Spontaneous time-translation symmetry breaking is the mechanism by which a system's lowest-energy or steady state becomes less symmetric in time than the equations governing it. It is the temporal analogue of the spatial symmetry breaking that produces ordinary crystals, and it is the central mechanism of the Time Crystal.
The spatial analogy
The source dossier explains that the existence of crystals in nature is a manifestation of spontaneous symmetry breaking, which occurs when the lowest-energy state of a system is less symmetrical than the equations governing the system. In the crystal ground state, the continuous translational symmetry in space is broken and replaced by the lower discrete symmetry of the periodic crystal.
The dossier lists three characteristics of this broken symmetry:
- the system has a lower symmetry than the underlying arrangement of the crystal,
- the system exhibits spatial and temporal long-range order (unlike a local and intermittent order in a liquid near the surface of a crystal),
- it is the result of interactions between the constituents of the system, which align themselves relative to each other.
The temporal case
Because the laws of physics are symmetrical under continuous translations in time as well as space, the question arose in 2012 as to whether it is possible to break symmetry temporally. The dossier distinguishes two regimes:
- Discrete time-translation symmetry breaking — realized in periodically driven systems, where the initial symmetry t → t + nT with n = 1 is broken to a lower discrete symmetry with n > 1. This is the Discrete Time Crystal (DTC).
- Continuous time-translation symmetry breaking — realized under continuous drive, where the system oscillates at an intrinsic frequency with a time phase taking random values between 0 and 2π. This is the Continuous Time Crystal (CTC).
The dossier states that breaking of time symmetry can occur only in non-equilibrium systems.
Relation to conservation laws
The dossier links the symmetry to Noether's Theorem: symmetries in nature lead directly to conservation laws. Time-translation symmetry — the statement that the laws of nature applying today applied in the past and will apply in the future — implies the conservation of energy. This is why the thermodynamics discussion of time crystals matters: the dossier asserts that time crystals do not violate energy conservation or the second law.
Distinguishing criteria
The dossier is explicit that many systems show spontaneous time-translation symmetry breaking without being time crystals: convection cells, oscillating chemical reactions, aerodynamic flutter, subharmonic response such as the Faraday instability, NMR spin echoes, parametric down-conversion, and period-doubled nonlinear dynamical systems. The distinguishing criteria for a discrete time crystal are broken symmetry, Crypto-Equilibrium, and long-range order, with the order resulting from many-body interactions.
Related work
- Time Crystal — the concept this mechanism produces.
- Noether's Theorem — the symmetry–conservation-law link.
- Crypto-Equilibrium — the condition distinguishing DTCs from other oscillating systems.
Source notes & attribution
- https://rexresearch.com/WilczekTimeCrystals/WILCZEKTimeCrystals.html