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

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 = (dc -e) expansions to O(e), obtained by two different analytic continuations of our 2D model to higher spatial dimensions: a hard continuation which has dc = 73 , and a soft continuation with dc = 52. 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 that we predict for them to be highly accurate.

Automated summary

We study the moving phase of two-dimensional incompressible polar active fluids in the presence of both quenched and annealed disorder. We show that long-range polar order persists in this defect-ridden two-dimensional system. By employing three distinct dynamic renormalization group schemes, we obtain the scaling laws of velocity fluctuations at large distances and long times. Surprisingly, the quenched and annealed parts of the velocity correlation function exhibit the same anisotropy exponent and the relaxational and propagating parts of the dispersion relation have the same dynamic exponent in the nonlinear theory, despite being distinct in the linearized theory. This is attributed to anomalous hydrodynamics. Furthermore, the three renormalization schemes yield similar universal exponents, indicating the high accuracy of the numerical values predicted.