[Paper Review] Detecting Time Crystal and Classifying Quantum Phases with Time Order

Tie-Cheng Guo, Li You|arXiv (Cornell University)|Aug 24, 2020

Advanced Thermodynamics and Statistical Mechanics6 citations

TL;DR

This paper proposes a novel classification scheme for quantum phases based on time order, defined via the two-time auto-correlation of a symmetry order operator acting on the ground state. It identifies symmetry-protected time-ordered phases as signatures of continuous time crystals (CTCs), demonstrating this with a phase diagram for a spin-1 atomic Bose-Einstein condensate.

ABSTRACT

Understanding phases of matter is of fundamental importance. Prior to the widespread appreciation and acceptance of topological order, the paradigm of spontaneous symmetry breaking, formulated along the Landau-Ginzburg-Wilson (LGW) dogma, has been central to the understanding of phases as well as phase transitions between order parameters with distinct symmetries. This Letter proposes to classify ground state phases of quantum matter based on temporal properties in terms of time order. More specifically, we define time order with twisted vector: the ground state acted on by a symmetry order operator, whose two-time auto-correlation function detects possible existence of nontrivial temporal structure. A (symmetry protected) time ordered phase thus implicates the presence and essence of continuous time crystal (CTC). As an example, time order phase diagram for a spin-1 atomic Bose-Einstein condensate (BEC) is presented.

Motivation & Objective

• To extend the Landau-Ginzburg-Wilson paradigm by introducing time order as a new criterion for classifying quantum phases.
• To address the lack of systematic classification for phases beyond spontaneous symmetry breaking, particularly those involving time-translation symmetry.
• To establish a framework that identifies and characterizes continuous time crystals (CTCs) through temporal order parameters.
• To provide a concrete realization of time-ordered phases in a physically realizable system: a spin-1 atomic Bose-Einstein condensate (BEC).

Proposed method

• Define time order using a twisted vector formed by applying a symmetry order operator to the ground state.
• Compute the two-time auto-correlation function of the time-ordered operator to detect nontrivial temporal structures.
• Use the long-time behavior of the two-time correlation function to identify the presence of continuous time crystal order.
• Construct a time order phase diagram by analyzing the correlation function across different parameter regimes of the spin-1 BEC Hamiltonian.
• Apply the method to a specific model of spin-1 BEC to demonstrate the emergence of time-ordered phases.
• Relate the existence of non-zero long-time correlations to the presence of a (symmetry-protected) time-ordered phase.

Experimental results

Research questions

• RQ1 Can time order, defined via two-time correlations, serve as a robust order parameter for classifying quantum phases?
• RQ2 How does time order distinguish between trivial and nontrivial quantum phases in systems with continuous time-translation symmetry?
• RQ3 What is the connection between time-ordered phases and the emergence of continuous time crystals (CTCs)?
• RQ4 In what parameter regimes of a spin-1 BEC does time order become non-zero, indicating a CTC phase?
• RQ5 Can the proposed time order framework be applied to classify phases beyond the standard LGW paradigm?

Key findings

• The two-time auto-correlation function of the time-ordered operator reveals nontrivial temporal order, signaling the presence of a continuous time crystal (CTC).
• A symmetry-protected time-ordered phase is identified as a distinct quantum phase characterized by long-lived temporal correlations.
• The time order phase diagram for a spin-1 BEC shows clear regions where time order is non-zero, indicating CTC phases.
• The method successfully detects CTC order without relying on conventional order parameters, offering a new route to phase classification.
• Nonzero long-time correlations in the two-time function confirm the existence of persistent temporal order, a hallmark of CTCs.
• The framework provides a systematic way to classify quantum phases based on temporal properties, extending beyond the traditional LGW paradigm.