Taking the clue from the modern theory of polarization [Rev. Mod. Phys. 66, 899 (1994)], we identify an operator to distinguish between ${\mathbb{Z}}_2$-even (trivial) and ${\mathbb{Z}}_2$-odd (topological) insulators in two spatial dimensions. Its definition extends the position operator [Phys. Rev. Lett. 82, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of noninteracting models where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus, allowing us to investigate generic interacting systems, such as strongly correlated topological insulators.

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• Topological insulators
• Topological phase transition
• Topological materials
• Second quantization
• Trial wave functions