I. Spontaneous formation of space crystals II. Spontaneous breaking of continuous time translation symmetry
II.1 Wilczek model
II.2 Lack of time crystal formation by systems in the ground states
II.3 Excited eigenstates and time crystals formation
II.4 Superfluid phase and Josephson oscillations and their relation to time crystals
III.Discrete time crystals
III.1 Proposals of discrete time crystals
III.1.1 Formation of discrete time crystals by atoms bouncing on an oscillating mirror
III.1.2 Formation of discrete time crystals by periodically driven spin systems
III.1.3 Different proposals of discrete time crystals III.2 Experimental realizations of discrete time crystals
IV. Spontaneous formation of quasi-crystals structures in time
V. Condensed matter physics in the time domain
V.1 Emergence of crystalline structures in time in periodically driven systems
V.2 Anderson localization in time
V.3 Topological time crystals
V.4 Quasi-crystals in time
V.5 Many-body time crystals with exotic interactions
V.6 Many-body localization induced by temporal disorder
V.7 Time crystals with properties of 2D or 3D space crystals
VI. Time engineering
VII. Time crystals in photonic materials
About the Author: Krzysztof Sacha is a world-renowned scientist, one of the pioneers of time crystal research and author of many prominent papers related to this topic. Among his scientific achievements belongs the prediction of the behavior of discrete time crystals in a periodically-driven many-body system. His other research themes include cold quantum gases, ionization in strong laser fields and quantum chaos. He is a Professor at Jagiellonian University in Krakow, Poland. Besides his academic work he is engaged in the popularization of science and regularly runs marathons.
