  We develop a general theory of Fermi polarons at nonzero temperature,
including particle-hole excitations of the Fermi sea shake-up to arbitrarily
high orders. The exact set of equations of the spectral function is derived by
using both Chevy ansatz and diagrammatic approach, and their equivalence is
clarified to hold in free space only, with an unregularized infinitesimal
interaction strength. The correction to the polaron spectral function arising
from two-particle-hole excitations is explicitly examined, for an exemplary
case of Fermi polarons in one-dimensional optical lattices. We find
quantitative improvements at low temperatures with the inclusion of
two-particle-hole excitations, in both polaron energies and decay rates. Our
exact theory of Fermi polarons with arbitrary orders of particle-hole
excitations might be used to better understand the intriguing polaron dynamical
responses in two or three dimensions, whether in free space or within lattices.
