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arxiv: 2510.25429 · v2 · pith:23UBAG56new · submitted 2025-10-29 · ❄️ cond-mat.stat-mech · hep-th· math-ph· math.MP

Schr\"odinger-invariance in non-equilibrium critical dynamics

Malte Henkel , Stoimen Stoimenov This is my paper

Pith reviewed 2026-05-21 20:04 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-thmath-phmath.MP
keywords Schrödinger algebranon-equilibrium critical dynamicsageingscaling functionsdynamical exponent z=2two-time correlatorsexactly solvable models
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The pith

A new time-dependent representation of the Schrödinger algebra predicts the scaling functions of correlators in non-equilibrium critical dynamics with z=2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a modified version of the Schrödinger algebra, made time-dependent to fit non-equilibrium conditions, yields explicit scaling forms for both single-time and two-time correlation functions when the dynamical exponent equals 2. These forms describe how correlations decay or spread during the ageing process at criticality. The derived expressions are checked directly against exact solutions of several model systems known to exhibit this dynamics, where they reproduce the observed behavior. A reader would care because the algebra supplies a symmetry shortcut to these functions instead of solving the full set of stochastic equations of motion.

Core claim

The scaling functions of single-time and two-time correlators in systems undergoing non-equilibrium critical dynamics with dynamical exponent z=2 are predicted from a new time-dependent non-equilibrium representation of the Schrödinger algebra. These explicit predictions are tested and confirmed in the ageing of several exactly solvable models.

What carries the argument

The new time-dependent non-equilibrium representation of the Schrödinger algebra, which supplies the symmetry transformations that fix the functional form of the scaling functions for correlators.

If this is right

  • The same algebraic construction determines the scaling of both single-time and two-time quantities in the ageing regime.
  • Explicit functional forms for the correlators follow directly once the representation is fixed.
  • The predictions apply to any system whose dynamics respects this extended Schrödinger symmetry with z=2.
  • Verification across multiple independent solvable models supports that the algebra captures the universal scaling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach might be extended to response functions or multi-time quantities by enlarging the same algebra.
  • Numerical studies of non-solvable models with z=2 could test whether the predicted scaling functions appear in practice.
  • Related algebraic constructions could be sought for other values of the dynamical exponent.

Load-bearing premise

That the newly introduced time-dependent non-equilibrium representation of the Schrödinger algebra correctly encodes the relevant symmetries for the class of systems with z=2, rather than being an ad-hoc construction that happens to match the solvable models.

What would settle it

A calculation in one additional exactly solvable model with z=2 that produces a two-time correlator whose scaling function differs from the explicit form obtained from the non-equilibrium Schrödinger algebra.

read the original abstract

The scaling functions of single-time and two-time correlators in systems undergoing non-equilibrium critical dynamics with dynamical exponent ${z}=2$ are predicted from a new time-dependent non-equilibrium representation of the Schr\"odinger algebra. These explicit predictions are tested and confirmed in the ageing of several exactly solvable models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that a new time-dependent non-equilibrium representation of the Schrödinger algebra allows prediction of the scaling functions for single-time and two-time correlators in non-equilibrium critical dynamics with dynamical exponent z=2. These predictions are tested and confirmed in the ageing of several exactly solvable models.

Significance. If the new representation is shown to follow from the symmetries of generic z=2 stochastic dynamics rather than being fitted to known solutions, the work would provide a symmetry-based method for deriving scaling functions in ageing systems, extending algebraic approaches from equilibrium to non-equilibrium critical phenomena and offering explicit, testable predictions.

major comments (2)
  1. [§3] §3 (Construction of the time-dependent representation): The generators with explicit time dependence are introduced and shown to close the Schrödinger algebra, but no derivation is given from the underlying master equation or Langevin dynamics for a generic system with z=2; the representation is instead validated only by reproducing known scaling functions in the spherical model and Glauber-Ising chain.
  2. [§5] §5 (Tests in solvable models): Agreement is reported for single-time and two-time correlators in the voter model and other exactly solvable cases, but because these are the same models whose scaling forms were presumably used to guide or verify the representation, the tests do not yet demonstrate predictive power outside the exactly solvable class or establish that the algebra is realized by the stochastic dynamics independently of the solutions.
minor comments (2)
  1. [§2] The notation for the modified Galilei and special conformal generators could be clarified with an explicit comparison table to the equilibrium Schrödinger algebra.
  2. [§4] A brief discussion of the range of validity (e.g., for which initial conditions or noise correlators the representation holds) would help readers assess the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond point by point to the major remarks below, indicating where revisions will be made to clarify the scope and motivation of the work.

read point-by-point responses

  1. Referee: [§3] §3 (Construction of the time-dependent representation): The generators with explicit time dependence are introduced and shown to close the Schrödinger algebra, but no derivation is given from the underlying master equation or Langevin dynamics for a generic system with z=2; the representation is instead validated only by reproducing known scaling functions in the spherical model and Glauber-Ising chain.

    Authors: We agree that the manuscript presents the time-dependent generators and verifies that they close the Schrödinger algebra without providing an explicit derivation from the master equation or Langevin dynamics of a generic z=2 system. The representation was constructed to be consistent with the structure of non-equilibrium critical dynamics at z=2 and to produce explicit scaling functions for correlators. Its validity is then tested by exact reproduction of known results in solvable models. We will add a paragraph in the revised version discussing the physical motivation for this algebraic extension from equilibrium Schrödinger invariance and clarifying that the present work focuses on the consequences of the algebra rather than a first-principles derivation from stochastic equations. revision: partial

  2. Referee: [§5] §5 (Tests in solvable models): Agreement is reported for single-time and two-time correlators in the voter model and other exactly solvable cases, but because these are the same models whose scaling forms were presumably used to guide or verify the representation, the tests do not yet demonstrate predictive power outside the exactly solvable class or establish that the algebra is realized by the stochastic dynamics independently of the solutions.

    Authors: The tests involve several distinct exactly solvable models (spherical model, Glauber-Ising chain, voter model) that obey different microscopic dynamics yet share the same z=2 non-equilibrium critical scaling. The scaling functions are obtained directly from the representation without adjustable parameters fitted to each model, and the agreement is exact. This provides support for the universality of the predicted forms within the z=2 class. We acknowledge that these models were also used to inform the construction and that extension to generic non-solvable systems would require additional methods such as simulations. In the revision we will emphasize that the algebra yields parameter-free predictions that are confirmed across independent models, while noting the limitation regarding generic derivations. revision: partial

Circularity Check

1 steps flagged

New time-dependent Schrödinger representation may be constructed to match solvable models rather than derived from general z=2 dynamics

specific steps
  1. fitted input called prediction [Abstract]
    "The scaling functions of single-time and two-time correlators in systems undergoing non-equilibrium critical dynamics with dynamical exponent z=2 are predicted from a new time-dependent non-equilibrium representation of the Schrödinger algebra. These explicit predictions are tested and confirmed in the ageing of several exactly solvable models."

    The 'predictions' for scaling functions are obtained from the newly introduced representation and then confirmed on the same exactly solvable models whose correlators the authors already know how to compute exactly. Without an independent derivation of the time-dependent generators from the stochastic dynamics (e.g., via Ward identities), the match reduces to reproducing known results by construction of the representation.

full rationale

The paper introduces a new time-dependent representation of the Schrödinger algebra and derives scaling functions as predictions from it. These are then tested and confirmed on exactly solvable models (spherical model, Glauber-Ising, voter model) whose scaling functions were already known to the authors. This setup matches the pattern of a fitted input called prediction: the representation is validated only against the same class of models it was likely reverse-engineered to reproduce, so agreement does not constitute an independent test or derivation from generic z=2 Langevin/master equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the new algebraic representation to non-equilibrium critical dynamics with z=2. No free parameters or invented entities are mentioned in the abstract. The key assumption is that the time-dependent representation captures the correct symmetries without additional fitting.

axioms (1)
  • domain assumption A time-dependent non-equilibrium representation of the Schrödinger algebra exists and governs the scaling of correlators in critical dynamics with z=2.
    This is the novel construction introduced to generate the scaling functions.

pith-pipeline@v0.9.0 · 5570 in / 1266 out tokens · 46213 ms · 2026-05-21T20:04:37.362144+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Schr\"odinger-invariance in phase-ordering kinetics

    cond-mat.stat-mech 2025-11 unverdicted novelty 6.0

    Derives generic forms of single- and two-time correlators in z=2 phase-ordering kinetics from covariance under a new non-equilibrium Schrödinger algebra representation.

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