Nearsightedness in Materials with Indirect Band Gap

TL;DR

The paper reveals that density-matrix nearsightedness in linear tight-binding models with indirect band gaps is controlled by the indirect gap $\mathsf{gap}_+$, yielding exponential decay with rate $\eta_{+}$ even when $\mathsf{gap}_-$ is small. It provides a sharp bound $|\rho^{\mathrm{ref}}_{\ell k, ab}| \le \frac{C_1}{\mathsf{gap}_+} e^{-\eta_{+} r_{\ell k}}$ and shows $\eta_{+}$ scales with $\min\{h_0, \gamma_0^d \mathsf{gap}_+\}$, refining previous results and explaining strong locality in small-gap semiconductors. The work extends these locality results to finite-energy lattice perturbations, with an additional decay term governed by $\mathsf{gap}_-$ and the perturbation norm, and analyzes the derivatives of the density matrix, which depend on $\mathsf{gap}_-$. A complementary strong locality discussion and numerical experiments (1D toy model and Mg$_2$Si, C) indicate that strong locality does not improve with $\mathsf{gap}_+$, and the observed decay can exhibit square-root behavior in $\mathsf{gap}_-$, highlighting the nuanced difference between nearsightedness and interatomic-force locality. Overall, the indirect-gap focus provides sharper locality estimates that justify linear-scaling approaches even in some small-gap materials, while clarifying the limits of strong locality under perturbations.

Abstract

We investigate the nearsightedness property in the linear tight binding model at zero Fermi-temperature. We focus on the decay property of the density matrix for materials with indirect band gaps. By representing the density matrix in reciprocal space, we establish a qualitatively sharp estimate for the exponential decay rate in homogeneous systems. An extending result under perturbations is also derived. This work refines the estimates presented in (Ortner, Thomas & Chen 2020), particularly for systems with small band gaps.

Figures (9)

• Figure 1: Schematic of the direct and indirect band gaps.
• Figure 2: The band structure of $\rm{Mg_2Si}$ and $\rm{C}$. In Carbon, we have $\mathsf{gap}_- \approx \mathsf{gap}_+$, whereas in $\rm{Mg_2Si}$, we observe $\mathsf{gap}_- \ll \mathsf{gap}_+$.
• Figure 3: A schematic plot of the contour $\mathscr{C}$ circling the whole spectrum of valence bands denoted by the green strip, and the conduction bands denoted in blue.
• Figure 4: 1D chain: Nearsightedness in homogeneous system with fixed $\mathsf{gap}_+ = 2.0$ (left) and $\mathsf{gap}_- = 0.01$ (right); cf. \ref{['thm:nearsight']}.
• Figure 5: 1D chain: the nearsightedness with local perturbations; cf. \ref{['thm:prtb']}

Theorems & Definitions (9)

• proof : Sketch of the Proof of Theorem \ref{['thm:nearsight']}
• proof : Sketch of the Proof
• proof
• proof : Proof of \ref{['thm:nearsight']}
• proof
• proof : Proof of \ref{['thm:prtb']}
• proof
• proof : Proof of Proposition \ref{['prop: str_loc']}
• proof