Characterization of two-dimensional fermionic insulating states

Inspired by the duality picture between superconductivity (SC) and insulator in two spatial dimension (2D), 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 arbitrary two-dimensional 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 noninteracting metal and/or insulator wave function, including gapped mean-field order wave function. Based on this expression, we observed analytically that it indeed vanishes for a wide class of band metals in the thermodynamic limit. On the other hand, it takes a finite value in insulating states, which is quantified by the localization length or the real-valued gauge invariant two form dubbed as the quantum metric tensor. When successively applied along a closed loop, our disorder operator plays the role of twisting the boundary condition of a periodic system. We argue this point by highlighting the Aharonov-Bohm phase associated with this nonlocal operator.