  Spontaneous symmetry breaking and elementary excitation are two of the
pillars of condensed matter physics that are closely related to each other. The
symmetry and its spontaneous breaking not only control the dynamics and
spectrum of elementary excitations, but also determine their underlying
structures. In this paper, we study the excitation properties of a
non-equilibrium quantum matter: a discrete time crystal phase that
spontaneously breaks the temporal translational symmetry. It is shown that such
an intriguing symmetry breaking allows an instanton-like excitation that
represents a tunneling between two "degenerate" time crystal phases.
Furthermore, we also observe a dynamical transition point at which the
instanton "size" diverges, a reminiscence of the critical slowing down
phenomenon in nonequilibrium statistic physics. A phenomenological theory has
been proposed to understand the phase dynamics of the proposed system and the
experimental realization and detection have also been discussed.
