Unfolding Bloch States in Disordered Systems

Abstract

In crystalline solids, disorder breaks translational symmetry and obscures k-resolved Bloch states, limiting an accurate description of wavefunction-based observables. In this work, we present a method that unfolds not only the band structures but also the corresponding Bloch states in disordered systems, going beyond conventional band-unfolding techniques. As a prototype application, we study defective graphene and demonstrate the capabilities by capturing key wavefunction-level responses, including disorder-driven redistribution of Berry curvature.

Figures (4)

• Figure 1: Workflow for unfolding Bloch states in disordered graphene. (a) Diagonalization of the pristine Hamiltonian $\hat{H}_0$ at each $\bm{k}$-point in the primitive-cell (PC) Brillouin zone (BZ), yielding the pristine Bloch eigenstate $|\bm{k},\pm\rangle$. (b) Representation of the defect Hamiltonian $\hat{H}_{DG}$ using $|\bm{k},\pm\rangle$, i.e., $\langle \bm{k},s|\hat{H}_{DG}|\bm{k'},s'\rangle$ with $s,s' \in \{+,-\}$. (c) Block-diagonalization of $\hat{H}_{DG}$ for a given $\bm{k}$ to find the dressed Bloch energies $\tilde{\varepsilon}_{\bm{k},n}$ and dressed Bloch states $|\widetilde{\bm{k},\pm}\rangle$. (d) Evaluating the spectral weight $\Gamma_{\bm{k},n}$ using the calculated dressed Bloch states $|\widetilde{\bm{k},\pm}\rangle$.
• Figure 2: Disordered-graphene models and their unfolded electronic structures. (a) Symmetry-breaking (SB) disorder, where positive (red) and negative (blue) on-site energies are assigned exclusively to the A and B sublattices, respectively. (b) Symmetry-preserving (SP) disorder, where each sublattice contains equal numbers of the positive and negative on-site energy. (c,d) Dressed band structure $\tilde{\varepsilon}_{\bm{k}, n}$ for (c) SB and (d) SP disorder (magenta), shown together with the pristine graphene bands (gray) for comparison. (e) Unfolded band structure for the SP case including spectral broadening ($\rho = 5\%$).
• Figure 3: Disorder-induced scattering and spectral broadening in disordered graphene (SP disorder). Momentum-resolved scattering rate $\Gamma^{d}_{\bm{k},-\rightarrow \bm{k'},-}$ from a fixed initial dressed state $|\widetilde{\bm{k},-}\rangle$ chosen near the $\overline{K}$ valley (indicated by the yellow arrow) for defect concentrations (a) $\rho = 1\%$ and (b) $\rho = 5\%$. The arrows in (a) highlight representative intra- and inter-valley scatterings. Distribution of the total spectral broadening $\Gamma_{\bm{k} ,n}$ over the first Brillouin zone for (c) $\rho = 1\%$ and (d) $\rho = 5\%$.
• Figure 4: Distribution of Berry curvature in the Brillouin zone for the SB and SP disorders. Panels (a,c) show SB disorder and panels (b,d) show SP disorder at defect concentrations (a,b) $\rho = 1\%$ and (c,d) $\rho = 5\%$. The color scale indicates the sign and magnitude of the Berry curvature, with contributions concentrated near the valley points K and $\overline{K}$.