Spin-Induced Non-Markovian Time-Crystal-Like Dynamics and Fractal Scaling in the Bateman Dual Oscillator

TL;DR

This paper demonstrates that a closed quantum system can exhibit persistent, non-Markovian, time-crystal-like dynamics and fractal scaling due to spin-induced effects in a Bateman dual oscillator framework, without external driving.

Contribution

It introduces a novel mechanism for emergent time-crystal-like behavior in a closed quantum system through non-Markovian dynamics and fractal scaling, bypassing traditional no-go theorems.

Findings

Persistent oscillations without external driving

Emergence of fractal scaling and self-similar trajectories

Connection between dissipative dynamics and temporal ordering

Abstract

Can a closed quantum system generate persistent time-crystal-like dynamics without external driving? Within the Bateman dual oscillator framework, we show that the answer is affirmative. We consider a nonrelativistic (2+1)-dimensional system in which spin-induced spatial deformation generates an effective Bateman oscillator structure. After quantization, the system is governed by a time-independent Hermitian Hamiltonian describing coherent coupling between damped and amplified oscillator sectors while preserving the total energy of the global doubled system. Tracing over the amplified sector, we derive an effective non-Markovian reduced dynamics for the observable subsystem. The resulting memory effects sustain persistent oscillations of subsystem observables and generate emergent time-crystal-like temporal ordering without external periodic driving or equilibrium spontaneous symmetry breaking. Since the oscillatory behavior originates from nonequilibrium reduced subsystem dynamics rather than equilibrium expectation values of the full Hamiltonian, the mechanism lies outside the assumptions of conventional no-go theorems for equilibrium time crystals. The same dynamics further exhibits logarithmic-spiral trajectories and self-similar fractal scaling, revealing a direct connection between coherent dissipative dynamics, non-Markovian memory effects, and emergent temporal ordering in a globally unitary quantum system. In this specific sense, "watching the growth" of these self-similar structures corresponds to observing the gradual formation of time-crystal-like ordering.