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, $\sin \theta \sin \theta \mathrm{/}2$, 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.