Exotic phases in condensed matter systems : space-time crystals and moiŕe superlattices

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This Thesis unfolds a comprehensive exploration of two distinct yet equally captivating realms within condensed matter systems. While each part stands independently, together they paint a vivid picture of the intricate and multifaceted world of solid state physics. Part I: Space-Time Crystals and Floquet-Bloch Theory– Space-time crystals have emerged as a novel and intriguing concept in quantum mechanics, uniquely distinguished by their periodicities in both space and time. We first argue that, within specific experimental setups, like that of a monoatomic crystal driven by coherent sound waves, these space-time periodic structures have shown a remarkable ability to convert energy quanta of sound waves into potential energy of DC electric fields. This energy conversion process is deeply tied to the intricate topologies of quasi-energy bands. To decode and understand these phenomena, we used the Floquet-Bloch (FB) theory which represents a significant advancement beyond the traditional Bloch theorem by adeptly incorporating time-periodic Hamiltonians, thereby revealing insights into the dynamic evolution of band structures under time-varying perturbations. Furthermore, detailed studies on DC current generation in periodically driven Bloch systems, when analyzed again under the FB framework, have unveiled two distinct types of currents: the intrinsic currents, which are in harmony with the system's quasi-equilibrium state even under continuous driving, and the extrinsic currents, which emerge from shifts between different equilibrium states. These insights not only deepen our understanding of space-time crystal dynamics but also pave the way for potential applications in energy and quantum technologies. Finally, we provide a semiclassical point of view for the electronic dynamics and responses which offers a systematic treatment of the FB systems. Part II: Moiré Superlattices and Beyond– Lattice dynamics in moiré systems, particularly in twisted bilayer graphene (TBG), have been a focal point of investigation. An underlying mismatch symmetry in TBG has been identified, which exists for any twist angle, providing insight into the connection between moiré phonons and phasons. This understanding extends to twisted multilayer graphene (TMG), emphasizing its unique lattice dynamics and the emergence of specific moiré phonon modes. Moreover, a novel perspective on moiré systems has been presented through periodically strained graphene. This innovative setup mimics the dynamics of twisted moiré structures without a twist, offering a promising platform for exploring strong correlating physics such as quantum anomalous Hall states, and fractional Chern insulator phases.



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