Phase Structure of Driven Quantum Systems

Abstract

Clean and interacting periodically-driven systems are believed to exhibit a single, trivial "infinitetemperature" Floquet-ergodic phase. In contrast, here we show that their disordered Floquet manybody localized counterparts can exhibit distinct ordered phases delineated by sharp transitions. Some of these are analogs of equilibrium states with broken symmetries and topological order, while others -genuinely new to the Floquet problem -are characterized by order and non-trivial periodic dynamics. We illustrate these ideas in driven spin chains with Ising symmetry.Introduction: Extending ideas from equilibrium statistical mechanics to the non-equilibrium setting is a topic of perennial interest. We consider a question in this vein: Is there a sharp notion of a phase in driven, interacting quantum systems? We find an affirmative answer for Floquet systems 1-3 whose Hamiltonians depend on time t periodically, H(t + T ) = H(t). Unlike in equilibrium statistical mechanics, disorder turns out to be an essential ingredient for stabilizing different phases; moreover, the periodic time evolution allows for the existence (and diagnosis) of phases without any counterparts in equilibrium statistical mechanics.Naively, Floquet systems hold little promise of a complex phase structure. In systems with periodic Hamiltonians, not even the basic concept of energy survives, being replaced instead with a quasi-energy defined up to arbitrary shifts of 2π/T . Indeed, interacting Floquet systems should absorb energy indefinitely from the driving field, as suggested by standard linear response reasoning wherein any nonzero frequency exhibits dissipation. This results in the system heating up to "infinite temperature", at which point all static and dynamic correlations become trivial and independent of starting state -thus exhibiting a maximally trivial form of ergodicity 4-6 . To get anything else requires a mechanism for energy localization wherein the absorption from the driving field saturates, and the long-time state of the system is sensitive to initial conditions. The current dominant belief is that translationally invariant interacting systems cannot generically exhibit such energy localization 4-6 , although there are computations that suggest otherwise 7-9 . The basic intuition is that spatially extended modes in translationally invariant systems interact with and transfer energy between each other.This can be different when disorder spatially localizes the modes, with individual modes exhibiting something like Rabi oscillations while interacting only weakly with distant modes. While the actual situation is somewhat more involved, several pieces of work 10-12 have made a convincing case for the existence of Floquet energy localization exhibiting a set of properties closely related to those exhibited by time-independent many-body localized 13 (MBL) systems 14 . In the following we show that such Floquet-MBL sys-

Observation of a discrete time crystal

“…These two states evolve at different rates corresponding to their respective quasi-energies, giving rise to the sub-harmonic periodic oscillations of physical observables. Such oscillations are expected to persist at increasingly long times as the system size increases 7,8,10 .In summary, we present the first experimental observation of discrete time translation symmetry breaking into a DTC. We measure persistent oscillations and synchronizations of interacting spins in a chain and show that the discrete time crystal is rigid, or robust to perturbations in the drive.…”

“…These two states evolve at different rates corresponding to their respective quasi-energies, giving rise to the sub-harmonic periodic oscillations of physical observables. Such oscillations are expected to persist at increasingly long times as the system size increases 7,8,10 .…”

“…Analogous to crystals in space, the breaking of translation symmetry in time and the emergence of a "time crystal" was recently proposed 2,3 , but later shown to be forbidden in thermal equilibrium [4][5][6] . However, nonequilibrium Floquet systems subject to a periodic drive can exhibit persistent time-correlations at an emergent sub-harmonic frequency [7][8][9][10] . This new phase of matter has been dubbed a "discrete time crystal" (DTC) 10,11 .…”

“…We apply a periodic Hamiltonian to the system under many-body localization (MBL) conditions, and observe a sub-harmonic temporal response that is robust to external perturbations. Such a time crystal opens the door for studying systems with long-range spatial-temporal correlations and novel phases of matter that emerge under intrinsically non-equilibrium conditions 7 .…”

“…The proposal of such a "time crystal" 2 for time-independent Hamiltonians has led to much discussion, with the conclusion that such structures cannot exist in the ground state or any thermal equilibrium state of a quantum mechanical system [4][5][6] . A simple intuitive explanation is that quantum equilibrium states have time-independent observables by Time (100 Floquet periods) construction; thus, time translation symmetry can only be spontaneously broken in non-equilibrium systems [7][8][9][10] . In particular, the dynamics of periodically-driven Floquet systems possesses a discrete time translation symmetry governed by the drive period.…”