We show that generic kinetic growth processes with surface relaxations can exhibit a hitherto unexplored crumpled phase with short-range orientational order at dimensions $d<4$. A sufficiently strong spatially nonlocal part of the chemical potential associated with the particle current above a threshold in the system can trigger this crumpling. The system can also be in a perturbatively accessible rough phase with long-range orientational order but short-range positional order at $d<4$ with known scaling exponents. Intriguingly, in $d>4$ we argue that there is no crumpling transition; instead, there is a roughening transition from a smooth to a rough phase for large enough nonlocal particle chemical potential. Experimental and theoretical implications of these results are discussed.

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• Nonequilibrium statistical mechanics
• Nonequilibrium systems
• Kardar–Parisi–Zhang equation
• Renormalization