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Burgers-Kardar-Parisi-Zhang (KPZ) scaling has recently (re-) surfaced in avariety of physical contexts, ranging from anharmonic chains to quantum systemssuch as open superfluids, in which a variety of random forces may beencountered and/or engineered. Motivated by these developments, we here providea generalization of the KPZ universality class to situations with long-rangedtemporal correlations in the noise, which purposefully break the Galileaninvariance that is central to the conventional KPZ solution. We compute thephase diagram and critical exponents of the KPZ equation with $1/f$-noise(KPZ$_{1/f}$) in spatial dimensions $1\leq d < 4$ using the dynamicrenormalization group with a frequency cutoff technique in a one-looptruncation. Distinct features of KPZ$_{1/f}$ are: (i) a genericallyscale-invariant, rough phase at high noise levels that violatesfluctuation-dissipation relations and exhibits hyperthermal statistics {\iteven in d=1}, (ii) a fine-tuned roughening transition at which the flowfulfills an emergent thermal-like fluctuation-dissipation relation, thatseparates the rough phase from (iii) a {\it massive phase} in $1< d < 4$ (in$d=1$ the interface is always rough). We point out potential connections tononlinear hydrodynamics with a reduced set of conservation laws and noisyquantum liquids.

Dynamic criticality far-from-equilibrium: one-loop flow of Burgers-Kardar-Parisi-Zhang systems with broken Galilean invariance