Revisiting magnetoelectric response in collinear antiferromagnetic zigzag chains: A downfolding approach beyond conventional low-energy models
Magnetoelectric (ME) effects in antiferromagnets provide a fertile platform for exploring symmetry-driven cross-correlated responses. However, their microscopic origin remains elusive and is often obscured in simplified low-energy descriptions. In this study, we revisit the microscopic mechanism of the ME effect in a collinear antiferromagnetic zigzag chain by employing a multi-orbital tight-binding model that explicitly includes both $s$- and $p$-orbital degrees of freedom. Using analytical and numerical calculations based on the Kubo formula, we demonstrate that the ME response is governed by orbital degrees of freedom activated through $s$--$p$ hybridization, while the spin contribution vanishes due to spin conservation. To elucidate the low-energy description, we derive an effective Hamiltonian projected onto the $s$-orbital subspace using the Schur complement. We show that a naive application of the Kubo formula within this effective model fails to capture the ME response. This issue is resolved by systematically incorporating vertex corrections in terms of orbital hybridization into the response functions. Furthermore, by introducing a quasiparticle renormalization scheme, we formulate a renormalized Kubo formula that preserves conservation laws and accurately reproduces the full multi-orbital results. Our analysis revisits the conventional low-energy perspective and reveals that the ME effect originates from virtual interorbital processes encoded in vertex corrections, rather than from the bare low-energy Hamiltonian. The effective framework developed here provides a unified microscopic understanding of orbital-driven ME responses and offers a systematic route to incorporate hybridization effects beyond simple low-energy models.