Erratum: Discrete Time Crystals: Rigidity, Criticality, and Realizations [Phys. Rev. Lett. 118 , 030401 (2017)]

Abstract

This corrects the article DOI: 10.1103/PhysRevLett.118.030401.

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems

“…In contrast to the continuous TTS breaking [27][28][29] that has turned out to be impossible at thermal equilibrium [30,31], the discrete TTS breaking has been theoretically proposed [32][33][34][35][36] and experimentally demonstrated [37,38]. Phases with broken discrete TTS feature discrete time-crystalline (DTC) order characterized by periodic oscillations of physical observables with period nT , where T is the Floquet period and n = 2, 3, · · · .…”

“…The DTC order is expected to be stabilized by many-body interactions against variations of driving parameters. Note that the system is assumed to be in a localized phase [33,34,36,37] or to have long-range interactions [38][39][40]. Otherwise, the DTC order only exists in a prethermalized regime [41,42] since the system will eventually be heated to a featureless infinite-temperature state due to persistent driving [43][44][45].…”

Time-translation-symmetry breaking in a driven oscillator: From the quantum coherent to the incoherent regime

“…Oscillations with period 2t F due to simultaneously initialized protected boundary states were studied in photonic quantum walks [3]; period-two oscillations can also be expected from the coexistence of Floquet Majorana fermions with quasienergies 0 and π/t F in a cold-atom system [4]. The onset of periodtwo phases was predicted and analyzed [5][6][7][8][9][10] in Floquet many-body localized systems, and the first observations of oscillations at multiples of the driving period in disordered systems were reported [11,12].In systems coupled to a thermal bath, on the other hand, the effect of period doubling has been well-known. A textbook example is a classical oscillator modulated close to twice its eigenfrequency and displaying vibrations with period 2t F [13].…”

Learning phase transitions from dynamics

“…This is a slight variation of the model studied in Ref. [46], where we took a different distribution for the bond terms J i . As we explain below, this model features an inherently dynamical phase that cannot be studied in a static setting.…”

“…We then apply our method to a periodically driven model, featuring among its three phases one which is unique to the time-dependent setting, namely, a time crystal [44][45][46][47][48][49][50]. Indeed, the method distinguishes between the time-crystalline, Floquet-ergodic, and Floquet-MBL [51][52][53] phases of this model.…”