On the Renormalization Group Flow of Active Flocks
Abstract
We study the statistical field-theoretic renormalization of active flocks via the Martin-Siggia-Rose-De Dominicis-Janssen (MSRDJ) formulation of stochastic dynamics, focusing on the Toner-Tu theory of Malthusian flocks, or polar-ordered, momentum non-conserving active fluids in which density fluctuations relax sufficiently rapidly to be eliminated as hydrodynamic degrees of freedom. Working in the limit of isotropic diffusion in two spatial dimensions, we perform an all orders analysis of the perturbative renormalization of propagators and interaction vertices. We show that a nonlinearly realized diagonal rotation/shift symmetry gives rise to Ward identities that protect the trigonometric structure of the convective derivative, ensuring that the sine and cosine interaction potentials retain their functional form under renormalization up to a rescaling of their overall couplings. We demonstrate that the freedom to renormalize the dynamical critical exponent in this boost non-invariant theory prevents the determination of the precise values of the critical exponents and anomalous dimensions in the theory, highlighting parallels and crucial differences with sine-Gordon and XY models. We express all anomalous dimensions as functions of the dynamical critical exponent, which remains undetermined due to the absence of a relation ordinarily forced upon boost invariant systems. We revisit and contextualize several claims and counterclaims in the recent literature in light of these results.
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BibTeX
@misc{grosvenor2026renormalization,
title = {{On the Renormalization Group Flow of Active Flocks}},
author = {Kevin T. Grosvenor and Subodh P. Patil},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.20552},
eprint = {2606.20552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.20552},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Renormalization Group Flow of Active Flocks
AU - Kevin T. Grosvenor
AU - Subodh P. Patil
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2606.20552
UR - https://arxiv.org/abs/2606.20552
AB - We study the statistical field-theoretic renormalization of active flocks via the Martin-Siggia-Rose-De Dominicis-Janssen (MSRDJ) formulation of stochastic dynamics, focusing on the Toner-Tu theory of Malthusian flocks, or polar-ordered, momentum non-conserving active fluids in which density fluctuations relax sufficiently rapidly to be eliminated as hydrodynamic degrees of freedom. Working in the limit of isotropic diffusion in two spatial dimensions, we perform an all orders analysis of the perturbative renormalization of propagators and interaction vertices. We show that a nonlinearly realized diagonal rotation/shift symmetry gives rise to Ward identities that protect the trigonometric structure of the convective derivative, ensuring that the sine and cosine interaction potentials retain their functional form under renormalization up to a rescaling of their overall couplings. We demonstrate that the freedom to renormalize the dynamical critical exponent in this boost non-invariant theory prevents the determination of the precise values of the critical exponents and anomalous dimensions in the theory, highlighting parallels and crucial differences with sine-Gordon and XY models. We express all anomalous dimensions as functions of the dynamical critical exponent, which remains undetermined due to the absence of a relation ordinarily forced upon boost invariant systems. We revisit and contextualize several claims and counterclaims in the recent literature in light of these results.
ER -