  Inspired by the duality picture between superconductivity and insulator in
two spatial dimension, we conjecture that the order parameter, suitable for
characterizing 2D fermionic insulating state, is the disorder operator, usually
known in the context of statistical transformation. Namely, the change of the
phase of the disorder operator along a closed loop measures the particle
density accommodating inside this loop. Thus, identifying this (doped) particle
density with the dual counterpart of the magnetic induction in 2D SC, we can
naturally introduce the disorder operator as the dual order parameter of 2D
insulators. The disorder operator has a branch cut emitting from this
``vortex'' to the single infinitely far point. To test this conjecture against
an arbitrary 2D lattice models, we have chosen this branch cut to be compatible
with the periodic boundary condition and obtain a general form of its
expectation value for non-interacting metal/insulator wavefunction, including
gapped mean-field order wavefunction. Based on this expression, we observed
analytically that it indeed vanishes for a wide class of band metals in the
thermodynamic limit. In insulating states, on the other hand, it is quantified
by the localization length or the real-valued gauge invariant 2-from dubbed as
the quantum metric tensor.
