AI 中文总结
研究离散 $Z_q$ 模型手性在连续非线性动力学中的表现，通过将其哈密顿量嵌入连续相能景观，揭示手性与连续相相互作用奇数部分的关系，在振荡器网络中实现相关相移并验证，表明可表示为连续动力学中可调时域不对称性。
AI 中文摘要
离散 $Z_q$ 自旋相互作用中的手性可区分顺时针和逆时针相位差，但其在连续非线性动力学中的表现尚不清楚。我们表明，任何成对的 $Z_q$ 哈密顿量都允许唯一的保平衡嵌入到连续相能景观中，该景观与 $q$ 态相格上的离散能量匹配，且每个格点都是静止的。这种嵌入揭示，当且仅当弛豫的正弦分量消失时，$Z_q$ 核是非手性的。离散 $Z_q$ 模型的手性因此恰好是连续相相互作用的奇数部分。在诱导的非线性相位动力学中，该奇数部分在多谐波耦合中变为与取向相关的相移，手性反转会翻转此相移同时保持耦合幅度。在自持振荡器网络中，该相移进一步表现为与方向相关的延迟。晶体管级环形振荡器模拟验证了预测的锁相和定向相位偏置的反转。这些结果表明，离散自旋哈密顿量中的代数手性可表示为连续非线性动力学中可调的时域不对称性。
英文摘要
Chirality in a discrete $Z_q$ spin interaction distinguishes clockwise from counterclockwise phase differences, but its manifestation in continuous nonlinear dynamics is unclear. We show that any pairwise $Z_q$ Hamiltonian admits a unique equilibrium-preserving embedding into a continuous phase-energy landscape that matches the discrete energy on the $q$-state phase grid, where every grid point is stationary. This embedding reveals that a $Z_q$ kernel is nonchiral if and only if the sine components of the relaxation vanish. Chirality of the discrete $Z_q$ model is therefore exactly the odd part of the continuous phase interaction. In the induced nonlinear phase dynamics, this odd part becomes an orientation-dependent phase shift in the multi-harmonic coupling, and chiral reversal flips this shift while preserving the coupling magnitudes. In self-sustaining oscillator networks, the shift is further realized as a direction-dependent delay. Transistor-level ring-oscillator simulations validate the predicted phase locking and reversal of directed phase bias. These results show that algebraic handedness in a discrete spin Hamiltonian can be represented as tunable time-domain asymmetry in continuous nonlinear dynamics.
