We show that flexural phonons mediate tunable spin-spin interactions beyond nearest neighbors in one-dimensional systems. While longitudinal phonons in a minimal lattice model generate only nearest-neighbor couplings, the interaction mediated by pure flexural modes is infrared divergent, and tensile strain or transverse confinement regularizes it, yielding, respectively, an exponentially decaying interaction and an oscillatory interaction with an exponentially decaying envelope. In the weak-strain and weak-confinement regimes, the interaction scales as the inverse square root of the tensile strain and as the inverse fourth root of the confinement strength. Using experimentally extracted phonon dispersions together with finite-element simulations for a hexagonal boron nitride ribbon, we estimate interaction strengths of order 10 MHz. Flexural phonons thus offer a purely mechanical route to controllable multiqubit operations in solid-state spin-based quantum devices.

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• Magnetic interactions
• Spin-phonon coupling
• 1-dimensional systems
• Boron nitride
• Nanoribbon
• Exact diagonalization