[Paper Review] A Brief History of Time Crystals

Vedika Khemani, Roderich Moessner+1 more|arXiv (Cornell University)|Oct 23, 2019

Quantum many-body systems18 references120 citations

TL;DR

A comprehensive review arguing that discrete time-translation symmetry breaking occurs in periodically driven, many-body localized quantum systems, defining Floquet time crystals and detailing their diagnostics, theory, and experiments.

ABSTRACT

The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the perpetuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization. Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics --- that of an intrinsically out-of-equilibrium many-body phase of matter. We include a compendium of necessary background, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. We formalize the notion of a time-crystal as a stable, macroscopic, conservative clock --- explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. We also cover a range of related phenomena, including various types of long-lived prethermal time-crystals, and expose the roles played by symmetries -- exact and (emergent) approximate -- and their breaking. We clarify the distinctions between many-body time-crystals and other ostensibly similar phenomena dating as far back as the works of Faraday and Mathieu. En route, we encounter Wilczek's suggestion that macroscopic systems should exhibit TTSB in their ground states, together with a theorem ruling this out. We also analyze pioneering recent experiments detecting signatures of time crystallinity in a variety of different platforms, and provide a detailed theoretical explanation of the physics in each case. In all existing experiments, the system does not realize a `true' time-crystal phase, and we identify necessary ingredients for improvements in future experiments.

Motivation & Objective

• Clarify the concept and definitions of time translation symmetry breaking in quantum systems.
• Explain why time crystals require out-of-equilibrium, many-body settings and infinite-system limits.
• Survey Floquet many-body localized time crystals and their diagnostic tools.
• Discuss prethermal time crystals and symmetry-protected time crystals as extensions.
• Review experimental realizations and their theoretical interpretations.

Proposed method

• Provide precise definitions of time-translation symmetry breaking and spatiotemporal order in many-body systems.
• Discuss the role of many-body localization in preventing heating under periodic driving.
• Present diagnostics for TTSB including temporal/spatio-temporal order and susceptibility.
• Analyze specific Floquet Ising-type models and the π spin-glass phase as concrete realizations of DTCs.
• Survey prethermal and symmetry-protected time crystals as extensions beyond ideal Floquet MBL systems.
• Compare theoretical predictions with experimental observations across platforms.

Experimental results

Research questions

• RQ1 What constitutes a rigorous definition of time-translation symmetry breaking in quantum many-body systems?
• RQ2 Under what conditions can a periodically driven (Floquet) system realize a stable time crystal without heating?
• RQ3 How do many-body localization and emergent integrals of motion enable Floquet time crystals?
• RQ4 What are the diagnostic signatures of TTSB and spatiotemporal order in eigenstates and dynamics?
• RQ5 How do prethermal and symmetry-protected time crystals extend or modify the Floquet MBL framework?

Key findings

• Discrete time-translation symmetry breaking can occur in Floquet systems that are many-body localized, yielding a stable, macroscopic clock.
• Time crystals exhibit spatiotemporal order, breaking both discrete time translation and spatial symmetries in certain phases like the π spin-glass.
• Diagnostics such as temporal and spatiotemporal order, susceptibility, and non-trivial late-time evolution are essential to identify TTSB.
• Experimental realizations across trapped ions, NV centers, and NMR platforms show signatures compatible with time-crystal physics, though true asymptotic TCs remain challenging.
• Prethermal time crystals and symmetry-protected time crystals broaden the scope beyond ideal Floquet MBL time crystals.