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.
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.
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.
Abstract 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.
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.
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.
Exotic phases of matter represent a paradigm shift in condensed matter physics, transcending classical thermodynamic states and embracing quantum mechanical principles. These phases, including topological insulators, quantum spin liquids, and time crystals, exhibit emergent properties driven by symmetry breaking, entanglement, and quantum coherence. This paper explores the fundamental principles governing these novel phases, recent breakthroughs in their realization, and their implications for quantum computing and materials science. Emphasis is placed on theoretical models such as the Kitaev honeycomb lattice and experimental advances in ultracold atom traps and strongly correlated electron systems. By reviewing current challenges and prospects, this study outlines a roadmap for navigating this new frontier in physics.
The concept of additional spatial dimensions has long been invoked to explain phenomena in high‑energy physics and cosmology. In this work we propose a fundamentally different paradigm: the presence of a hidden temporal coordinate within the response functions of quantum materials. We argue that the intrinsic dynamics of electrons, phonons, and collective excitations are governed not only by the observable laboratory time t but also by an auxiliary time parameter a_h that is not directly accessible to conventional measurement. This hidden time coordinate manifests itself in the structure of the Green's functions and in the evolution of many‑body states, leading to observable consequences such as anomalous dispersion, non‑Markovian relaxation, and time‑crystalline behaviour. We develop a formalism that incorporates the hidden time into the standard path‑integral and operator approaches, derive modified equations of motion, and discuss potential experimental signatures in ultrafast spectroscopy. The framework suggests that quantum materials can serve as a laboratory for exploring higher‑dimensional temporal physics, opening pathways for discovering new time‑related phases and for achieving unprecedented control over quantum dynamics.