  In recent years, new phases of matter that are beyond the Landau paradigm of
symmetry breaking are mountaining, and to catch up with this fast development,
new notions of global symmetry are introduced. Among them, the higher-form
symmetry, whose symmetry charges are spatially extended, can be used to
describe topologically ordered phases as the spontaneous breaking of the
symmetry, and consequently unify the unconventional and conventional phases
under the same conceptual framework. However, such conceptual tools have not
been put into quantitative test except for certain solvable models, therefore
limiting its usage in the more generic quantum manybody systems. In this work,
we study Z2 higher-form symmetry in a quantum Ising model, which is dual to the
global (0-form) Ising symmetry. We compute the expectation value of the Ising
disorder operator, which is a non-local order parameter for the higher-form
symmetry, analytically in free scalar theories and through unbiased quantum
Monte Carlo simulations for the interacting fixed point in (2+1)d. From the
scaling form of this extended object, we confirm that the higher-form symmetry
is indeed spontaneously broken inside the paramagnetic, or quantum disordered
phase (in the Landau sense), but remains symmetric in the ferromagnetic/ordered
phase. At the Ising critical point, we find that the higher-form symmetry is
also spontaneously broken, even though the 0-form symmetry is preserved. We
discuss examples where both the global 0-form symmetry and the dual higher-form
symmetry are preserved, in systems with a codimension-1 manifold of gapless
points in momentum space. These results provide non-trivial working examples of
higher-form symmetry operators, including the first computation of one-form
order parameter in an interacting conformal field theory, and open the avenue
for their generic implementation in quantum many-body systems.
