultaneously.

• To solve this, the team used Neural Quantum States.
• The Analogy: Instead of trying to calculate every single molecule's path with a standard calculator, they trained an AI (a neural network) to "guess" the shape of the wave function. This AI learned the patterns of the critical state and then watched how the system evolved after the "shout," allowing them to simulate up to 256 quantum spins with high accuracy.

Summary

The paper claims to have found a new universal law for how systems behave when they are violently shaken at a critical point.

1. They found that order-parameter fluctuations collapse into a single pattern across different sizes and strengths.
2. This pattern requires a new dynamical exponent ($w$) to explain.
3. This behavior is "universal" but only appears in systems above a certain effective dimension (it works in 2D quantum and 3D/4D classical, but not in lower dimensions).
4. This suggests that far-from-equilibrium physics has hidden, simple rules that we are just beginning to uncover, distinct from the rules that govern gentle, near-equilibrium changes.

The paper does not claim this applies to medical treatments, climate change, or specific future technologies yet; it strictly identifies this new mathematical behavior in theoretical models of magnets and quantum spins.

Technical Summary

Technical Summary: Universal Symmetry-Breaking Dynamics at Continuous Phase Transitions

Problem and Context
Understanding universal dynamics in matter far from equilibrium remains a central challenge in many-body physics. While universal scaling is well-established for specific protocols—such as the Kibble–Zurek mechanism for slow ramps, initial-slip scaling following sudden quenches from disordered states, and non-thermal fixed points in isolated systems—dynamical behavior in regimes where fluctuations dominate and linear-response theory fails remains largely unexplored. Specifically, the dynamics following a sudden symmetry-breaking quench at a continuous phase transition, where the initial state is already critical and the perturbation couples directly to the order parameter, presents an open question. Such scenarios are relevant for identifying fundamental organizing principles of nonequilibrium dynamics and are accessible in current experimental platforms like quantum simulators.

Methodology
The authors investigate this problem using Ising models on periodic hypercubic lattices. The protocol involves preparing a system in a critical state (either the ground state of a quantum critical point or a Gibbs state at a thermal critical point) and performing a sudden quench by switching on a uniform longitudinal field $h$ that couples to the order parameter $M$. The post-quench Hamiltonian is $H = H_{\text{crit}} - hM$.

The study covers:

• Quantum Models: 1D and 2D transverse-field Ising models (TFIM). The 2D case poses a significant numerical challenge as the initial state is a strongly correlated critical many-body state. The authors employ a Neural Quantum State (NQS) approach, representing the wave function $|\psi_\theta\rangle = \sum_s \psi_\theta(s)|s\rangle$ using a translation-equivariant complex-valued convolutional ansatz. Time evolution is computed via the time-dependent variational principle (TDVP), allowing simulations of up to $16 \times 16 = 256$ spins.
• Classical Models: 3D and 4D Ising models with Glauber dynamics, initialized at their respective thermal critical points.

The primary observable is the order-parameter fluctuation $\langle M^2(t) \rangle$. The analysis utilizes dynamic finite-size scaling (DFSS) to examine how these fluctuations scale with system size $L$, quench strength $h$, and time $t$.

Key Contributions and Results
The central finding is the identification of a previously unrecognized form of universal far-from-equilibrium scaling. The authors observe that the rescaled order-parameter fluctuations $L^{-\kappa}\langle M^2(t) \rangle$ exhibit a compelling "temporal collapse" across a wide range of system sizes and quench strengths when plotted against a single combined variable $\hat{h}\hat{t}^w$.

• Emergent Single-Variable Scaling: While standard DFSS predicts a two-parameter scaling form $F(\hat{t}, \hat{h})$, the data indicates that in the observed regime, this collapses onto a single-variable dependence $\tilde{F}(\hat{h}\hat{t}^w)$. This reduction requires the introduction of a new, so-far-unknown dynamical critical exponent $w$.
• Dimensional Dependence and Lower Critical Effective Dimension: The phenomenon is observed in:
  • The 2D quantum Ising model ($w \approx 1.86 \pm 0.05$).
  • The 3D and 4D classical Ising models ($w \approx 0.854 \pm 0.015$ and $w \approx 0.936 \pm 0.017$, respectively).
  • Crucially, this collapse is absent in the 1D quantum and 2D classical cases. In the 1D quantum model, the apparent scaling regime shrinks with increasing system size, indicating a finite-size artifact rather than true universality. In the 2D classical model, curves intersect locally ("single crossing") but do not exhibit a global collapse.
  • This pattern suggests a lower critical effective dimension for this specific universal scaling regime.
• Nature of the Regime: The protocol drives the system beyond linear response, generating superextensive energy fluctuations $(\Delta E_0)^2 \sim h^2 L^\kappa$. The observed scaling is distinct from hydrodynamic modes, initial-slip scaling (which concerns the mean order parameter), and soft-quench scaling controlled by equilibrium exponents.

Significance and Claims
The paper claims to have identified a new universal scaling regime that emerges after symmetry-breaking quenches at continuous transitions. The significance lies in:

1. New Exponent: The necessity of an additional dynamical exponent $w$ to describe the collapse, which is not accounted for by standard equilibrium or near-equilibrium scaling theories.
2. Universality Beyond Ising: While demonstrated here for Ising models, the authors suggest via general scaling arguments that this behavior may characterize systems with non-conserved order parameters more broadly.
3. Experimental Relevance: The dynamics described are directly accessible in current programmable quantum platforms, such as Rydberg atom arrays and trapped ions, offering a pathway for experimental verification.

The authors remain modest regarding the microscopic origin of the exponent $w$, noting it remains an open question whether this reflects a broader organizing principle of far-from-equilibrium critical dynamics. They also state that it is currently unknown whether similar single-variable reductions occur for observables other than order-parameter fluctuations.