Focusing on semiclassical systems, we show that the parametrically long exponential growth of outof-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos. Indeed, scrambling can simply result from the presence of unstable fixed points in phase space, even in an integrable model. We derive a lower bound on the OTOC Lyapunov exponent which depends only on local properties of such fixed points. We present several models for which this bound is tight, i.e. for which scrambling is dominated by the local dynamics around the fixed points. We propose that the notion of scrambling be distinguished from that of chaos.
We study nonlinear optical responses in superconducting systems with inversion (I) symmetrybreaking order parameters. We first show that any superconducting system with I and time-reversal (T ) symmetries requires an I-breaking order parameter to support optical transitions between particle-hole pair bands. We then use a 1D toy model of an I-breaking superconductor to numerically calculate linear and nonlinear conductivities, including shift current and second harmonic generations (SHG) responses. We find that the magnitude of the signal is significantly larger in shift current/SHG response compare to the linear response due to the matrix element effect. We also present various scaling behaviors of the SHG signal, which may be relevant to the recent experimental observation of SHG in cuprates [1]. Finally, we confirm the generality of our observations regarding nonlinear responses of I-breaking superconductors, by analyzing other models including a 1D three-band model and 2D square lattice model.
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