  As an unconventional realization of topological orders with an exotic
interplay of topology and geometry, fracton (topological) orders feature
subextensive topological ground state degeneracy and subdimensional excitations
that are movable only within a certain subspace. It has been known in the
exactly solvable three-dimensional X-cube model that universally represents the
type-I fracton orders, that mobility constraints on subdimensional excitations
originate from the absence of spatially deformable string-like operators. To
unveil the interplay of topology and geometry, in this paper, we study the
dynamical signature in the X-cube model in the presence of external Zeeman
fields via large-scale quantum Monte Carlo simulation and stochastic analytic
continuation. We compute both real-space correlation functions and dynamic
structure factors of subdimensional excitations (i.e., fractons, lineons, and
planons) in the fracton phase and their evolution into the trivial paramagnetic
phase by increasing external fields. We find in the fracton phase, that the
correlation functions and the spectral functions show clear anisotropy exactly
caused by the underlying mobility constraints. On the other hand, the external
fields successfully induce quantum fluctuations and offer mobility to
excitations along the subspace allowed by mobility constraints. These numerical
results provide the evolution of a dynamical signature of subdimensional
particles in fracton orders, indicating that the mobility constraints on local
dynamical properties of subdimensional excitations are deeply related to the
existence of fracton topological order. The results will also be helpful in
potential experimental identifications in spectroscopy measurements such as
neutron scattering and nuclear magnetic resonance.
