  We study a Bose-Einstein condensate (BEC) at low energy limit and show that
their collective dynamics exhibit interesting topological behavior. The system
undergoes dynamical topological phase transition at its global periods if its
dispersion relation is strictly linear and when all of the bosonic mode are
properly displaced from their equilibrium position. We corroborate the
occurrence of dynamical phase transition by calculating Fisher zeros of the
Loschmidt amplitude for sudden quench dynamics. A connection is established
between the order of nonanalycity in the accumulated geometric phase and the
spectral density of the system. Furthermore, it is shown that a power law
scaling holds at all critical times for various displacement spectra, whose
dynamical exponent equals unity. Eventually, a scheme for the quantum
simulation of such dynamical phase transition is proposed. The scheme is based
on the vibrational spectrum of a free-standing membrane of a two-dimensional
material. To induce a displacement in the system and for tracking its
collective geometrical dynamics, we propose to employ a spin that properly
couples to the modes. When appropriate geometrical and boundary conditions are
applied to the membrane, a spectrum with linear dispersion is attainable, then
decoherence dynamics of the spin unveils occurrence of a dynamical topological
phase transition.
