Verallgemeinerte Symmetrien in Quantentensornetzwerken

Solid, liquid, and gas are commonly recognized as the classical phases of matter, and transitions between them occur through adjustments in temperature and pressure. However, it is essential to note that liquid and gas constitute the same phase, while there exists more than one solid phase. The explanation for these distinctions lies in the role of symmetries in matter. Liquids and gases lack inherent symmetry, and in solids, various crystalline structures correspond to diverse lattice symmetries. The classification of these variations is crucial in understanding classical phases of matter. The complexity intensifies when delving into the quantum realm at tiny scales. In this domain, different phases emerge even at absolute zero temperature, with phase transitions driven by quantum fluctuations. Nevertheless, this complexity presents opportunities, as quantum phases of matter exhibit exotic phenomena that can be harnessed for applications such as topological quantum computation. In 2016, the Nobel Prize in Physics honored J. Michel Kosterlitz, David J. Thouless, and Duncan Haldane for discovering exotic quantum phases of matter. Their work, among other contributions, highlighted that the same symmetry can yield different quantum phasesa departure from conventional phases where the crystalline structure dictates the entire phase, as seen in the example of ice. The distinction between quantum phases with identical symmetries arises from the entanglement between the particles, i.e., quantum correlations. This seemingly counterintuitive property, referred to by Albert Einstein as `Spooky action at a distance,` was experimentally demonstrated by Austrian physicist Anton Zeilinger, among others. Zeilinger, Alain Aspect, and John F. Clauser jointly received the Physics Nobel Prize in 2022 for their experiments with entangled photons, establishing the violation of Bell inequalities, and pioneering quantum information science. Our project aims to explore generalized symmetries in quantum spin chains. These symmetries act cohesively on the entire system, unlike ordinary symmetries that operate independently on individual particles. Tensor networks serve as our primary technical tool, enabling the modeling of generalized symmetries and states invariant under them. Tensor networks are advantageous as they exhibit `little entanglement` between particles, a realistic approximation considering local interactions in nature. This framework allows to build extensive objects by using smaller building blocks which facilitates the study of these systems revealing the emerging global properties from local ones. In particular, the different symmetric phases are encoded in the local entanglement. Moreover, tensor networks provide an general formalism to attack different kind of symmetries avoiding particular realizations that could obscure the relevant features.