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 $d=(d_c-\epsilon )$ expansions to $O\left(\epsilon \right)$, obtained by two different analytic continuations of our 2D model to higher spatial dimensions: a “hard” continuation which has $d_c=\frac{7}{3}$, and a “soft” continuation with $d_c=\frac{5}{2}$. 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 that we predict for them to be highly accurate.

Figure 1

Renormalization group flows in $d=2$ in the plane of $g^{\left(\mathrm{unc}\right)}$ and $g_{\mu }^{\left(\mathrm{unc}\right)}$ (50). There are three unstable fixed points (indicated by the triangle, square, and diamond symbols), and one stable fixed point at $(g^{\left(\mathrm{unc}\right)},g_{\mu }^{\left(\mathrm{unc}\right)})=(1,1)$ (indicated by the red circle), which is our focus here.

Figure 2

Plot of the exponents $z_{1,2}$ and ${\zeta }_{1,2}$ versus dimension in the hard continuation to $O(\epsilon =\frac{7}{3}-d$). The exponents $z_2$ and ${\zeta }_2$ become anomalous (i.e., depart from the values $z_2=1$ and ${\zeta }_2=2/3$ below $d=3$). The two $z$'s and the two $\zeta$'s lock onto equality below $d=7/3$. At $d=7/3, z_{1,2}={\zeta }_{1,2}=\frac{2}{3}$; at $d=2, z_{1,2}=\frac{13}{27}, {\zeta }_{1,2}=\frac{7}{9}$. There are small slope discontinuities in $z_2\left(d\right)$ and ${\zeta }_2\left(d\right)$ at $d=7/3$; the slope of $z_2$ changes from $1/2$ to $5/9$, while that of ${\zeta }_2$ changes from $-1/4$ to $-1/3$. Both changes are so small as to be invisible to the naked eye, but are present in this plot.

Figure 3

Feynman diagram representation of the formal solution (A2) of (A1). The circle with an interior cross represents the quenched noise, while the solid circle represents the annealed noise. The meaning of the various other elements in this figure are given in Fig. 4.

Figure 4

Definitions of the elements in the Feynman diagrams: (a) $=u_i\left(\tilde{\mathbf{q}}\right)$, (b) $=\frac{q_y(1-2{\delta }_{ix})}{q_i}G\left(\tilde{\mathbf{q}}\right)f_{{}_Q}^i\left(\tilde{\mathbf{q}}\right)$, (c) $=\frac{q_y(1-2{\delta }_{ix})}{q_i}G\left(\tilde{\mathbf{q}}\right)f_A^i\left(\tilde{\mathbf{q}}\right)$, (d) $=C_{{}_Q}^{ij}\left(\tilde{\mathbf{q}}\right)\delta \left(\omega \right)$, (e) $=C_{{}_A}^{ij}\left(\tilde{\mathbf{q}}\right)$, (f) the circle $=-\alpha (1-\frac{1}{2}{\delta }_{yi})[{\delta }_{xi}+P_{yx}\left(\mathbf{q}\right){\delta }_{yi}]$, (g) the square $=-\alpha /2$.

Figure 5

Feynman diagram representation of the EOM (A7). This amounts to “amputating” the leftmost leg of each of the Feynman diagrams in Fig. 3.

Figure 6

A diagrammatic expansion of the “fast” component of $u_y\left(\tilde{\mathbf{q}}\right)$ after partitioning $u_y\left(\tilde{\mathbf{q}}\right)$ into “slow” components $u_y^{<}\left(\tilde{\mathbf{q}}\right)$ and “fast” components $u_y^{>}\left(\tilde{\mathbf{q}}\right)$.

Figure 7

A diagrammatic expansion of $G^{-1}\left(\tilde{\mathbf{q}}\right)u_y^{<}\left(\tilde{\mathbf{q}}\right)$ after partitioning $u_y\left(\tilde{\mathbf{q}}\right)$ into “slow” components $u_y^{<}\left(\tilde{\mathbf{q}}\right)$ and “fast” components $u_y^{>}\left(\tilde{\mathbf{q}}\right)$.

Figure 8

The graphical representations of the correction to $D_Q$.

Figure 9

Two particular graphical corrections to $D_{{}_A}$. They correspond to the graphs (a) and (b) in Fig. 8 but with one of the quenched noise averages replaced by the annealed noise average.

Figure 10

The graphical representations of potential corrections to $G^{-1}\left(\tilde{\mathbf{q}}\right)u_y\left(\tilde{\mathbf{q}}\right)$.

Figure 11

Two particular graphical corrections to $D_{{}_A}$ purely from the annealed fluctuations. They correspond to the graphs (a) and (b) in Fig. 8 but with both the quenched noise averages replaced by the annealed noise averages.