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arXiv:2008.10188 · Frontiers in Physics

Quantum Phases of Time Order in Many-Body Ground States

Tie-Cheng Guo, Li You

Published Mar 22, 2022• 3 citations
arXivHTMLPDF

Abstract

Abstract Understanding phases of matter is of both fundamental and practical 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, is central to understanding phases associated with order parameters of distinct symmetries and transitions between phases. This work proposes to identify ground state phases of quantum many-body system in terms of time order, which is operationally defined by the appearance of nontrivial temporal structure in the two-time auto-correlation function of a symmetry operator (order parameter). As a special case, the (symmetry protected) time crystalline order phase detects continuous time crystal (CTC). Time order phase diagrams for spin-1 atomic Bose-Einstein condensate (BEC) and quantum Rabi model are fully worked out. Besides time crystalline order, the intriguing phase of time functional order is discussed in two non-Hermitian interacting spin models.

DOI
10.3389/fphy.2022.847409

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Abstract Understanding phases of matter is of both fundamental and practical 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, is central to understanding phases associated with order parameters of distinct symmetries and transitions between phases. This work proposes to identify ground state phases of quantum many-body system in terms of time order, which is operationally defined by the appearance of nontrivial temporal structure in the two-time auto-correlation function of a symmetry operator (order parameter). As a special case, the (symmetry protected) time crystalline order phase detects continuous time crystal (CTC). Time order phase diagrams for spin-1 atomic Bose-Einstein condensate (BEC) and quantum Rabi model are fully worked out. Besides time crystalline order, the intriguing phase of time functional order is discussed in two non-Hermitian interacting spin models.
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.
Abstract Spontaneous symmetry breaking is related to the appearance of emergent phenomena, while a non-vanishing order parameter has been viewed as the sign of turning into such symmetry-breaking phase. We study the spontaneous symmetry breaking in the conventional superconductor and Bose–Einstein condensation with a continuous measure of symmetry by showing that both the many-body systems can be mapped into the many spin model. We also formulate the underlying relation between the spontaneous symmetry breaking and the order parameter quantitatively. The degree of symmetry stays unity in the absence of the two emergent phenomena, while decreases exponentially at the appearance of the order parameter which indicates the inextricable relation between the spontaneous symmetry and the order parameter.
The quantum phase of a physical system is defined as its ground state. As examples of quantum phases we mention super uids, superconductors, Bose- Einstein condensates and quantum Hall states. Different phases of a system originate from the different ways a systems' constituents can be organised or, in other words, the different orders of the system. Landau had the insight that the difference between two phases can be attributed to the corresponding orders having different symmetries [1]. Also, the transition between phases can be associated with the breaking of symmetries. This insight led to Ginzburg and Landau describing orders using local order parameters and developing a general theory of phase transitions based on symmetry breaking. This theory was successful in describing a large number of observed phases of matter. However, phases of matter have been discovered which cannot be described using a local order parameter and Landau's theory of symmetry breaking known as topological phases.
Time crystals are nonequilibrium phases of matter characterized by the emergence of temporal ordering, in which an interacting many-body system develops robust structure in its time evolution that is not trivially dictated by the external driving or environment. While related phenomena have long been studied in classical nonlinear systems, their realization in entangled quantum matter represents a distinct frontier. The theoretical understanding of discrete time crystals has substantially advanced, yet recent experiments using modern quantum devices and quantum processors reveal regimes beyond established paradigms. These developments call for an extended classification of time-crystalline phases according to both their stabilization mechanisms and their physical character, including discrete and continuous, closed and open, critical, topological, quasiperiodic, and controlled realizations. We review recent implementations of time crystals on quantum platforms and propose such a classification framework, identifying promising directions for the discovery of novel time-crystalline phases of matter.
Spontaneous symmetry breaking plays a pivotal role in physics ranging from the emergence of elementary particles to the phase transitions of matter. The spontaneous breaking of continuous time translation symmetry leads to a novel state of matter named continuous time crystal (CTC). It exhibits periodic oscillation without the need for periodic driving, and the relative phases for repetitively realized oscillations are random. However, the mechanism behind the spontaneous symmetry breaking in CTCs, particularly the random phases, remains elusive. Here we propose and experimentally realize two types of CTCs based on distinct mechanisms: manifold topology and near-chaotic motion. We observe both types of CTCs in thermal atomic ensembles by artificially synthesizing spin-spin nonlinear interactions through a measurement-feedback scheme. Our work provides general recipes for the realization of CTCs, and paves the way for exploring CTCs in various systems.
The study of phases is useful for understanding novel states of matter. One such state of matter is time crystals which constitute periodically driven interacting many-body systems that spontaneously break time translation symmetry. Time crystals with arbitrary periods (and dimensions) can be realized using the model of Bose-Einstein condensates bouncing on periodically driven mirror(s). In this work, we identify the different phases that characterize the two-dimensional time crystal. By determining the optimal initial conditions and value of system parameters, we provide a practical route to realize a specific phase of the time crystal. These different phases can be mapped to the many-body states existing on a two-dimensional Hubbard lattice model, thereby opening up interesting opportunities for quantum simulation of many-body physics in time lattices.