RKKY interaction in altermagnets with adiabatic electron-phonon coupling

Altermagnets, characterized by time-reversal symmetry breaking without net magnetization and momentum-dependent spin-split bands, offer a promising platform for spintronics due to their anisotropic spin textures and potential for tunable magnetic interactions. Here, we theoretically investigate the slow phonon-renormalized Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in two-dimensional Rashba $d$-wave altermagnets, incorporating arbitrary Rashba spin-orbit coupling (RSOC) strength and spin-dependent static-Holstein electron-phonon coupling (EPC). Using a continuum model Hamiltonian that captures altermagnetic anisotropy, RSOC, and adiabatic lattice distortions, we compute the noncollinear RKKY exchange tensor via second-order perturbation theory and numerical momentum-space integration. Our results reveal that static lattice distortions provide a versatile tuning knob for engineering the magnitude, anisotropy, and chirality of RKKY couplings: moderate EPC suppresses long-range coherence while inducing component-specific phase shifts and sign reversals, particularly in Dzyaloshinskii-Moriya and compass-like terms, enabling control over ferromagnetic/antiferromagnetic alignments and clockwise/counterclockwise chiralities. Systematic parameter scans demonstrate enhanced oscillatory complexity with altermagnetic order, RSOC, and doping. These findings establish slow phonons as a low-energy tuning knob that systematically controls the magnitude, anisotropy, phase, and chirality of noncollinear RKKY couplings - even at arbitrary Rashba strength - thereby providing a practical route to engineer ferromagnetic/antiferromagnetic alignments and clockwise/counterclockwise DM chiralities in altermagnet-based heterostructures.