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. Our goal is to establish a symmetry-based framework linking local geometry to ME responses. Numerical calculations based on the Kubo formula reveal that the nonmagnetic-driven ME response is maximized at an apex angle of $\theta \approx 0.6\pi$. 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 formulation provides a unified description of geometry-induced responses in terms of a scattering framework. 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, $\sin{\theta}\sin{\theta/2}$, peaks at $\theta = 2\tan^{-1}\sqrt{2} \approx 0.608\pi$, 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 provide a microscopic framework for geometry-induced multipole phenomena.