Theoretical Studies On Symmetry-protected Topological States Of Matter

Topological states of matter have emerged as one of the most prominent topics within the field of condensed matter physics,as their innovative concepts and potential applications have ignited significant research interests.Initially,topological phases were classified into ten symmetry classes,based on the presence or absence of time-reversal symmetry,particle-hole symmetry,and chiral symmetry.Subsequently,crystallographic symmetries were incorporated into this topological classification,leading to the development of concepts such as topological crystalline insulators and higher-order topological insulators.Typically,there are two distinct categories of symmetries:symmorphic and nonsymmorphic symmetries.The former refers to symmetries that maintain at least one fixed point under the symmetry operation,such as mirror and rotational symmetries.The latter refers to symmetries that alter the position of all points,including glide plane and screw axis symmetries.These symmetries are intimately intertwined with topological states of matter and forming a vital component of the condensed matter physics.In this dissertation,we systematically investigate various properties of numerous topological phases without intrinsic topological order,including electronic,ferroelectric,magnetic,and topological properties,highlighting the crucial role of symmetries in topological states of matter and providing a theoretical reference for further experimental research and potential applications.The dissertation is divided into five chapters,arrangement is as follows.In Chapter 1,we outline the development of topological states of matter,primarily discussing the research progress on several distinct topological phases,including the Hall effect,quantum Hall effect,quantum anomalous Hall effect,quantum spin Hall effect,topological insulators,topological crystalline insulators,and the recently proposed higher-order topological insulators.In Chapter 2,we briefly introduce the first-principles calculation methods,including density functional theory,tight-binding method,Wilson loop method,as well as several software programs.In Chapter 3,we focus on the theoretical study of symmorphic symmetry-protected topological phases,encompassing mirror symmetry-protected dual topological insulators,topological crystalline insulators,topological nodal-line semimetals,and rotational symmetryprotected higher-order topological insulators.In Chapter 4,we concentrate on the theoretical study of nonsymmorphic symmetry-protected topological phases,specifically the antiferromagnetic topological insulators protected by screw axis symmetry and the orbital shiftinduced boundary obstructed topological materials protected by glide plane symmetry.Chapter 5 summarizes the entire dissertation and provides the outlook for further research.Detailed research contents are as follows:(1)By exploring the evolution of electronic structure and topological properties under the influence of magnetic and electric field,we summarize the phase transition conditions for topological crystalline insulators,topological nodal-line semimetals,and higher-order topological insulators,as well as reveal the nonvolatile ferroelectric control of higher-order topological corner states.Based on this,we predict two ideal two-dimensional dual topological insulators,Na2MgPb and Na2CdSn,which exhibit the coexistence of topological crystalline insulators and topological insulators under the protection of time-reversal and mirror symmetry.Next,we propose a method utilizing an external magnetic field to break time-reversal symmetry and introduce the topological nodal-line semimetal phase.Importantly,the nodal lines protected by mirror symmetry are robust even in the presence of strong spin-orbit coupling.Subsequently,we predict that ferromagnetic NpSb monolayer serves as an ideal candidate material for achieving a topological phase transition between the topological crystalline insulator and the higher-order topological insulator.When the mirror symmetry is preserved for out-of-plane magnetization,the NpSb monolayer exhibits a nonzero mirror Chern number and manifests a pair of gapless edge states.It is noteworthy that upon rotating the magnetization into the plane,the edge states are gapped,and in-gap topological corner states emerge,thus rendering it to be a higher-order topological insulator.Finally,we map out the emergence of nontrivial corner states in two-dimensional ferroelectrics and propose a concept of two-dimensional ferroelectric higher-order topological insulators.Implementing density functional theory,we identify a series of experimentally synthesized two-dimensional ferroelectric materials,including both traditional ferroelectrics,such as In2Se3 and SnS monolayer,and sliding ferroelectrics represented by BN bilayer.Our work demonstrates that both quantized and non-quantized inplane polarizations could trigger the emergence of higher-order topology,and that external electric field could control the position of corner states between layers.Overall,these works establish a theoretical foundation for a better understanding and explanation of the relationships among ferromagnetic order,ferroelectric order,and topological states of matter.(2)We reveal the initial relationship between boundary polarization and topological states,and provide a general principle to generate surface states,hinge states,and corner states.Additionally,we point out the impact of nonsymmorphic symmetries on surface band structures and two-dimensional antiferromagnetic topology.Consequently,we put forward the realization of orbital shift-induced boundary obstructed topological materials.Under periodic boundary condition,these materials exhibit trivial symmetry indicators and trivial real-space invariants.However,under certain open boundary condition,at least one of the Wannier orbitals deviates from the original Wyckoff position and shifts into another occupied Wyckoff position,giving rise to metallic surface states,hinge states,and corner states.More importantly,the surface spectra exhibit complex degeneracy under the influence of glide plane symmetry,ultimately resulting in exotic wallpaper fermions,such as Dirac fermions,hourglass fermions,and Mobius fermions.Through the first-principles calculations,we systematically study 454 trivial materials in the I4/mcm space group,predict 10 orbital shift-induced boundary obstructed topological materials and firstly discover the emergence of electronic third-order topological insulators.Additionally,we find that screw axis symmetry could effectively protect the twodimensional antiferromagnetic topology.Subsequently,we investigate 57 antiferromagnets preserving the screw axis symmetry,and identify 9 ideal candidate materials as twodimensional antiferromagnetic topological insulators characterized by quantized spin Hall conductivity and gapless edge states.These works not only deepen our understanding of the physical phenomena induced by nonsymmorphic symmetries,but also provide potential candidates for theoretical and experimental research on higher-order topological insulators.