Characterization of two-dimensional fermionic insulating states

TL;DR

This paper proposes a duality-based disorder operator as a novel order parameter for 2D fermionic insulators, and analytically demonstrates its effectiveness in distinguishing insulating states from metals.

Contribution

It introduces a disorder operator as a dual order parameter for 2D insulators and derives its expectation value for various lattice models, validating its ability to differentiate insulators from metals.

Findings

Disorder operator expectation value vanishes in metals in the thermodynamic limit.

In insulators, the expectation value relates to localization length and quantum metric tensor.

The approach is compatible with periodic boundary conditions and applies to non-interacting wavefunctions.

Abstract

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.