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You can find the original article here: https://arxiv.org/abs/1202.2539 

Quantum Time Crystals – Summary and Knowledge Base Entry #

Author: Frank Wilczek, MIT Center for Theoretical Physics
Published: February 12, 2012 (arXiv:1202.2539v1 [quant-ph])
Type: Theoretical Physics Paper

Overview #

Frank Wilczek’s Quantum Time Crystals (2012) introduces the theoretical framework for a new phase of matter where time-translation symmetry is spontaneously broken, much like spatial symmetry is broken in ordinary crystals. This idea implies that a system’s lowest-energy (ground) state could still exhibit periodic motion — a radical departure from traditional assumptions in quantum mechanics.

Key Concepts #

1. Symmetry and Time Translation #

• The paper challenges the long-standing belief that the ground state of a quantum system must be static.

• Wilczek explores whether a closed quantum system could spontaneously break time translation symmetry, producing periodic motion without external energy input.

2. Ring Particle Model #

• The initial model uses a charged particle on a ring threaded by magnetic flux.

• When the flux is a non-integer multiple of the quantum flux, the system’s ground state supports a persistent current, demonstrating motion in its lowest energy configuration.

• This forms the quantum analog of a “time crystal,” where motion recurs cyclically.

3. Symmetry Breaking and Observability #

• The study connects observability and spontaneous symmetry breaking, showing that physical states with different time phases become effectively distinct when extended to macroscopic systems.

• This framework parallels how spatial crystals arise from broken translational symmetry.

4. Soliton Model #

• Wilczek extends the idea to many interacting particles using a nonlinear Schrödinger equation.

• The system forms a moving soliton (localized wave packet) on the ring, representing a collective oscillating ground state that breaks time symmetry.

5. Imaginary Time Crystals #

• The concept extends into imaginary time within the path integral formalism of quantum mechanics.

• This predicts potential periodic behaviors in thermodynamic quantities related to temperature (1/T periodicity), hinting at measurable experimental effects.

Significance #

• Conceptual Leap: Introduces the first theoretical framework for spontaneous time-translation symmetry breaking.

• Foundation for Experimental Work: Inspired later physical realizations of discrete time crystals in trapped ions, spin systems, and superconducting circuits.

• Cross-Disciplinary Impact: Bridges condensed matter, quantum field theory, and thermodynamics.

Limitations and Future Questions #

• The model is theoretical and requires idealized conditions (non-local Hamiltonians, perfect isolation).

• Experimental realizations require open systems with periodic driving rather than true spontaneous symmetry breaking.

• Open questions include finite-temperature transitions, quantum coherence times, and classification of space-time ordered phases.

Conclusion #

Wilczek’s Quantum Time Crystals established a new paradigm for understanding dynamical phases of matter, showing that time itself can exhibit crystalline order under quantum mechanics. The paper remains a cornerstone of theoretical physics, influencing quantum computation, condensed matter, and fundamental symmetry studies.

Reference:
Wilczek F. Quantum Time Crystals. arXiv:1202.2539 [quant-ph], 2012.