  We introduce a many-body topological invariant, called the topological
disorder parameter (TDP), to characterize gapped quantum phases with global
internal symmetry in (2+1)d. TDP is defined as the constant correction that
appears in the ground state expectation value of a partial symmetry
transformation applied to a connected spatial region $M$, the absolute value of
which scales generically as $\exp(-\alpha l+\gamma)$ where $l$ is the perimeter
of $M$ and $\gamma$ is the TDP. Motivated by a topological quantum field theory
interpretation of the operator, we show that $e^\gamma$ can be related to the
quantum dimension of the symmetry defect, and provide a general formula for
$\gamma$ when the entanglement Hamiltonian of the topological phase can be
described by a (1+1)d conformal field theory (CFT). A special case of TDP is
equivalent to the topological R\'enyi entanglement entropy when the symmetry is
the cyclic permutation of the replica of the gapped phase. We then investigate
several examples of lattice models of topological phases, both analytically and
numerically, in particular when the assumption of having a CFT edge theory is
not satisfied. We also consider an example of partial translation symmetry in
Wen's plaquette model and show that the result can be understood using the edge
CFT. Our results establish a new tool to detect quantum topological order.
