RKKY quadratic and biquadratic spin-spin interactions in twisted bilayer graphene

We study the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) quadratic and biquadratic spin-spin interactions of two magnetic impurities in twisted bilayer graphene away from the magic angle. We apply the Bistritzer-MacDonald model of two graphene layers twisted with respect to each other by a small angle. By reducing the model to the Dirac-type one with modified Fermi velocity, we derive expressions for the RKKY quadratic and biquadratic spin interactions using perturbation theory for the free energy. The biquadratic interaction is suppressed by a larger power of the interaction constant and decreases faster with the distance between impurities compared to the quadratic one. Nevertheless, due to the different period of the oscillations with the impurity separation distance, chemical potential, twist angle, and temperature, it is possible to fine-tune the system to the regime of dominating biquadratic interaction. The existence of such a fine-tuned regime might provide a promising opportunity to observe nonconventional spin ordering.