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Diversity of critical phenomena in the ordered phase of polar active fluids

Patrick Jentsch, Chiu Fan Lee

TL;DR

This work provides a comprehensive linear stability analysis of the Toner–Tu model in the ordered phase of polar active fluids, yielding two generic instability types distinguished by how growth rates scale with wavevector and deriving exact instability criteria. By applying two-parameter criticality conditions, the authors identify four previously unreported critical points, including two nonequilibrium universality classes that already show linear-scale deviations from known classes. They map the $C_2$ and $C_3$ points to Lifshitz-type criticality for transverse Goldstone modes, and split the $C_4$ point into two cases with distinct linear exponents, revealing rich critical behavior within the TT framework. Finally, they construct explicit hydrodynamic models realizing each critical point, establishing physical attainability and providing a concrete platform for future RG analyses and microscopic investigations, thus offering a roadmap for discovering new universality classes in active matter.

Abstract

We present a comprehensive analytical linear stability analysis of the Toner-Tu model for polar active fluids in the ordered phase. Our results provide exact instability criteria and demonstrate that all generic hydrodynamic instabilities fall into two fundamental categories, distinguished by their scaling with the wavevector magnitude. By applying a general criticality condition, we show that each instability can give rise to a critical point by fine-tuning only two parameters. We identify four previously unreported critical points of the Toner-Tu model, two of which already display nonequilibrium critical behavior that extends beyond known universality classes at the linear level. We further construct explicit hydrodynamic models that realize each newly identified critical point, establishing their physical attainability and providing concrete targets for future renormalization-group analyses and microscopic model studies. Altogether, our framework offers a unified theoretical foundation and a practical roadmap for the systematic discovery of new universality classes in active matter.

Diversity of critical phenomena in the ordered phase of polar active fluids

TL;DR

This work provides a comprehensive linear stability analysis of the Toner–Tu model in the ordered phase of polar active fluids, yielding two generic instability types distinguished by how growth rates scale with wavevector and deriving exact instability criteria. By applying two-parameter criticality conditions, the authors identify four previously unreported critical points, including two nonequilibrium universality classes that already show linear-scale deviations from known classes. They map the and points to Lifshitz-type criticality for transverse Goldstone modes, and split the point into two cases with distinct linear exponents, revealing rich critical behavior within the TT framework. Finally, they construct explicit hydrodynamic models realizing each critical point, establishing physical attainability and providing a concrete platform for future RG analyses and microscopic investigations, thus offering a roadmap for discovering new universality classes in active matter.

Abstract

We present a comprehensive analytical linear stability analysis of the Toner-Tu model for polar active fluids in the ordered phase. Our results provide exact instability criteria and demonstrate that all generic hydrodynamic instabilities fall into two fundamental categories, distinguished by their scaling with the wavevector magnitude. By applying a general criticality condition, we show that each instability can give rise to a critical point by fine-tuning only two parameters. We identify four previously unreported critical points of the Toner-Tu model, two of which already display nonequilibrium critical behavior that extends beyond known universality classes at the linear level. We further construct explicit hydrodynamic models that realize each newly identified critical point, establishing their physical attainability and providing concrete targets for future renormalization-group analyses and microscopic model studies. Altogether, our framework offers a unified theoretical foundation and a practical roadmap for the systematic discovery of new universality classes in active matter.

Paper Structure

This paper contains 15 sections, 27 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Linear stability of the TT ordered phase
  3. Two types of instabilities
  4. Phase separation vs. pattern formation
  5. Criticality criteria
  6. Critical points from $C_2$ and $C_3$
  7. Two critical points from $C_4$
  8. Case $\theta_c = 0$
  9. Case $\theta_c \neq 0$
  10. Realizing the critical points in hydrodynamic models
  11. $C_1$ criticality
  12. $C_2$ & $C_3$ criticality
  13. $C_{4a}$ criticality
  14. $C_{4b}$ criticality
  15. Summary and outlook

Figures (1)

  • Figure 1: Linear instability regions and associated critical points in the ordered phase of the Toner--Tu model. Colored regions indicate long-wavelength (hydrodynamic) instabilities in the $q \to 0$ limit; the color encodes the direction of the most unstable wavevector. Hatched regions mark Type I instabilities. Critical points are labeled by the numbers used in the text: (a) “0” denotes the order--disorder transition associated with $C_0$nesbitt_njp21bertrand_prr22; “1” marks the critical point recently reported in miller_pre24. The remaining points are predictions of our stability analysis: (b) $C_2$ (see Sec. \ref{['sec:C23']}); (c) $C_{4a}$ with $\theta_c = 0$ based on model Eq. \ref{['eq:kapmodel']}; and (d) $C_{4b}$ with $\theta_c \neq 0$ based on model Eq. \ref{['eq:model2']}. Since $C_3$ criticality occurs necessarily in conjunction with $C_{4a}$ criticality, it is not depicted here explicitly.