Quantum Walk over a triangular lattice subject to Pachner move

We present a $2\mathrm{-dimensional}$ quantum walker on curved discrete surfaces with dynamical geometry. This walker extends the quantum walker over the fixed triangular lattice introduced in [PRA, 97(6):062111, 2018]. We write the discrete equations of the walker on an arbitrary triangulation, whose flat spacetime limit recovers the Dirac equation in (2+1)-dimension. The geometry is changed through Pachner moves, allowing the surface to transform into any topologically equivalent surface, starting from a flat space. We present the first theoretical model which couple in a non-linear fashion the dynamics of the walker and the dynamical lattice. Numerical simulations show that both the number of wells and the local curvature grows as $t^a e^{-b t^2}$ and that in long time flatness emerge, in agreement with theory. We also prove that the global behaviour is invariant under random fluctuations of the discrete metrics.