Schrödinger-invariance in non-equilibrium critical dynamics

This paper proposes a new time-dependent non-equilibrium representation of the Schrödinger algebra to predict scaling functions for single-time and two-time correlators in systems with dynamical exponent $z=2$, and validates these predictions through tests on several exactly solvable ageing models.

The Gist

Imagine you have a giant, chaotic crowd of people in a room. If you suddenly shout "Freeze!" (a "quench"), the crowd doesn't just stop instantly; they slowly settle into a new pattern. In physics, this is called physical ageing. It happens when a system is jolted from a disordered state into a critical point where it's on the verge of changing phases, like water turning to ice but not quite there yet.

For decades, physicists have struggled to predict exactly how these systems behave over time because the math is incredibly complex. This paper by Malte Henkel and Stoimen Stoimenov offers a new, elegant way to solve this puzzle using a concept called Schrödinger-invariance.

Here is the breakdown of their discovery using simple analogies:

1. The Problem: The "Slow Motion" Crowd

When a system ages, it loses its memory of the past. If you ask, "How similar is the crowd's arrangement at 2:00 PM to how it was at 1:00 PM?", the answer depends entirely on when you ask.

• Time-Translation Invariance is broken: In normal physics, the laws of motion don't care if you start your stopwatch at noon or midnight. In these ageing systems, the "rules" change depending on how old the system is.
• The Challenge: Because the rules change, standard math tools fail. Scientists usually have to run massive, expensive computer simulations to guess what happens next.

2. The Solution: A New "Time Machine" for Math

The authors realized that these chaotic, ageing systems actually follow a hidden, rigid set of rules known as the Schrödinger algebra. You might know Schrödinger from quantum mechanics, but here, it's being used as a geometric symmetry for time and space.

Think of the Schrödinger algebra as a master blueprint.

• In the past, this blueprint only worked for systems in perfect equilibrium (like a calm lake).
• The authors created a new, time-dependent version of this blueprint. They essentially "tuned" the math to account for the fact that the system is getting older. They introduced a "dial" (represented by the symbol $\xi$) that adjusts the math to fit the slowing-down nature of ageing.

3. The Prediction: The "Crystal Ball"

By using this new blueprint, the authors didn't just guess; they derived exact formulas for how the system behaves.

• The Correlator (The "Similarity Score"): They predicted exactly how similar the system looks at two different times.
• The Result: They found that the shape of these "similarity scores" is universal. It doesn't matter if you are looking at a model of magnets, a growing surface (like sand piling up), or a chemical reaction. If they share the same underlying "symmetry," they all follow the same mathematical curve.

4. The Proof: Testing on "Exactly Solvable" Models

To prove their crystal ball works, they tested it against several famous models that are known to be solvable (meaning we already know the answers from other methods):

• The Voter Model: Imagine a grid of people where everyone copies their neighbor's opinion.
• The Spherical Model: A theoretical model of magnets where spins can point in any direction, not just up or down.
• The Edwards-Wilkinson Model: A model for how a rough surface (like a growing crystal or a sand dune) smooths out over time.
• The Arcetri Model: A variation of the surface growth model.
• Bosonic Contact Processes: Models of particles that multiply or die out.

The Verdict: In every single case, the authors' new formulas matched the known exact answers perfectly. They didn't just get the "big picture" right; they got the specific details of the curves right, including how they change based on the dimension of the space (1D, 2D, 3D, etc.).

5. The Big Takeaway

The paper claims that symmetry is the key. Even though these systems are far from equilibrium and seem chaotic, they are governed by a deep, hidden symmetry (the Schrödinger algebra).

• What this means: You don't need to simulate every single particle in a complex system to know how it ages. If you know the system's "symmetry class" (its specific parameters like mass and scaling dimensions), you can write down the exact formula for its behavior.
• The "Universal" Aspect: Just as all circles look the same regardless of size, all these different physical models (magnets, surfaces, chemicals) look the same mathematically when viewed through this new lens. They all collapse onto the same "master curve."

Summary

Henkel and Stoimenov took a complex, messy problem (how systems age out of equilibrium) and solved it by finding a hidden geometric order. They showed that by applying a "time-tuned" version of a classic physics symmetry, you can predict the exact behavior of these systems without needing a supercomputer. It's like realizing that while a crowd of people seems chaotic, they are actually all dancing to the same strict, predictable rhythm if you know the right beat.

Technical Summary

Technical Summary: Schrödinger-invariance in non-equilibrium critical dynamics

Problem and Context
Many-body systems with strongly interacting degrees of freedom often exhibit collective properties that are not evident from single-particle considerations. While few such systems are analytically solvable, they are frequently studied via numerical simulations requiring significant computational resources. The authors address the challenge of describing the dynamics of these systems, specifically those undergoing non-equilibrium critical dynamics after a quench from a disordered initial state to a critical point ($T=T_c$). The goal is to predict the scaling functions of single-time and two-time correlators and response functions without relying on model-specific details, but rather through the application of dynamical symmetries. The study focuses on systems with a dynamical exponent $z=2$, where conservation laws on the order parameter are excluded.

Methodology
The paper employs a field-theoretical approach based on the Janssen-de Dominicis action for systems far from equilibrium. The core methodology involves:

1. Non-equilibrium Representation of the Schrödinger Algebra: The authors utilize a new time-dependent representation of the Schrödinger Lie algebra. Unlike the equilibrium representation, which assumes time-translation invariance, this non-equilibrium representation ($X_{equi} \to X = e^{W(t)} X_{equi} e^{-W(t)}$ with $W(t) = \xi \ln t$) accounts for the breaking of time-translation invariance characteristic of physical ageing.
2. Covariance of Response Functions: The dynamics are described by the covariance of response functions under this new algebra. The authors derive explicit forms for two-time response functions $R(t,s;r)$, single-time correlators $C(t;r)$, and two-time auto-correlators $C(t,s;r)$ by mapping equilibrium scaling operators to non-equilibrium ones.
3. Scaling Hypotheses: The derivation relies on specific scaling hypotheses, including the assumption that the composite response field $\tilde{\phi}^2$ behaves as a simple power law in momentum space, introducing an exponent $\nu$.
4. Exact Solvability and Verification: To test these theoretical predictions, the authors apply the derived scaling forms to several exactly solvable models: the Voter model, the Spherical model, the Edwards-Wilkinson (EW) model, the Arcetri model, and bosonic reaction-diffusion systems (Bosonic Contact Process and Bosonic Pair-Contact Process).

Key Contributions and Results
The primary contribution is the derivation of explicit, universal scaling functions for non-equilibrium correlators and responses, which are then confirmed against exact solutions of specific models.

• Predicted Scaling Forms: The paper provides closed-form expressions for the scaling functions involving Kummer/Tricomi confluent hypergeometric functions ($U$) and Appell functions ($F_1$). These forms depend on a small set of non-equilibrium exponents: $\xi$ (related to time-translation breaking), $\tilde{\xi}$, $\tilde{\delta}_2$, and $\tilde{\xi}_2$, which characterize the composite field $\tilde{\phi}^2$.
• Verification in Solvable Models: The theoretical predictions are tested against:
  • Voter Model: Exact results for $d=1, 2, 3$ and general $d$ show complete agreement with the Schrödinger-invariance predictions. The upper critical dimension is identified as $d^*=2$.
  • Spherical Model: The exact solutions for $2 < d < 4$ and $d > 4$ align with the predictions, identifying specific scaling dimensions listed in Table 1.
  • Arcetri Model: This model, related to surface growth and the KPZ equation, yields correlators identical to the Spherical model in $d+2$ dimensions, confirming the universality of the scaling forms.
  • Edwards-Wilkinson and Bosonic Models: The EW model and the Bosonic Contact Process (BCPD) are shown to belong to the same universality class, while the Bosonic Pair-Contact Process (BPCPD) at its multi-critical point exhibits distinct scaling dimensions.
• Exponent Identification: The authors successfully identify the universal scaling dimensions $(\xi, \tilde{\xi}, \tilde{\delta}_2, \tilde{\xi}_2)$ for all considered models. A consistent finding across all models is that the exponent $\nu = 0$.
• Universality Classes: The analysis reveals that the EW universality class acts as a mean-field theory for the Voter, Spherical, and Arcetri models, though the dimension $d$ at which this regime is reached varies. The BPCPD at its multi-critical point possesses a distinct mean-field theory.

Significance
The paper claims to realize a long-standing ambition in theoretical physics: predicting the form of time-dependent observables in interacting many-body systems out of equilibrium solely from dynamical symmetry. By demonstrating that the Schrödinger algebra, in its new non-equilibrium representation, dictates the functional form of ageing correlators and responses, the authors provide a powerful analytical tool that bypasses the need for model-specific calculations. The work confirms that despite the diversity of physical mechanisms (magnetic spin flips, surface growth, particle reactions), the underlying non-equilibrium critical dynamics share a common symmetry structure. The paper concludes that this framework suggests the existence of further exactly solved systems within this symmetry class and notes that extensions to quantum hydrodynamics remain a potential area for future exploration.