Time crystals: From Schr\"odinger to Sisyphus

TL;DR

This paper explores the concept of Hamiltonian time crystals across various systems, including quantum equations, molecular chains, and molecules, revealing spontaneous symmetry breaking and novel rotational behaviors.

Contribution

It introduces new examples of time crystals in molecular and classical systems, demonstrating spontaneous symmetry breaking and complex rotational dynamics without angular momentum.

Findings

Time crystals can emerge in quantum and classical systems.

Molecular dynamics shows spontaneous symmetry breaking in cyclopropane.

Rotational motion can resemble Sisyphus dynamics at short observation intervals.

Abstract

A Hamiltonian time crystal can emerge when a Noether symmetry is subject to a condition that prevents the energy minimum from being a critical point of the Hamiltonian. A somewhat trivial example is the Schr\"odinger equation of a harmonic oscillator. The Noether charge for its particle number coincides with the square norm of the wave function, and the energy eigenvalue is a Lagrange multiplier for the condition that the wave function is properly normalized. A more elaborate example is the Gross-Pitaevskii equation that models vortices in a cold atom Bose-Einstein condensate. In an oblate, essentially two dimensional harmonic trap the energy minimum is a topologically protected timecrystalline vortex that rotates around the trap center. Additional examples are constructed using coarse grained Hamiltonian models of closed molecular chains. When knotted, the topology of a chain can support a time crystal. As a physical example, high precision all-atom molecular dynamics is used to analyze an isolated cyclopropane molecule. The simulation reveals that the molecular D$_{3h}$ symmetry becomes spontaneously broken. When the molecule is observed with sufficiently long stroboscopic time steps it appears to rotate like a simple Hamiltonian time crystal. When the length of the stroboscopic time step is decreased the rotational motion becomes increasingly ratcheting and eventually it resembles the back-and-forth oscillations of Sisyphus dynamics. The stroboscopic rotation is entirely due to atomic level oscillatory shape changes, so that cyclopropane is an example of a molecule that can rotate without angular momentum. Finally, the article is concluded with a personal recollection how Frank's and Betsy's Stockholm journey started.