Rough or crumpled: Strong coupling phases of a generalized Kardar-Parisi-Zhang surface

We study a generalized Kardar-Parisi-Zhang (KPZ) equation [Jana et al., Phys. Rev. E 109, L032104 (2024)] that sets the paradigm for universality in roughening of growing nonequilibrium surfaces without any conservation laws but with competing local and nonlocal nonlinear effects. This equation in two dimensions exhibits two distinct strong coupling regimes: a rough phase and a crumpled phase, in addition to a weak coupling phase. The conformation fluctuations of such a rough surface are given by nonuniversal scaling exponents, with orientational long-range order and positional short-range order, whereas the crumpled phase has positional and orientational short-range order.