A simple position operator for periodic systems

We present a position operator that is compatible with periodic boundary conditions (PBCs). It is a one-body operator that can be applied in calculations of correlated materials by simply replacing the traditional position vector by the new definition. We show that it satisfies important fundamental as well as practical constraints. To illustrate the usefulness of the PBC position operator, we apply it to the localization tensor, a key quantity that is able to differentiate metallic from insulating states. In particular, we show that the localization tensor given in terms of the PBC position operator yields the correct expression in the thermodynamic limit. Moreover, we show that it correctly distinguishes between finite precursors of metals and insulators.

Figure 1

The OBC localization tensor $\lambda \left(N\right)$ (solid lines) and the PBC localization tensor ${\lambda }_L\left(N\right)$ (dashed lines) as a function of the number of electrons $N$ for various values of the dimerization parameter $\delta$ in a tight-binding model [see Eqs. (16) and (17)]. The dots and numbers next to each curve are the values of the PBC localization tensor in the thermodynamic limit. They were obtained from Eq. (22). Inset: ${\lambda }_{\left(L\right)}\left(N\right)$ for $\delta =0.04$ and $\delta =0.1$ for large $N$.

Figure 2

The OBC polarizability $\overline{\alpha }\left(N\right)$ (solid lines) and the PBC polarizability ${\overline{\alpha }}_L\left(N\right)$ (dashed lines) as a function of the number of electrons $N$ for various values of the dimerization parameter $\delta$ in a tight-binding model. The dots and numbers next to each curve are the values of the PBC localization tensor in the thermodynamic limit. They were obtained from Eq. (24). Inset: $\overline{\alpha }\left(N\right)$ and ${\overline{\alpha }}_{\left(L\right)}\left(N\right)$ for $\delta =0.04$ and $\delta =0.1$ for large $N$.

Figure 3

The PBC localization tensor ${\lambda }_L\left(N\right)$ as a function of the number of electrons $N$ for various values of the electronegativity difference $\gamma$ for a square lattice in a tight-binding model.

Figure 4

The PBC polarizability ${\overline{\alpha }}_L\left(N\right)$ as a function of the number of electrons $N$ for various values of the electronegativity difference $\gamma$ for a square lattice in a tight-binding model.

Figure 5

The OBC spin localization tensor ${\lambda }^S\left(N\right)$ (solid lines) and the PBC spin localization tensor ${\lambda }_L^S\left(N\right)$ (dashed lines) as a function of the number of electrons $N$ for various values of the dimerization parameter $\delta$ in a Heisenberg model.