Nonreciprocal and Geometric Frustration in Dissipative Quantum Spins

Nonreciprocal interactions create conflicting dynamical objectives that cannot be simultaneously satisfied, leading to nonreciprocal frustration. Geometric frustration, on the other hand, arises when conflicting static objectives in energy minimization cannot be satisfied. In this work, we show that nonreciprocal interaction among three collective quantum spins, mediated by a damped cavity, induces not only nonreciprocal frustration, intrinsic to nonreciprocity, but also geometric frustration with a remarkable robustness against disorder. It therefore ensures that the accidental degeneracy for steady states remains intact even when the system is perturbed away from a fine-tuned point of enhanced symmetry, in sharp contrast to the equilibrium case. Leveraging this finding, we identify a nonreciprocal phase transition driven by both geometric and nonreciprocal frustration. It gives rise to a time-dependent state, which shows a chiral dynamics along a geometry shaped by the geometric frustration and dynamically restores the broken discrete symmetries. Moreover, it constitutes a time-crystalline order, with multiple harmonics set by an emergent time scale that exhibits critical slowing down. We discuss the potential experimental realizations using a three-component spinor BEC-cavity system, where our finding manifests as a geometric frustration in the structural phase transition and chiral dynamics of the frustrated self-organized BECs. We demonstrate the feasibility of experimental observation under realistic conditions.