  A Plastic Quantum Walk admits both continuous time and continuous spacetime.
The model has been recently proposed by one of the authors in
\cite{molfetta2019quantum}, leading to a general quantum simulation scheme for
simulating fermions in the relativistic and non relativistic regimes. The
extension to two physical dimensions is still missing and here, as a novel
result, we demonstrate necessary and sufficient conditions concerning which
discrete time quantum walks can admit plasticity, showing the resulting
Hamiltonians. We consider coin operators as general $4$ parameter unitary
matrices, with parameters which are function of the lattice step size
$\varepsilon$. This dependence on $\varepsilon$ encapsulates all functions of
$\varepsilon$ for which a Taylor series expansion in $\varepsilon$ is well
defined, making our results very general.
