Half-integer topological invariant and its application in condensed matter systems

Abstract

In this thesis, we investigate quantum anomaly in condensed matter, in- troducing quantum anomalous semimetals and half-integer topological invariants, and exploring their behavior and potential applications. We generalize the topological invariants from integer to half-integer num- bers in systems with Fermi surfaces and develop a topological classifica- tion. This enables us to uncover unique properties of quantum anoma- lous semimetals, particularly in bulk-edge correspondence, transport prop- erty, and charge pumping process, and discover anomalous boundary current phenomena in these systems and semi-magnetic topological in- sulator thin films. Our study reveals that the boundary current arises from the collec- tive behavior of all occupied gapless bulk states, with half-quantized Hall conductance distinguishing them from conventional integer topo- logical materials. We also uncover half-charge pumping in half-integer topological systems, providing new perspectives on their behavior. We show that finite-frequency Hall conductance in integer Hall conductance systems can illustrate a half-quantized plateau indicating the parity anomaly that is characterized by half-quantized Hall conductance. This suggests new experimental techniques for verifying quantum anomalies in con- densed matter systems and probing related topological properties. Additionally, we generalize 1D Wilson fermion to the temporal di- mension, forming a discrete time crystal with unique dynamical sym- metry and topology. Our research provides an exact solution to the time-dependent model, revealing its dynamic behavior and perfect sub- harmonic response to external driving forces. We demonstrate that this response is caused by nonsymmorphic dynamical symmetry, offering a deeper understanding of the relationship between dynamical symme- try and time-dependent non-equilibrium Floquet systems. We further investigate many-body effects in these novel dynamical systems, reveal- ing their inherent resilience and stability of discrete time crystal phase, even under imperfect driving conditions and in the presence of perturba- tions. This robustness makes them well-suited for practical applications requiring stability across various conditions. Our work emphasizes the importance of understanding the inter- play between symmetry, topology, quantum anomaly, and electronic prop- erties in condensed matter systems, revealing the half-integer topologi- cal invariants and their application in condensed matter physics. Our findings advance condensed matter physics knowledge of topology and hold promise for future developments in advanced materials, electronic devices, and quantum computers.