Time 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.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