Power-law correlation in the homogeneous disordered state of anisotropically self-propelled systems

Self-propelled particles display unique collective phenomena, due to the intrinsic coupling of density and polarity. For instance, the giant number fluctuation appears in the orientationally ordered state, and the motility-induced phase separation appears in systems with repulsion. Effects of strong noise typically lead to a homogeneous disordered state, in which the coupling of density and polarity can still play a significant role. Here we study universal properties of the homogeneous disordered state in two-dimensional systems with uniaxially anisotropic self-propulsion. Using hydrodynamic arguments, we propose that the density correlation and polarity correlation generically exhibit power-law decay with distinct exponents (-2 and -4, respectively) through the coupling of density and polarity. Simulations of self-propelled lattice gas models indeed show the predicted power-law correlations, regardless of whether the interaction type is repulsion or alignment. Further, by mapping the model to a two-component boson system and employing non-Hermitian perturbation theory, we obtain the analytical expression for the structure factors, the Fourier transform of the correlation functions. This reveals that even the first order of the interaction strength induces the power-law correlations.