Many universality classes in an interface model restricted to non-negative heights

We present a simple one dimensional stochastic model with three control parameters and a surprisingly rich zoo of phase transitions. At each (discrete) site $x$ and time $t$, an integer $n(x,t)$ satisfies a linear interface equation with added random noise. Depending on the control parameters, this noise may or may not satisfy the detailed balance condition, so that the growing interfaces are in the Edwards-Wilkinson (EW) or in the Kardar-Parisi-Zhang (KPZ) universality class. In addition, there is also a constraint $n(x,t)\ge 0$. Points $x$ where $n>0$ on one side and $n=0$ on the other are called fronts". These fronts can be pushed" or ``pulled", depending on the control parameters. For pulled fronts, the lateral spreading is in the directed percolation (DP) universality class, while it is of a novel type for pushed fronts, with yet another novel behavior in between. In the DP case, the activity at each active site can in general be arbitrarily large, in contrast to previous realizations of DP. Finally, we find two different types of transitions when the interface detaches from the line $n=0$ (with $⟨n(x,t)⟩\to$ const on one side, and $\to \mathrm{\infty }$ on the other), again with new universality classes. We also discuss a mapping of this model to the avalanche propagation in a directed Oslo rice pile model in specially prepared backgrounds.