  Broken symmetries in solids involving higher order multipolar degrees of
freedom are historically referred to as "hidden orders" due to the formidable
task of detecting them with conventional probes. Examples of such hidden orders
include spin-nematic order in quantum magnets, and quadrupolar or higher
multipolar orders in various correlated quantum materials. In this work, we
theoretically propose that the study of magnetostriction provides a powerful
and novel tool to directly detect higher-order multipolar symmetry breaking $-$
such as the elusive octupolar order $-$ by examining its scaling behaviour with
respect to an applied magnetic field $h$. As an illustrative example, we
examine such key scaling signatures in the context of Pr-based cage compounds
with strongly correlated $f$-electrons, Pr(Ti,V,Ir)$_2$(Al,Zn)$_{20}$, whose
low energy degrees of freedom are composed of purely higher-order multipoles:
quadrupoles $\mathcal{O}_{20,22}$ and octupole $\mathcal{T}_{xyz}$. Employing a
symmetry-based Landau theory of multipolar moments coupled to lattice strain
fields, we demonstrate that a magnetic field applied along the [111] direction
results in a length change with a distinct linear-in-$h$ scaling behaviour,
accompanied by hysteresis, below the octupolar ordering temperature. We show
that the resulting "magnetostriction coefficient" is directly proportional to
the octupolar order parameter, providing the first clear access to this subtle
order parameter. Along other field directions, we show that the field
dependence of the magnetostriction provides a window into quadrupolar orders.
Our work provides a springboard for future experimental and theoretical
investigations of multipolar orders and their quantum phase transitions in a
wide variety of systems.
