A Spatial Localizer for Electrons in Insulators

Abstract

The location of electrons governs phenomena ranging from chemical bonding and electric polarization to the topological classification of band insulators and the emergence of correlated states in quantum matter. While a prescription exists for finding local state representations of electrons in one-dimensional insulators, no comparably general theory exists in higher dimensions. Here, we introduce a general framework for finding the location of electrons in insulators in two and three dimensions based on the spectral properties of quantum-mechanical operators that we term Spatial Localizers. This framework naturally extends the notion of Wannier centers to insulators with boundaries, defects, and disorder, which we use to establish a position-space formulation of the bulk-defect correspondence for electronic charge. This framework also yields maximally localized electronic states. As two representative examples, we show that these states reduce to maximally localized Wannier functions in atomic insulators, whereas in Chern insulators they form coherent states that mirror the coherent-state structure of Landau levels in the quantum Hall effect.

Figures (4)

• Figure 1: Spatial Localizer in the obstructed atomic insulator $\mathrm{WSe_2}$. (a) Atomic structure of $\mathrm{WSe_2}$; the grey hexagon denotes a unit cell centered at a W atom. The W and Se atoms are colored grey and yellow, respectively. (b) LIF for the effective $\mathrm{WSe_2}$ model. The red cross denotes the extraction point for the WF in (c). (c) WF constructed from the eigenstate of the Spatial Localizer $L({\bf r}^W)$ with minimal $\mu({\bf r}^W)$ at the center of the plot. The WF is purely real. The color bar uses a linear color scaling. White dots with black edges in (b) and (c) indicate $\mathrm{W}$ atoms for reference. (d) LIF around a WC and its Wannier monopole as a topological charge (sphere).
• Figure 2: Wannier centers in insulators with equal band representations. LIF over the unit cell of an insulator in two phases protected by the space group F222. A trivial insulator (a) and an OAL (b). An octant of the unit cell has been removed to reveal its center, $\frac{1}{2} (\mathbf{a_1} + \mathbf{a_2} + \mathbf{a_3})$. Both phases have identical band representations, and thus their Wannier centers cannot be determined from symmetry considerations.
• Figure 3: Bulk-defect correspondence for electronic charge around a topological defect. LIF for a crystalline insulator with a disclination and two electrons per unit cell in an OAL phase (a) and a trivial phase (b). Both figures follow the color bar on the right. Grey lines denote unit cell boundaries. (c,d) Net charge densities (with $+2e$ ionic charges per unit cell) quantized to $e/2$ in (c) and zero in (d). Excess edge charge is not shown for clarity.
• Figure 4: The Spatial Localizer in a Chern insulator. (a) and (b) Magnitude of WFs and coherent states as a function of distance from the mean, $|\mathbf{r}|$ for $|\mathbf{r}|>0.5$. In (a), the fit line is proportional to $1/r^{2}$. (c) and (d) Berry connection $\mathcal{A}({\bf k})$ over the Brillouin zone obtained from the coherent state in (a) and WF in (b). The color on (c) and (d) corresponds to the magnitude of the Berry connection. The left (right) column corresponds to the topological (trivial) phase. All states are calculated on an $N\times N$ square lattice of unit cells with $N=50$. Insets: LIF for a single unit cell with $N=24$. Details on the insets are provided in Appendix F.