Hydrodynamic theory of two-dimensional incompressible polar active fluids with quenched and annealed disorder

TL;DR

This paper investigates the long-range order and scaling laws of 2D incompressible polar active fluids with disorder, revealing universal exponents through multiple renormalization group approaches.

Contribution

It provides the first detailed analysis of the hydrodynamics of 2D polar active fluids with quenched and annealed disorder using three distinct RG schemes.

Findings

Long-range polar order persists despite disorder.

Universal scaling exponents are consistent across different RG methods.

Quenched and annealed disorder contribute similarly to anisotropy and dynamics.

Abstract

We study the moving phase of two-dimensional (2D) incompressible polar active fluids in the presence of both quenched and annealed disorder. We show that long-range polar order persists even in this defect-ridden two-dimensional system. We obtain the large-distance, long-time scaling laws of the velocity fluctuations using three distinct dynamic renormalization group schemes. These are an uncontrolled one-loop calculation in exactly two dimensions, and two $d=(d_c-\epsilon)$-expansions to $O(\epsilon)$, obtained by two different analytic continuations of our 2D model to higher spatial dimensions: a ``hard" continuation which has $d_c={7\over 3}$, and a ``soft" continuation with $d_c={5\over 2}$. Surprisingly, the quenched and annealed parts of the velocity correlation function have the same anisotropy exponent and the relaxational and propagating parts of the dispersion relation have the same dynamic exponent in the nonlinear theory even though they are distinct in the linearized theory. This is due to anomalous hydrodynamics. Furthermore, all three renormalization schemes yield very similar values for the universal exponents, and, therefore, we expect the numerical values we predict for them to be highly accurate.