Truncated dynamics, ring molecules, and mechanical time crystals

Abstract

In applications of mechanics, including quantum mechanics, we often consider complex systems, where complete solutions of the underlying "fundamental" equations is both impractical and unnecessary to describe appropriate observations accurately. For example, practical chemistry, including even precision first-principles quantum chemistry, is never concerned with the behavior of the subnuclear quarks and gluons. Instead, we often focus on a few key variables, and construct a so-called effective theory for those. Such effective theories can become complicated and non-local, even for fairly simple systems But in many circumstances, when there is a separation of scales, we can treat the reduced set of variables as a conventional dynamical system in its own right, governed by an energy conserving Lagrangian or Hamiltonian, in a useful approximation. The structure of that emergent description can display qualitatively new features, notably including reduced dimensionality, manifested through unconventional Poisson brackets. Here we discuss the physical meaning and consequences of such truncated dynamics. We propose physically realizable toy models of molecular rings, wherein time crystals emerge at the classical level. We propose that such behavior occurs in the effective theory of highly diamagnetic aromatic ring molecules, and could be widespread.

Provenance of classical Hamiltonian time crystals

“…Thus, for a generic given c the minimal values that h(x, y) attains along the curve g(x, y) = c are not among its critical points on the sphere. It is even possible that this is the case for all those values of c that are allowed by the structure of g(x, y) Recently, explicit examples of classical Hamiltonian time crystals have been presented [24]. The examples go around the No-Go arguments in the manner that we have outlined: they are Hamiltonian systems with conserved quantities, but the numerical values of the conserved charges are constrained.…”

“…In our first example we follow [24] and analyze timecrystalline dynamics in the context of a Hamiltonian system with time evolution determined by a Lie-Poisson bracket [25]. The Hamiltonian function we use has been originally introduced in [31][32][33], in connection of membrane stability analysis.…”

“…We start by explaining how the Lie-Poisson structure of [24] fits in our general framework: a Lie-Poisson bracket commonly describes the way how a Poisson manifold, i.e. a manifold that is equipped with a Poisson bracket, becomes foliated into symplectic leaves.…”

“…The two-spheres are the orbits of SU(2) representations, and each two-sphere is equipped with a symplectic two-form that corresponds to the Poisson bracket (3.11), with cos θ and φ a canonical pair. We proceed to describe a physical scenario where N such vectors n i appear as dynamical degrees of freedom, each equipped with its own Lie-Poisson bracket (3.5) [24]. For this we interpret the vectors as links that connect the N + 1 vertices x i of a piecewise linear polygonal string in R 3 n…”

Time crystals: From Schrödinger to Sisyphus

“…Here I describe how, in spite of the No-Go arguments, genuine (semi)classical and quantum Hamiltonian time crystals do exist [16][17][18]. They could even be widespread.…”