  We study the magnetoresistance of two-dimensional bosonic Anderson
insulators. We describe the change in spatial decay of localized excitations in
response to a magnetic field, which is given by an interference sum over
alternative tunnelling trajectories. The excitations become more localized with
increasing field (in sharp contrast to generic fermionic excitations which get
weakly delocalized): the localization length \xi(B) is found to change as
\xi^{-1}(B)-\xi^{-1}(0)\sim B^{4/5}. The quantum interference problem maps onto
the classical statistical mechanics of directed polymers in random media
(DPRM). We explain the observed scaling using a simplified droplet model which
incorporates the non-trivial DPRM exponents. Our results have implications for
a variety of experiments on magnetic-field-tuned superconductor-to-insulator
transitions observed in disordered films, granular superconductors, and
Josephson junction arrays, as well as for cold atoms in artificial gauge
fields.
