  Without Lorentz invariance, spontaneous global symmetry breaking can lead to
multicritical Nambu-Goldstone modes with a higher-order low-energy dispersion
$\omega\sim k^n$ ($n=2,3,\ldots$), whose naturalness is protected by polynomial
shift symmetries. Here we investigate the role of infrared divergences and the
nonrelativistic generalization of the Coleman-Hohenberg-Mermin-Wagner (CHMW)
theorem. We find novel cascading phenomena with large hierarchies between the
scales at which the value of $n$ changes, leading to an evasion of the "no-go"
consequences of the relativistic CHMW theorem.
