Fig. 2: Time-translation symmetry breaking for nonlocal observables with B = 0.
From: Long-lived topological time-crystalline order on a quantum processor
a Dynamics of nonlocal observables. Auto-correlation function for the three-body string operators \(\{{Z}_{{L}_{i}}\}\) (thick black lines) and instantaneous expectation values for the six-body string operators \(\{{X}_{{L}_{i}}\}\) (thick red lines) are shown in the upper six and lower three panels, respectively. Experimental data points (dots) are obtained from averaging over 24 random realizations, with error bars representing the standard error of the statistical mean. The numerical results (lines) are computed by taking into account qubit decoherence and gate errors (Supplementary Note 3). Whereas the instantaneous expectation values for \(\{{X}_{{L}_{i}}\}\) remain zero, the auto-correlators for \(\{{Z}_{{L}_{i}}\}\) exhibit stable subharmonic oscillations for up to 20 cycles (see Supplementary Note 2.F for measurement details). b Fourier spectra of time-domain signals observed in a, where a stable subharmonic frequency peak appears for \(\{{Z}_{{L}_{i}}\}\) but not \(\{{X}_{{L}_{i}}\}\). c Dynamics of the auto-correlation function for local observables \(\{{\sigma }_{k}^{z}\}\). Such auto-correlations decay quickly to zero, in sharp contrast to those of the string operators \(\{{Z}_{{L}_{i}}\}\).
