  Synthetic gauge fields have provided physicists with a unique tool to explore
a wide range of fundamentally important phenomena. However, most experiments
have been focusing on synthetic vector gauge fields. The very rich physics
brought by coupling tensor gauge fields to fracton phase of matter remain
unexplored in laboratories. Here, we propose schemes to realize synthetic
tensor gauge fields that address dipoles instead of single-particles. A lattice
tilted by a strong linear potential and a weak quadratic potential yields a
rank-2 electric field for a lineon formed by a particle-hole pair. Such a
rank-2 electric field leads to a new type of Bloch oscillations, which modulate
the quadrupole moment and preserve the dipole moment of the system. In higher
dimensions, the interplay between interactions and vector gauge potentials
imprints a phase to the ring-exchange interaction and thus generates synthetic
tensor gauge fields for planons. Such tensor gauge fields make it possible to
realize a dipolar Harper-Hofstadter model in laboratories. The resultant
dipolar Chern insulators feature chiral edge currents of dipoles in the absence
of net charge currents.
