Hydrodynamic theory of <i>p-</i>atic liquid crystals

We formulate a comprehensive hydrodynamic theory of two dimensional liquid crystals having p-fold rotational symmetry often referred to as
p-atics. The p-atic phase can be realized in p-fold symmetric colloidal suspensions at the interface, in two-dimensional melting as an intermediate state between crystalline solids and isotropic liquids, and also in biological tissues. We construct the p-atic tensor order parameter Q_p, which embodies the rotational symmetry of the p-atic phase, and identify novel couplings between p-atic order and flow using a phenomological approach, which can be expressed in terms of linear and nonlinear functions of the strain rate tensor. We uncover shear dependent flow alignment of p-atics due to these nonlinear couplings, not found in nematic liquid crytals. Finally using the hydrodynamic formulation, we show that a shear flow of arbitrary finite shear rate can remarkably give rise to long ranged orientational order of p-atics in equilibrium.