Anisotropic magnetoresistance of 2D Rashba films with in-plane Zeeman field and short-range disorder

Technical Summary: Anisotropic Magnetoresistance of 2D Rashba Films with In-Plane Zeeman Field and Short-Range Disorder

Problem Statement
The paper addresses the theoretical controversy regarding the existence of anisotropic magnetoresistance (AMR) in two-dimensional (2D) Rashba electron gases subjected to an in-plane Zeeman field and short-range (delta-correlated) scalar disorder. While an in-plane field breaks spin-rotation symmetry, deforms helicity Fermi contours, and rotates spin textures, it remains unclear whether these effects alone generate a finite AMR in the leading quasiclassical conductivity. Previous literature presents conflicting results: some calculations using relaxation-time approximations or specific numerical treatments find finite AMR, while others suggest vanishing AMR for point-like disorder. The central question is whether the diagonal quasiclassical conductivity in the two-sheet metallic regime depends on the magnitude or direction of the in-plane field.

Methodology
The authors employ a diagrammatic Kubo formalism within the quasiclassical limit ($E_F \tau_0 \gg 1$) for a continuum 2D Rashba model with an in-plane Zeeman field. The study focuses on the leading diagonal intraband contribution, treating short-range disorder via the Born approximation and including impurity vertex corrections (ladder diagrams).

The analysis proceeds in two main steps:

1. Self-Energy and Lifetime: The authors derive the Born self-energy for short-range scalar disorder. They project this matrix onto local helicity eigenstates to determine the quasiparticle lifetime. A key theoretical tool is the derivation of a density Ward identity, which relates the spin-vector part of the Born self-energy to the derivative of the total particle density with respect to the Zeeman field.
2. Conductivity Calculation: The diagonal intraband Kubo conductivity is evaluated using the dressed current vertex. The authors demonstrate that the impurity ladder cancels the anomalous Rashba velocity component, reducing the effective transport vertex to the parabolic velocity. The resulting conductivity integral is then evaluated as a geometric contour integral over the Fermi surfaces.

Key Contributions and Results

• Vanishing Spin-Vector Self-Energy: The paper establishes that for a continuum model with both helicity bands occupied, the total area enclosed by the two Rashba-Zeeman Fermi sheets is invariant under the displacement of the Rashba degeneracy point caused by the in-plane field. Consequently, the derivative of the total density with respect to the field vanishes ($\partial n / \partial B = 0$). Via the Ward identity, this forces the spin-vector part of the on-shell Born self-energy to zero. The resulting quasiparticle lifetime is a scalar, isotropic, and field-independent quantity ($\tau_0$), contradicting models that assume a field-dependent or angle-dependent scalar lifetime.
• Geometric Origin of Conductivity: With the scalar lifetime established and the impurity ladder cancelling the anomalous velocity, the diagonal Kubo conductivity reduces to a sum of the areas enclosed by the two Fermi contours. Since the total occupied area is independent of the in-plane field, the longitudinal conductivity $\sigma_{xx}$ (and $\sigma_{yy}$) is strictly independent of the field magnitude and direction.
• Resolution of AMR: The authors prove that $\sigma_{xx}(B) = \sigma_{xx}(0)$ and $\sigma_{xx} - \sigma_{yy} = 0$. Therefore, point-like nonmagnetic impurities do not produce AMR in the leading quasiclassical two-sheet sector.

Significance and Claims
The paper claims to resolve the "short-range-disorder quasiclassical problem" by demonstrating that the apparent AMR found in some previous works arises from inconsistencies in evaluating the density of states and spin projectors on the same helicity Fermi sheet, or by considering regimes outside the protected two-sheet diagonal sector (e.g., single-band regimes, high-field/broadening effects, or finite-range disorder).

• Clarification of Previous Work: The results confirm the ladder cancellation of the anomalous Rashba velocity identified by Raimondi et al. but argue that their finite numerical AMR belongs to a broader calculation involving finite broadening and high fields, not the protected diagonal two-sheet contribution. The paper supports the zero-AMR finding of Wang and Pang for delta-disorder but clarifies that their work did not explicitly prove the field-independence of the diagonal conductivity itself.
• Mechanism for Non-Zero AMR: The paper posits that a non-zero AMR requires physics beyond the protected short-range quasiclassical mechanism. This includes finite-range (remote) disorder, which introduces momentum-dependent form factors that evade the Ward identity, single-band regimes where the area invariance no longer holds, or interband mixing.
• Theoretical Constraint: The study serves as a rigorous constraint on analytical treatments: any model claiming a field-dependent scalar relaxation time for point-like disorder in the continuum two-sheet Rashba model is incorrect due to the exact cancellation enforced by the density Ward identity and the geometric area invariance.

In summary, the paper concludes that within the specific limits of a continuum two-sheet Rashba model with short-range scalar disorder, the leading quasiclassical conductivity is isotropic and field-independent, and any observed AMR must originate from mechanisms outside this specific theoretical framework.