Shape dependence of Edelstein and magnetoelectric effects in the V-shaped model

We theoretically investigate the shape dependence and microscopic mechanism of the magnetoelectric (ME) effect, including both nonmagnetic (Edelstein-type) and magnetic origins, in a V-shaped one-dimensional chain model. Numerical calculations based on the Kubo formula reveal that the nonmagnetic-driven ME response is maximized at an apex angle of θ ≈ 0.6 π . To clarify its origin, we derive a low-energy effective Hamiltonian in the s -orbital subspace and demonstrate that the polarity induced by the V-shaped geometry manifests as an effective spin-orbit interaction. An analytical derivation of the Green’s function shows that the geometric effect can be described as a T -matrix contribution associated with local symmetry breaking. This scattering-theoretical representation enables us to identify the contributions of local geometric symmetry breaking to the ME response. Using a multipole-basis representation, we identify symmetry-based selection rules for the ME tensor and show that the coupling between the effective spin-orbit interaction and the orbital angular momentum generated across the apex plays an essential role. The resulting angular dependence, sin θ sin θ / 2 , peaks at θ = 2 tan − 1 2 ≈ 0.608 π , in good agreement with the numerical results. We also analyze a ferromagnetic V-shaped model including the Zeeman interaction and show that the magnetic-driven ME response originates from the spin magnetization induced by the coupling between the electric-field-driven charge-potential gradient and the Zeeman term. These results reveal distinct ME mechanisms depending on the presence or absence of time-reversal symmetry and clarify the microscopic origin of shape-dependent ME responses in V-shaped chains.