本文探討了時間晶體的物理學，時間晶體是一種在周期性驅動下表現出離散時間平移對稱性破缺的物質相。特別是，基於多體局域化（MBL）物理的時間晶體可以抵抗熱化並長時間保持初始狀態。我們通過 Floquet 特徵譜的精確對角化來分析時間晶體的頑健性，並討論它們是如何被多體交互作用“融化”。本文研究了各種交互作用對時間晶體的影響，並提供了未來的研究方向。

This article explores the physics of time crystal, a phase of matter that exhibits discrete time translation symmetry breaking under periodic driving. In particular, time crystals based on many-body localization (MBL) physics can resist thermalization and maintain the initial state information for extended periods of time. We analyze the robustness of time crystals through exact diagonalization of the Floquet eigenspectrum and discuss how they can be "melted" by many-body interactions. This paper investigates the impact of various interactions on time crystals and provides future research directions.

Acknowledgements iii
摘要 iv
Abstract v
1 Preview 1
2 From equilibrium to non-equilibrium 5
2.1 Thermalization of classical system . . . . . . . . . . . . . . . . . . . . 5
2.2 Thermalization of quantum system . . . . . . . . . . . . . . . . . . . . 6
2.3 Anderson localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Many Body Localization(MBL) . . . . . . . . . . . . . . . . . . . . . 9
2.5 Localized protect quantum order . . . . . . . . . . . . . . . . . . . . . 14
3 Periodically driven systems 17
3.1 Rabi-oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Floquet theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Single particle Floquet system . . . . . . . . . . . . . . . . . . . . . . 20
4 Time crystal 25
4.1 From transverse field Ising model to time crystal . . . . . . . . . . . . 25
4.2 Floquet-MBL model . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . 34
5 Stability of time crystal 39
5.1 Perturbation in Floquet TFIM . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Perturbation with Z2 symmetry . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Perturbation without Z2 symmetry . . . . . . . . . . . . . . . . . . . . 43
5.4 Breakdown of MBL by weak interaction . . . . . . . . . . . . . . . . . 47
6 Conclusion 49
References 51

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