A diffuse-interface theory of active nematic interfaces: transport mechanisms and modal structure

We develop a long-wavelength theory for the linear stability of a flat interface between an active nematic and an isotropic fluid. Starting from a diffuse-interface Cahn--Hilliard--Landau--de Gennes description coupled to Brinkman-screened Stokes hydrodynamics, we project the linearized dynamics onto a small set of interfacial degrees of freedom: the conserved translation, or height, mode; a scalar profile distortion or amplitude mode; and a transverse orientational mode associated with director rotations. Eliminating the gapped scalar profile mode gives a reduced interfacial operator coupling the conserved height mode to the transverse orientational mode. The main result is that activity generates, in the screened diffuse-interface regime, a direct local contribution proportional to $q^2$ in the height sector. This term competes with the passive local diffusive capillary relaxation, which enters at order $q^4$, and defines a local active interfacial channel controlled by the internal structure of the diffuse interface. This mechanism is distinct from the non-analytic $|q|$ and $|q|q^2$ terms characteristic of weakly screened Hele--Shaw/Saffman--Taylor-type transport, which are controlled by long-ranged momentum transport in the surrounding fluid. This framework identifies a diffuse-interface route to active interfacial instability that can operate while the homogeneous active nematic remains linearly stable because of hydrodynamic screening. It also provides a basis for distinguishing local diffuse-interface instabilities, bulk-flow-driven hydrodynamic instabilities, and mixed regimes in active nematic--isotropic interfaces.