Unfolding Bloch States in Disordered Systems

1. Problem Statement

In crystalline solids, the introduction of defects, impurities, or alloying breaks translational symmetry. This necessitates the use of Supercells (SC) in first-principles calculations to model disorder. However, using SCs reduces the Brillouin zone (BZ), causing the primitive cell (PC) bands to fold into a smaller, dense, and overlapping dispersion. This obscures the underlying physics and hinders direct comparison with $k$-resolved experimental data (e.g., ARPES).

While band-unfolding techniques have been developed to restore an effective PC band structure by projecting SC eigenstates onto PC Bloch states, they suffer from a critical limitation:

• Current Limitation: Conventional unfolding only provides energy dispersions and spectral weights (broadening). It does not yield the corresponding unfolded Bloch eigenstates (wavefunctions).
• Consequence: Without well-defined $k$-resolved wavefunctions, it is impossible to calculate wavefunction-based observables essential for modern condensed matter physics, such as:
  • Optical transition matrix elements.
  • Orbital magnetic moments.
  • Nonlinear optical coefficients (e.g., shift current).
  • Geometric and topological quantities (Berry connection, Berry phase, and Berry curvature).

2. Methodology

The authors propose a theoretical framework that reverses the conventional unfolding workflow to directly obtain unfolded Bloch states. The method relies on a $k$-resolved decomposition of the disorder Hamiltonian.

Key Theoretical Steps:

1. Hamiltonian Decomposition:
   The total Hamiltonian is written as $\hat{H} = \hat{H}_0 + \hat{H}_{dis}$, where $\hat{H}_0$ is the pristine Hamiltonian and $\hat{H}_{dis}$ is the disorder potential.
   The disorder Hamiltonian is decomposed into:

   • Intra-$k$ blocks ($\hat{H}^k_{dis}$): Diagonal in crystal momentum $k$. These dominate the physics.
   • Inter-$k$ blocks ($\hat{H}^{k',k}_{dis}$): Off-diagonal terms coupling different momenta ($k \neq k'$).
   • Assumption: For randomly distributed disorder, the inter-$k$ terms vanish under ensemble averaging or are parametrically small ($||\hat{H}^{k',k}_{dis}|| \ll ||\hat{H}^k_{dis}||$) in the thermodynamic limit.

2. Reversed Workflow:

   • Step 1 (Projection): Instead of diagonalizing the large SC Hamiltonian first, the authors project the total Hamiltonian $\hat{H}$ onto the pristine PC Bloch basis $|k, n\rangle$.
   • Step 2 (Block Diagonalization): Because the dominant part ($\hat{H}_0 + \hat{H}^k_{dis}$) is block-diagonal in $k$, the large matrix is broken into small independent blocks (one for each $k$).
   • Step 3 (Diagonalization): Each small $k$-block is diagonalized independently to yield:
     • Dressed Bloch Energies ($\tilde{\varepsilon}_{k,n}$): Renormalized band energies.
     • Dressed Bloch States ($|\tilde{k}, n\rangle$): Well-defined, $k$-resolved wavefunctions in the PC BZ.

3. Spectral Weight Calculation:
   The spectral broadening (spectral weight) is calculated using Fermi's Golden Rule. The inter-$k$ coupling terms ($\hat{H}^{k',k}_{dis}$) are treated as a perturbation that induces scattering between dressed states. The scattering rate $\bar{\Gamma}_{k,n}$ determines the linewidth of the unfolded states.

3. Key Contributions

• Unified Framework: The first method to simultaneously unfold band structures and Bloch wavefunctions in disordered systems.
• Computational Efficiency: By diagonalizing small $k$-blocks rather than a single large SC matrix, the method offers substantial computational savings.
• Access to Topological Observables: It enables the direct calculation of geometric quantities (like Berry curvature) in the proper primitive-cell $k$-space, avoiding the artifacts of mapping SC $k$-space results onto the PC BZ.
• Physical Insight: It distinguishes between the renormalization of states (intra-$k$) and the scattering/broadening (inter-$k$) within a single consistent formalism.

4. Results: Application to Disordered Graphene

The authors applied the method to graphene with two types of on-site disorder (concentrations $\rho = 1\%$ and $5\%$):

1. Symmetry-Breaking (SB) Disorder: Positive and negative potentials are assigned exclusively to A and B sublattices, respectively.
   • Result: Induces a bandgap (~0.05 eV).
   • Berry Curvature (BC): The BC becomes spatially broadened around the $K$ and $K'$ valleys, losing sharp localization due to the broken inversion symmetry.
2. Symmetry-Preserving (SP) Disorder: Equal numbers of positive and negative potentials on both sublattices.
   • Result: Remains gapless; the dispersion closely follows the pristine graphene band.
   • Berry Curvature (BC): The BC remains sharply localized near the valleys, preserving the robust $\pi$ Berry phase.
3. Spectral Broadening:
   • Calculated scattering rates ($\Gamma$) show that broadening increases with defect concentration.
   • Broadening is minimal near the $K$ points but increases significantly near the $M$ point (where the density of states is higher).
   • The method successfully captures both intravalley and intervalley scattering features.

5. Significance

This work bridges a critical gap in the theoretical description of disordered materials. By providing access to unfolded Bloch states, the method allows researchers to:

• Calculate wavefunction-dependent properties (optical absorption, nonlinear optics, magnetic moments) in alloys and defected materials, which were previously inaccessible via standard unfolding.
• Perform direct geometric and topological analysis (e.g., Berry curvature, anomalous Hall conductivity) in the correct $k$-space, ensuring the physical interpretation of these quantities remains valid.
• Offer a practical, computationally efficient route for studying complex disordered systems beyond the limitations of supercell approximations.

In essence, the paper transforms the treatment of disorder from a purely spectral (energy-based) problem to a full wavefunction-based problem, opening new avenues for designing and analyzing topological and optoelectronic materials with defects.