In this paper, we introduce the concept of the Quantum Energy-Time Lattice (QETL), a theoretical framework proposing that discrete energy exchange events in non-equilibrium open quantum systems can naturally form a structured lattice in the temporal and energetic domains. Unlike conventional discrete time crystals that break continuous time-translation symmetry through periodic driving of spatial or spin configurations, the QETL framework posits that periodicity emerges intrinsically within the energy exchange profile such that E(t + nT_E) equals E(t), where T_E represents the fundamental energy-exchange period and n is an integer. Crucially, we demonstrate that a system exhibiting this temporal-energetic lattice structure does not necessarily manifest as an ordinary time crystal, as the underlying quantum state may retain internal phase fluctuations or decoherence channels that prevent standard sub-harmonic synchronization. We formulate the mathematical dynamics governing QETL, analyze its characteristic observable signals where the energy exchange spectrum exhibits sharp peaks at omega_n equal to nomega_E, and explore the profound implications of establishing a dual energy-time lattice symmetry. This paradigm opens new avenues for exploring non-equilibrium quantum matter and offers deep insights into the fundamental interplay between energy dissipation, temporal order, and quantum coherence.
In this paper, we introduce the concept of the Quantum Energy-Time Lattice (QETL), a theoretical framework proposing that discrete energy exchange events in non-equilibrium open quantum systems can naturally form a structured lattice in the temporal and energetic domains. Unlike conventional discrete time crystals that break continuous time-translation symmetry through periodic driving of spatial or spin configurations, the QETL framework posits that periodicity emerges intrinsically within the energy exchange profile such that E(t + nT_E) equals E(t), where T_E represents the fundamental energy-exchange period and n is an integer. Crucially, we demonstrate that a system exhibiting this temporal-energetic lattice structure does not necessarily manifest as an ordinary time crystal, as the underlying quantum state may retain internal phase fluctuations or decoherence channels that prevent standard sub-harmonic synchronization. We formulate the mathematical dynamics governing QETL, analyze its characteristic observable signals where the energy exchange spectrum exhibits sharp peaks at omega_n equal to nomega_E, and explore the profound implications of establishing a dual energy-time lattice symmetry. This paradigm opens new avenues for exploring non-equilibrium quantum matter and offers deep insights into the fundamental interplay between energy dissipation, temporal order, and quantum coherence.
Traditional condensed matter physics relies heavily on spatial periodicity to classify states of matter using crystal lattices and momentum space. In recent years, the paradigm has expanded to include temporal dimensions, notably through the theoretical and experimental exploration of discrete time crystals. However, searching strictly for discrete time crystals imposes rigid constraints on phase rigidity and symmetry breaking. In this paper, we propose a broader and more fundamental paradigm: Quantum Material Temporal Lattice Spectroscopy (QMTLS). Rather than seeking an isolated time crystal phase, QMTLS focuses on identifying internal temporal structures characterized by periodic sequences t_n = nT formed spontaneously or under driving within quantum materials. We formalize the theoretical foundation of temporal lattices in condensed matter, define the spectral density function S(omega) containing discrete temporal lattice peaks, and outline the microscopic mechanisms driving these phenomena. By avoiding traditional experimental validation sections and focusing on rigorous theoretical foundations, this framework opens a promising pathway toward discovering novel non-equilibrium states of matter with potential profound implications for fundamental physics.
Boundary Time Crystals (BTCs) represent a distinctive form of nonequilibrium temporal order in which persistent oscillations emerge in open dissipative quantum systems without external periodic driving. Although their dynamical characterization is well established through the spectral properties of the Liouvillian, the thermodynamic meaning of persistent temporal order remains less systematically articulated. This work develops a nonequilibrium-thermodynamic framework for interpreting BTCs as emergent dissipative structures sustained by irreversible energy and entropy exchange. The framework connects Liouvillian temporal dynamics with thermodynamic forces and fluxes, entropy production, dissipative organization, and self-organization, thereby providing a common language for describing temporal persistence and irreversible activity within open quantum systems. In addition, phenomenological measures are introduced to characterize the balance between coherent temporal dynamics and dissipative relaxation, together with a normalized entropy-production measure for comparing dynamical regimes. A concrete numerical demonstration based on the established collective-spin BTC model illustrates the framework by contrasting non-BTC and BTC regimes, finite-size temporal decay, and the associated dissipative activity. The results support a thermodynamic interpretation in which persistent temporal order is maintained as a nonequilibrium organized state rather than as a consequence of equilibrium thermodynamics. The proposed framework is complementary to microscopic quantum-thermodynamic treatments of BTCs and does not replace the established Liouvillian criterion for identifying temporal-crystalline order. Instead, it provides a broader conceptual bridge between temporal symmetry breaking, irreversible thermodynamics, and dissipative self-organization, offering a basis for future quantitative studies of the thermodynamic structure and stability of time-crystalline phases.
Discrete time crystals exhibit robust subharmonic oscillations, spontaneously breaking the discrete time-translation symmetry of a periodic drive. Since their emergence, the question of whether other types of temporal many-body order can exist has attracted considerable attention. Here we identify discrete time liquids -- a new form of spatiotemporal order characterized by short-range temporal coherence as a result of time-solitons. We study a two-dimensional spin system coupled to a finite-temperature cavity, whose fluctuations provide a noisy, approximately periodic drive with only finite temporal coherence. Remarkably, the spins nevertheless form a metastable state with robust period-doubled dynamics over an extended parameter regime. Unlike a time crystal, this state has only finite temporal coherence: individual trajectories stochastically switch between the two symmetry-related temporal configurations of the period-doubled response. We identify these switches as time-solitons: topological defects in temporal order that preserve local period doubling while destroying long-range temporal coherence, thereby producing the time liquid state. We propose a minimal model which reproduces the distribution of temporal correlations of the many-body system, suggesting a general description of the time liquid as a time crystal with a finite density of time-solitons. Because these solitons can occur at arbitrary times, ensemble averaging restores continuous time-translation symmetry, raising the intriguing possibility that discrete time liquids can exist in thermal equilibrium.
The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the perpetuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization. Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics --- that of an intrinsically out-of-equilibrium many-body phase of matter. We include a compendium of necessary background, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. We formalize the notion of a time-crystal as a stable, macroscopic, conservative clock --- explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. We also cover a range of related phenomena, including various types of long-lived prethermal time-crystals, and expose the roles played by symmetries -- exact and (emergent) approximate -- and their breaking. We clarify the distinctions between many-body time-crystals and other ostensibly similar phenomena dating as far back as the works of Faraday and Mathieu. En route, we encounter Wilczek's suggestion that macroscopic systems should exhibit TTSB in their ground states, together with a theorem ruling this out. We also analyze pioneering recent experiments detecting signatures of time crystallinity in a variety of different platforms, and provide a detailed theoretical explanation of the physics in each case. In all existing experiments, the system does not realize a `true' time-crystal phase, and we identify necessary ingredients for improvements in future experiments.
Symmetries are well known to have had a profound role in our understanding of nature and are a critical design concept for the realization of advanced technologies. In fact, many symmetry-broken states associated with different phases of matter appear in a variety of quantum technology applications. Such symmetries are normally broken in spatial dimension, however, they can also be broken temporally leading to the concept of discrete time symmetries and their associated crystals. Discrete time crystals (DTCs) are a novel state of matter emerging in periodically driven quantum systems. Typically, they have been investigated assuming individual control operations with uniform rotation errors across the entire system. In this work we explore a new paradigm arising from nonuniform rotation errors, where two dramatically different phases of matter coexist in well defined regions of space. We consider a quantum spin network possessing long-range interactions where different driving operations act on different regions of that network. What results from its inherent symmetries is a system where one region is a DTC, while the second is ferromagnetic. We envision our work to open a new avenue of research on chimeralike phases of matter where two different phases coexist in space.
Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.