Schrödinger-invariance in phase-ordering kinetics

This paper derives the generic scaling forms of single-time and two-time correlators in non-equilibrium phase-ordering kinetics with dynamic exponent $z=2$ by establishing a new non-equilibrium representation of the Schrödinger algebra and utilizing the covariance of four-point response functions.

The Gist

The Big Picture: Cooling Down a Chaotic Crowd

Imagine you have a giant room full of people (atoms or molecules) who are all running around wildly, bumping into each other, and facing random directions. This represents a system at a high temperature where everything is disordered.

Now, imagine you suddenly turn off the heating and drop the temperature to freezing (a process physicists call a "quench"). The people stop running and start trying to find a comfortable spot. They begin to form small groups, then larger groups, until eventually, everyone in a specific area is facing the same direction. This process of forming order out of chaos is called phase-ordering.

The paper by Stoimenov and Henkel is about figuring out the universal rules that govern how these groups grow and how long it takes for the system to settle down, without needing to know the specific details of every single person in the room.

The Problem: It's Too Slow and Too Complex

When you watch this process, you notice three things:

1. It gets slower: The groups grow big, but the rate at which they grow slows down over time.
2. Time doesn't work like a clock: If you start watching at 1 minute, the system looks different than if you start watching at 100 minutes. The system "remembers" when it started.
3. It scales: If you zoom out, the pattern of groups looks the same regardless of the specific size of the room or the exact number of people.

Physicists have known these patterns for decades, but they usually have to run complex computer simulations to predict them. This paper asks: Can we predict these patterns just by using math and symmetry, like solving a puzzle?

The Secret Weapon: A New Kind of "Symmetry"

The authors use a mathematical concept called Schrödinger symmetry.

The Analogy:
Think of a movie.

• Standard Symmetry: If you play the movie forward, backward, or rotate the screen, the physics of the scene usually looks the same.
• Schrödinger Symmetry: This is a special rule for how things move and change over time. It's like a "magic lens" that tells us how a system behaves if we stretch time and space in a specific way.

Usually, this "magic lens" only works for systems that are already settled down (in equilibrium). But this paper claims that even for a system that is still cooling down and changing (out of equilibrium), we can use a modified version of this lens.

The "Recipe" Used in the Paper

The authors didn't just guess; they followed a specific recipe to prove their point:

1. The "Response" Trick: Instead of looking directly at the groups forming, they looked at how the system responds if you gave it a tiny nudge. In physics, there is a mathematical trick where you can calculate how two things are connected (correlated) by looking at how they respond to a push.
2. The Four-Point Connection: They looked at a complex interaction involving four points in time and space. Think of this as watching four different people in the room and seeing how their movements are linked.
3. The "New Lens": They applied their modified Schrödinger symmetry to these four points. They found that if you assume the system follows these symmetry rules, the messy, complex equations simplify into a neat, predictable pattern.

What They Discovered

By using this new "lens," they were able to derive the exact shapes of the curves that describe how the system ages.

• The "Soft" vs. "Hard" Groups: They explained why some systems form smooth, rounded groups (like a cloud) while others form sharp, jagged groups (like ice crystals). This depends on whether the "people" in the system are "soft" (can change shape easily) or "hard" (keep a rigid shape).
• The "Cusp" (The Sharp Point): For systems with rigid groups, the math predicts a sharp point in the data (called a "cusp"). The paper shows this matches a known rule called Porod's Law, which describes how light scatters off rough surfaces.
• Finite Rooms: They also figured out what happens if the room isn't infinite but has walls (a finite size). They predicted that once the groups grow large enough to hit the walls, the growth stops and levels off at a specific height.

The "Magic" Formula

The most important result is a new relationship between the size of the groups, the time passed, and the dimension of the space.

They found that the "ageing exponent" (a number that tells us how fast the system forgets its past) is directly linked to the scaling dimension (how the system looks when you zoom in or out).

In simple terms: The way the system grows is dictated by a hidden symmetry, just like the way a snowflake grows is dictated by the symmetry of the water molecule. Even though the snowflake looks chaotic, it follows a strict geometric rule. This paper proves that cooling materials follow a similar strict rule, and we can find it using Schrödinger's math.

Summary

• The Goal: To understand how materials organize themselves after being cooled down quickly.
• The Method: They used a special mathematical symmetry (Schrödinger-invariance) adapted for systems that are not yet settled.
• The Result: They successfully derived the standard rules of how these systems age and grow, proving that these complex behaviors are actually the result of deep, underlying mathematical symmetries.
• The Takeaway: You don't need to simulate every single atom to understand the big picture; if you understand the "symmetry rules" of the game, you can predict the outcome.

Technical Summary

Technical Summary: Schrödinger-invariance in phase-ordering kinetics

Problem Statement
Phase-ordering kinetics describes the growth of correlated microscopic clusters in a many-body system following a quench from a disordered high-temperature state to a low-temperature ordered phase ($T < T_c$). This process is characterized by "physical ageing," defined by slow dynamics, the absence of time-translation invariance, and dynamical scaling. The system evolves via a characteristic time-dependent length scale $\ell(t) \sim t^{1/z}$, where $z$ is the dynamical exponent. For non-conserved order parameters (Model-A dynamics), $z=2$. While the phenomenological scaling forms of single-time and two-time correlators ($C$) and response functions ($R$) are well-established, their derivation from fundamental dynamical symmetries remains a challenge. Specifically, standard equilibrium symmetries do not directly apply to non-equilibrium ageing regimes.

Methodology
The authors employ a field-theoretic approach based on the Janssen-de Dominicis formalism, adapted for non-equilibrium phase-ordering. The core methodology involves:

1. Functional Integral Representation: Expressing physical observables as functional integrals over the order parameter $\phi$ and the response field $\tilde{\phi}$. The action consists of a deterministic part $J_0$ and a term $J_b$ representing short-ranged initial correlations.
2. Schrödinger Covariance: Utilizing the Schrödinger algebra, the maximal finite-dimensional symmetry of the free Schrödinger equation, as a candidate for dynamical symmetry.
3. Non-Equilibrium Representation: Applying a specific postulate where the Lie algebra generators of an equilibrium system are transformed into non-equilibrium generators via a change of representation: $X^{equi}_n \to X_n = e^{\xi \ln t} X^{equi}_n e^{-\xi \ln t}$. This introduces a dimensionless parameter $\xi$ and modifies the scaling dimensions of the operators.
4. Four-Point Response Functions: Deriving the scaling forms of correlators not from direct two-point functions, but from the covariance of four-point response functions $\langle \phi \phi \tilde{\phi} \tilde{\phi} \rangle$. This is necessary because, in the non-equilibrium context, non-vanishing deterministic averages require equal numbers of $\phi$ and $\tilde{\phi}$ fields (Bargman superselection rules), and physical correlators like $\langle \phi \phi \rangle$ are generated by expanding the initial correlation term $J_b$.

Key Contributions and Results
The paper demonstrates that the generic scaling forms of phase-ordering kinetics can be derived from the covariance of multi-point response functions under these new non-equilibrium representations of the Schrödinger algebra.

• Scaling Functions: The authors derive the scaling forms for the two-time autocorrelator $C(t,s)$ and the single-time correlator $C(s;r)$. They show that the scaling functions depend on the ratio $y=t/s$ and the scaled distance $r/s^{1/2}$.
• Exponent Relations: By analyzing the four-point response function, the authors derive the relation between the autocorrelation exponent $\lambda$ and the scaling dimensions: $\lambda/2 = 2\delta - \xi$. Under the specific condition for phase-ordering where $\delta = \xi$, this simplifies to $\lambda = 2\delta$.
• New Scaling Relation: A distinct scaling relation is proposed for phase-ordering: $2\delta = d / (1 + \zeta_p) = \lambda$, where $\zeta_p$ is an exponent characterizing the onset of the scaling regime. This relation differs from that of non-equilibrium critical dynamics ($T=T_c$).
• Porod's Law and Cusp Behavior: The formalism successfully reproduces the small-distance behavior of the single-time correlator. For scalar order parameters with "hard" interfaces, the expansion of the scaling function yields a cusp at $r=0$, consistent with Porod's law ($C(s;r) \sim C_0 - C_1 |r|/s^{1/2}$).
• Finite-Size Scaling: The paper extends the analysis to finite systems of linear size $N$. It derives the scaling behavior of the plateau height of the autocorrelator in finite systems, showing $C^{(2)}_\infty \sim N^{-\lambda}$ (for fixed $s$) and $C^{(2)}_\infty \sim s^{\lambda/2}$ (for fixed $N$).
• Global Correlators: The authors derive the scaling for the global two-time correlator and the squared magnetisation, recovering known results such as $\langle m^2(s) \rangle \sim s^{d/2}$ and the slip exponent relation $\Theta = \frac{1}{2}(d - \lambda)$.

Significance and Claims
The paper claims that the complete phenomenology of phase-ordering kinetics, including well-established scaling laws and specific features like Porod's law, can be systematically derived from the covariance of multi-point response functions under a new non-equilibrium representation of the Schrödinger Lie algebra.

The authors emphasize that this approach unifies the treatment of single-time and two-time correlators on a single conceptual basis. While the effective equation of motion for phase-ordering is not strictly Schrödinger-invariant due to non-linear terms, the symmetry of the linear part (augmented by a $1/t$ potential) is sufficient to deduce the long-time scaling behavior. The work provides a theoretical framework that reproduces known "folklore" results and offers new scaling relations (specifically regarding the exponent $\zeta_p$ and the relation between $\delta$ and $\lambda$) that distinguish phase-ordering kinetics from critical dynamics at $T_c$. The authors note that while the method reproduces known bounds and behaviors, the specific values of exponents like $\lambda$ and $\xi$ still depend on the full non-linear equation of motion and cannot be calculated from first principles solely via this symmetry argument.