Over the past decade, the study of topological phases of matter in condensed matter physics has unveiled a wealth of exotic phenomena. Research has extensively explored how symmetry interacts with topology, leading to the development of symmetry-protected topological (SPT) phases in both non-interacting and interacting systems at equilibrium. In particular, strong correlations can give rise to phenomena such as fractionalization and anyonic particle statistics.
Despite these advancements, several crucial questions remain. Understanding how to measure and detect these topological phases is essential. Additionally, exploring non-equilibrium settings where topological phases exhibit unique properties not found in equilibrium, and developing methods to control and stabilize these phases to maintain quantum coherence, present significant challenges. Finally, investigating the interplay between magnetic double space groups and band topology in magnetic materials is a key focus of ongoing research. This thesis addresses these challenges through three studies, ranging from idealized models to real-world materials.
In the first part, we develop a method to detect topological phases in two-dimensional (2D) ultracold atomic clouds engineered to exhibit a fractional Chern insulator (FCI) phase in the Hofstadter model, both on an infinite cylinder and a finite square lattice by means of the density-matrix renormalization group (DMRG). This method accurately determines the fractional quantized Hall conductivity by observing the displacement of the atomic cloud under the action of a constant force, providing an experimentally realistic and measurable signal for detecting the topological nature of the state.
In the second part of the thesis, we move beyond equilibrium systems to explore topological phases in periodically driven Floquet systems that are far from equilibrium. Due to their time-dependent nature, these systems can exhibit exotic phases not found in conventional equilibrium physics. We identify novel topological phases arising from the interplay of dynamical space-time symmetries, which combine spatial and temporal symmetries. These phases introduce a new class of topological phases within Floquet systems, exhibiting gap-dependent, distinct topological classifications. This contrasts with existing Floquet topological phases protected by static symmetries, where the topological classifications across all quasi-energy gaps are characterized by the same Abelian group. We provide a formal mathematical framework based on group cohomology for the systematic analysis of higher-order space-time symmetries.
A significant challenge in Floquet systems with many-body interactions is the heating issue, where nonintegrable closed Floquet systems eventually relax to a featureless, trivial infinite-temperature state as a result of Floquet eigenstate thermalization hypothesis (Floquet-ETH). Recent studies have explored overcoming this problem using disorder and dissipation to sustain nontrivial steady states. Our study focuses on prethermalization, where quasi-steady states form as finite-temperature thermal states of the static effective Hamiltonian, known as the prethermal Hamiltonian, in the intermediate long-time regime before heating to featureless thermal states. We consider systems with dual energy scales of resonant and high-frequency drives, where the Hamiltonian consists of terms whose amplitudes are either comparable to or much smaller than the driving frequency. By starting with the dynamical space-time symmetry group of the original time-dependent Hamiltonian, we map out how to derive the enlarged static symmetry group of the prethermal Hamiltonian. This enlarged group can control phases and symmetries of quasi-steady states during the prethermal regime. Finally, we discuss methods for detecting dynamical symmetries and explore their intrinsic dynamical nature through the time evolution of states.
In the final chapter, we investigate the intertwined orders of magnetism and topology in real material. Specifically, we explore how selective tuning of symmetry and magnetism can influence and control the resulting topology in a 2D magnetic system, using the hypothetical ferromagnetic (FM) monolayer MnPSe$_3$ as an example. We systematically perturb selected symmetries to examine different topological phases, including the symmetry-protected semi-metallic (SM) phase and the quantum anomalous Hall insulator (QAHI) phase. This is achieved through magnetization tilts and appropriate chemical substitutions, revealing how different magnetic space groups affect the topological properties of the material.
