Inter-band coherence effects in disordered crystals: beyond the non-crossing approximation

Technical Summary: Inter-band coherence effects in disordered crystals: beyond the non-crossing approximation

Problem Statement
The paper addresses the theoretical challenge of accurately describing non-equilibrium transport in disordered crystalline systems, specifically focusing on phenomena where inter-band dynamics and quantum interference are critical. While standard semiclassical approaches and the non-crossing approximation (NCA) successfully capture leading-order disorder effects (scaling as $\tau^{-1}$), they fail to account for higher-order quantum interference terms that appear at zeroth order in the disorder strength ($\tau^0$). These terms, arising from coherent multiple scattering off impurities (crossed diagrams), are known to significantly alter response functions, particularly in the anomalous Hall effect (AHE), spin Hall effect, and orbital Hall effect. Existing diagrammatic techniques (Kubo, Keldysh) have identified these contributions (specifically X and $\Psi$ diagrams), but a corresponding formulation within the Density Matrix Kinetic Equation (DMKE) framework has been lacking. The DMKE is preferred for its ability to naturally separate Fermi surface and Fermi sea contributions and to handle band geometry (Berry curvature, quantum metric) and inter-band coherence transparently.

Methodology
The authors develop a quantum kinetic theory for Bloch electrons driven by a uniform DC electric field within the length gauge. The core of the methodology involves:

1. Iterative Solution of the DMKE: Starting from the quantum Liouville equation, the single-particle density matrix $\rho$ is decomposed into an impurity-averaged component $\langle \rho \rangle$ and impurity-induced fluctuations $g$. The collision integral $I = -i\hbar^{-1}\langle [V, g] \rangle$ is constructed by iteratively solving for $g$ in powers of the impurity potential $V$.
2. Fourth-Order Expansion: The authors explicitly solve for the collision integral up to the fourth order in the disorder potential ($V^4$). This order is necessary to capture the leading quantum interference effects (crossed impurity processes) in a Gaussian white noise model, which vanish at lower orders.
3. Handling Disconnected Terms: A crucial technical step involves the systematic removal of disconnected impurity pairings (terms that factorize into products of lower-order collisions, e.g., $I^{(2)}[I^{(2)}]$) to avoid double counting. This ensures the resulting collision integral contains only connected, coherent scattering processes.
4. Mapping to Keldysh Formalism: The authors establish a rigorous term-by-term dictionary between the DMKE commutator structure and the Keldysh contour expansion. They demonstrate that the nested commutators in the DMKE correspond to the placement of impurity vertices on the forward and backward branches of the Keldysh contour, reproducing standard self-energy, ladder (vertex correction), and crossed (X and $\Psi$) topologies.

Key Contributions

• Formulation of the Crossed Collision Integral: The paper provides the first explicit derivation of the collision integral terms corresponding to crossed impurity diagrams (X and $\Psi$) within the DMKE framework. This bridges the gap between diagrammatic perturbation theory and the density matrix approach for sub-leading disorder effects.
• Decomposition of Scattering Processes: The fourth-order collision operator is shown to separate into three distinct physical components: self-energy corrections, ladder-type vertex renormalizations, and crossed quantum interference contributions.
• Analytical Application to Massive Dirac Fermions: The formalism is applied to a two-dimensional massive Dirac fermion model with Gaussian white noise impurities. The authors derive analytical expressions for the single-particle lifetime ($\tau$), transport relaxation time ($\tau_{tr}$), and longitudinal conductivity ($\sigma_{xx}$) at the Born approximation level.
• Evaluation of Anomalous Hall Conductivity (AHE): The authors calculate the extrinsic contribution to the AHE arising from the crossed diagrams. They demonstrate that these terms generate an extrinsic contribution of order $\tau^0$ that coexists with the intrinsic Berry curvature term.

Results

• Longitudinal Transport: At the Born approximation level, the derived longitudinal conductivity matches standard Boltzmann transport results, with the transport relaxation time $\tau_{tr}$ correctly accounting for vertex corrections (ladder diagrams).
• Anomalous Hall Effect:
  • The intrinsic contribution is recovered as the standard Berry curvature term.
  • The extrinsic contribution from the X-diagram is found to be finite, yielding a non-zero $\tau^0$ term that competes with the intrinsic term.
  • The extrinsic contribution from the $\Psi$-diagram vanishes identically for the specific case of Gaussian white noise impurities due to statistical isotropy (symmetry under angle swapping). The authors note that this term would remain finite for anisotropic impurity potentials.
• Consistency: The results obtained via the DMKE are shown to be in exact agreement with those derived from the Keldysh formalism and previous Kubo diagrammatic derivations (specifically Refs. [12, 13, 14]).

Significance and Claims
The paper claims to provide a "blueprint" for incorporating band geometry and crossed disorder corrections into multi-band transport theory using the DMKE. Its primary significance lies in:

1. Completing the DMKE Framework: It extends the DMKE beyond the non-crossing approximation, enabling a consistent description of steady-state phenomena where $\tau^0$ disorder corrections are comparable to intrinsic topological terms.
2. Unified Language: It establishes the DMKE as a natural language for describing inter-band coherence, band geometry, and extrinsic scattering on equal footing, facilitating the decomposition of responses into Fermi surface and Fermi sea contributions.
3. Applicability: The authors assert that this approach opens the door to a consistent description of spin, pseudospin, orbital, and valley-related phenomena in the steady state. They suggest the formalism can be extended to non-linear regimes and implemented in first-principles Wannier calculations for realistic materials, moving beyond the simplified Dirac model used for demonstration.

The work does not claim to resolve all controversies in the field but rather provides the necessary theoretical machinery to systematically evaluate these higher-order terms in complex, multi-band systems where semiclassical approximations fail.