Symmetry is fundamental to topological phases. In the presence of a gauge field, spatial symmetries will be projectively represented, which may alter their algebraic structure and generate novel topological phases. We show that the $\mathbb{Z}_2$ projectively represented translational symmetry operators adopt a distinct commutation relation, and become momentum dependent analogous to twofold nonsymmorphic symmetries. Combined with other internal or external symmetries, they give rise to many exotic band topology, such as the degeneracy over the whole boundary of the Brillouin zone, the single fourfold Dirac point pinned at the Brillouin zone corner, and the Kramers degeneracy at every momentum point. Intriguingly, the Dirac point criticality can be lifted by breaking one primitive translation, resulting in a topological insulator phase, where the edge bands have a Möbius twist. Our work opens a new arena of research for exploring topological phases protected by projectively represented space groups.
The Lieb-Schultz-Mattis theorem and its higher dimensional generalizations by Oshikawa and Hastings require that translationally invariant 2D spin systems with a half-integer spin per unit cell must either have a continuum of low energy excitations, spontaneously break some symmetries, or exhibit topological order with anyonic excitations. We establish a connection between these constraints and a remarkably similar set of constraints at the surface of a 3D interacting topological insulator. This, combined with recent work on symmetry-enriched topological phases (SETs) with on-site unitary symm…
Nodal-point and Nodal-line structures in the dispersion of electron energy bands are characterized by their high degeneracy in certain corners or lines in the Brillouin zone (BZ). These nodal structures can also exist in the dispersion of itinerant electrons in magnetically ordered materials whose symmetry groups are anti-unitary groups called the magnetic space groups. In the present work, we provide a complete list of magnetic space groups which can host symmetry-protected nodal-point/line band structures for spin-1/2 fermionic particles, where the degeneracies at the nodal points/lines are …
A projective symmetry group (PSG) has been regarded as a classification theory of spin liquids. However, it does not include a symmetry-protected topological order of fermionic spinon excitations, and thus the classification of gapped spin liquids is incomplete. We demonstrate the classification beyond PSG by utilizing the Kitaev model on the squareoctagon lattice, where two gapped spin liquids are distinguished by a topological $Z_2$ invariant. This $Z_2$ invariant can be defined solely by the time-reversal and translation symmetries on condition that the time-reversal symmetry is implemented…
Spin-orbit coupled materials have attracted revived prominent research interest as of late, especially due their direct connection with topological notions. Arguably, a hallmark of this pursuit is formed by the concept of the topological band insulator (TBI). In these incompressible systems band inversions, often driven by strong spin-orbit coupling, result in a state that is topologically distinct from the usual insulator as long as time reversal symmetry is maintained. More generally, topological states that arise by virtue of a protecting symmetry have resulted in a flourishing research fie…
We study coupled non-Hermitian Rice-Mele chains, which consist of Su-Schrieffer-Heeger (SSH) chain system with staggered on-site imaginary potentials. In two dimensional (2D) thermodynamic limit, the exceptional points (EPs) are shown to exhibit topological feature: EPs correspond to topological defects of a real auxiliary 2D vector field in k space, which is obtained from the Bloch states of the non-Hermitian Hamiltonian. As a topological invariant, the topological charges of EPs can be $\pm$1/2, obtained by the winding number calculation. Remarkably, we find that such a topological character…
We study two coupled Su-Schrieffer-Heeger (SSH) chains system, which is shown to contain rich quantum phases associated with topological invariants protected by symmetries. In the weak coupling region, the system supports two non-trivial topological insulating phases, characterized by winding number N = +1 or -1, and two types of edge states. The boundary between the two topological phases arises from two band closing points, which exhibit topological characteristics in one-dimensional k space. By mapping Bloch states on a vector field in k space, the band degenerate points correspond to a pai…
Symmetry protected topological phases in non-interacting fermion systems
对称性保护拓扑相是具有一定对称性的带隙量子相，如果对称性被破坏，这些带隙量子相可以光滑地连接到同一个平凡乘积态。对于具有时间反演（T）、电荷共轭（C）和/或U（1）（N）对称性的非相互作用费米子系统，全对称群可以取决于这些对称性操作之间的关系，例如T N T^{-1}= N或T N T^{-1}= -N。结果表明，不同对称群的费米子系统的SPT相有不同的分类。在本文中，我们使用Kitaev的K-理论的方法来分类这些可能的对称群的带隙自由费米子相位。特别地，我们可以将U（1）视为自旋旋转。我们发现，在偶数维中，具有S_z自旋旋转对称性的超导体可以用Z分类，而在奇数维中，具有时间反转和S_z自旋旋转对称性的超导体可以用Z分类。我们表明，所有10类带隙自由费米子相位可以实现的电子系统具有一定的对称性。我们还指出，为了正确地描述费米子系统的对称性，我们需要指定它的完整对称群，包括费米子数宇称变换（-）^N。全对称群实际上是一个射影对称群。
A fundamental dichotomous classification for all physical systems is according to whether they are spinless or spinful. This is especially crucial for the study of symmetry-protected topological phases, as the two classes have distinct symmetry algebra. As a prominent example, the spacetime inversion symmetry $PT$ satisfies $(PT)^2=\pm 1$ for spinless/spinful systems, and each class features unique topological phases. Here, we reveal a possibility to switch the two fundamental classes via $\mathbb{Z}_2$ projective representations. For $PT$ symmetry, this occurs when $P$ inverses the gauge tran…
Topological crystalline phases (TCPs) are topological states protected by spatial symmetries. A broad range of TCPs have been conventionally studied by formulating topological invariants (symmetry indicators) at invariant momenta in the Brillouin zone, which leaves open the question of their stability in the absence of translational invariance. In this work, we show that robust basis-independent topological invariants can be generically constructed for TCPs using projected symmetry operators. Remarkably we show that the real-space topological markers of these invariants are exponentially local…
Unconventional Altermagnetism in Quasicrystals: A Hyperspatial Projective Construction
School of Physics, Peking University（北京大学物理学院）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
We consider weak topological insulators with a twofold rotation symmetry around the dark direction, and show that these systems can be endowed with the topological crystalline structure of a higher-order topological insulator protected by rotation symmetry. These hybrid-order weak topological insulators display surface Dirac cones on all surfaces. Translational symmetry breaking perturbations gap the Dirac cones on the side surfaces leaving anomalous helical hinge modes behind. We also prove that the existence of this topological phase comes about due to a hidden crystalline topological invari…
With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals, particularly focusing on one-dimensional (1D) and 2D systems. We first give a brief introduction to quasicrystalline structures. Then, we discuss topological phases in 1D quasicrystals where the topological nature is attributed to the synthetic dimensions associated with the quasiperiodic order of quasicrystals. We further present the generalization of various ty…
Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of $G$-crossed braided tensor categories ($G\times$BTCs)…
Collinear magnets in honeycomb lattices under the action of time-dependent strains are investigated. Given the limits of high-frequency periodically varying deformations, we derive an effective Floquet theory for spin system that results in the emergence of a spin chirality. We find that the coupling between magnons and spin chirality depends on the details of the strain such as the spatial dependence and applied direction. Magnonic fluctuations about the ferromagnetic state are determined, and it is found that spatially homogeneous strains drive the magnon system into topologically protected …
