  Landau levels of a band are conventionally believed to be bounded by its
upper and lower band edges at zero magnetic field so that no Landau level is
expected to appear in the gapped regions of the original band structure. Two
notable examples evading this expectation are topological bands with quantized
invariants and flat bands with singular band crossing. Here we introduce a
distinct class of flat band systems in which an isolated flat band exhibits
anomalous Landau level spreading (LLS) outside the zero-field energy bounds. In
particular, we demonstrate that the LLS is determined by the cross-gap Berry
connection measuring the wave function geometry of multi-bands. Moreover, we
find that symmetry puts strong constraints on the LLS of flat bands. Namely,
the LLS hows a quadratic magnetic field dependence in systems with
space-time-inversion symmetry while it completely vanishes when the system
respects chiral symmetry. Also, when the system respects time-reversal or
reflection symmetries at zero magnetic field, the maximum and minimum values of
the LLS have the same magnitude but with opposite signs. Our general mechanism
for the LLS of flat bands reveals the fundamental role of wave function
geometry in describing anomalous magnetic responses in solids.
