Session U54: Topological Crystalline Phases

Aperiodic Topological Crystalline Insulators

Topological crystalline insulators (TCIs) are usually described with topological protection imposed by the crystalline symmetry. In general, however, the existence of TCI states does not necessitate the periodicity of crystals as long as an essential lattice symmetry can be identified. Here we demonstrate the compatibility of TCIs with aperiodic systems, as exemplified by an octagonal quasicrystal. In contrast to most common topological insulators and TCIs, the proposed aperiodic TCIs are attributed to a different symmetry-invoked band inversion mechanism, which inverts states with the same parities but opposite eigenvalues of a specific symmetry (such as reflection). The nontrivial topology is characterized by a nonzero integer ``mirror Bott index''. Moreover, we demonstrate that the topological edge states and quantized conductance of the aperiodic TCI, which are robust against disorder, can be effectively manipulated by external electric fields. Our findings not only provide a better understanding of electronic topology in relation to symmetry but also extend the experimental realization of topological states to much broader material categories beyond crystals.   

Real Space Invariants: Spectral Flow of Fragile Topological State under Twisted Boundary Conditions

In this paper, we propose the twisted boundary condition as a generic approach to theoretically and experimentally detect the fragile topological state. We prove that, when the fragile phase can be written as a difference of a trivial atomic insulator and the so-called obstructed atomic insulator, the gap between the fragile phase and other bands must close under a specific twist of
the boundary condition of the system. We explicitly work out all the twisted boundary conditions that can detect all the 2D fragile phases implied by symmetry eigenvalues in all wallpaper groups. We develop the concept of real space invariants, which are local good quantum numbers in real space, and which also fully characterize the eigenvalue fragile phases. We show that the number of unavoidable level crossings under the twisted boundary condition is completely determined by the real space invariants. Possible realizations of the twisted boundary condition of the fragile band in metamaterial systems are also discussed.
  

Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop

We study the bulk and boundary properties of fragile topological insulators (TIs) protected by inversion symmetry, mostly focusing on the class A insulator. First, we propose an efficient method for diagnosing fragile band topology by using the symmetry data in momentum space. Second, we study the bulk-boundary correspondence of fragile TIs protected by inversion symmetry. In particular, we show that a minimal fragile TI with the filling anomaly corresponds to the dD (d+1)th-order TI. Although dD (d+1)th-order TIs have no in-gap states, the boundary mass terms carry an odd winding number along the boundary, which induces localized charges on the boundary at the positions where the boundary mass terms change abruptly. Also by studying the (nested) Wilson loop spectra, we show that all the spectral windings of the (nested) Wilson loop should be unwound to resolve the Wannier obstruction of fragile TIs. By counting the minimal number of bands required to unwind the spectral winding of the Wilson loop and nested Wilson loop, we determine the minimal number of bands to resolve the Wannier obstruction, which is consistent with our diagnosis method of fragile topology.   

Crystalline topological phases without symmetry indicators

Topological insulating phases of matter are quantum phases that cannot be deformed to atomic insulators. They generically feature protected anomalous surface states, which can be used for applications in spintronics and quantum computation.

Recently, two new types of topological phases protected by crystalline symmetries have been discovered. The first are higher-order topological insulators, featuring conducting states on the hinges [1], recently observed in bismuth [2]. The other are fragile topological states, which become trivial when occupying additional bands. Twisted bilayer graphene may be a realization of a fragile phase, which may help to explain superconductivity in this system [3].

Usually, crystalline topology can be detected by computing symmetry indicators. However, there exist topological phases which cannot be detected in this way, and we present the first classification of such states [4]. We show these crystalline invariants are also well-defined in quantum spin Hall systems and use this to construct a hybrid-order weak TI [5], which is both a weak and higher-order TI.

[1] Kooi et al. Phys. Rev. B 98, 245102
[2] Schindler et al. Nature Physics 14, 918–924 (2018)
[3] Zou et al. Phys. Rev. B 98, 085435
[4] Kooi et al. arXiv:1906.08695
[5] Kooi et al. arXiv:1908.00879   

Topology of chiral insulators

Building on direct analogies with electronic matter, the concepts of topological insulator and topological protection have been successfully applied to a host of different physical systems as diverse as photonic metamaterials, geophysical fluids and mechanical structures. In this work, we focus on materials, or metamaterials, having a chiral symmetry, such as crystals with a sub-lattice symmetry and all mechanical systems assembled from beads and springs.

Firstly, we show that the topology of waves in chiral materials is naturally encoded in their chiral polarization, which quantifies the spatial imbalance of localized Wannier states between the two sub-lattices. Secondly, we demonstrate that the chiral polarization originates both from topological properties of the Hamiltonian and of the frame on which this Hamiltonian is defined. Our results elucidate a long-standing ambiguity on the topological characterization of the simplest, and probably oldest example of a topological state: the Su-Schrieffer-Hegger (SSH) chain. Finally, we establish a generalized bulk-boundary correspondence for chiral systems and show how to effectively use the concept of chiral polarization to predict the existence of interfacial states at the boundary of chiral insulators.   

Boundary obstructed topological phases

Symmetry protected topological (SPT) phases are gapped phases of matter that cannot be deformed to a trivial phase without breaking the symmetry or closing the bulk gap. Here, we introduce a new notion of a topological obstruction that is not captured by bulk energy gap closings in periodic boundary conditions. More specifically, we say two bulk Hamiltonians belong to distinct boundary obstructed topological ‘phases’ (BOTPs) if they can be deformed to each other on a system with periodic boundaries, but not for symmetric open boundaries without closing the gap at a high symmetry region on the surface. BOTPs are not topological phases of matter in the standard sense since they are adiabatically deformable to each other on a torus but, similar to SPTs, they are associated with surface states or fractional corner charges in the open system. We show that the double-mirror quadrupole model of [Science, 357(6346), 2018] is a prototypical example of such phases, and present a detailed analysis of boundary obstructions in this model. In addition, we introduce several three-dimensional models having boundary obstructions, which are characterized either by surface states or fractional corner charges and discuss their general formulation in terms of Wannier band representations.   

Topological Crystalline States with Non-trivial Relative Group Cohomology

The method of relative homotopy group has been know useful to characterize the surface defects that are not trivial extensions of bulk defects. We develop the method of relative group cohomology to study novel topological crystalline states characterized by non-trivial relative group cohomology and their relation to fragile topological crystalline states.   

Topological Invariants of a Filling-Enforced Quantum Band Insulator

In the classical picture of insulators, electrons occupy localized Wannier (atomic-like) orbitals. Recently, insulators for which electron filling prohibits this atomic description have been theoretically proposed. The electronic wave functions of these insulators, termed filling-enforced Quantum Band Insulators (feQBIs), display a necessary degree of quantum entanglement due to the non-atomic filling. Currently, little is known about the relation between feQBIs and conventional topological invariants. In this work, we study such relations for a particularly interesting example of an feQBI realized with spinless fermions in space group 106.   

Fermions in ZN Gauge Fields: From Chern Insulators to Higher-Order Topological Insulators

Fermions coupled to gauge fields defined on the links of
lattices have received considerable attention due to their
importance in describing a variety of physical systems. Here,
we show how a higher-order topological insulator emerges in a
system of fermions coupled to Z_N gauge fields. By tuning the
filling of fermions and the strength of the coupling of
fermions to the gauge fields, we show that a variety of fermion
phases are stabilized for different parameter regimes. We map
out the phase diagram of this system and identify conditions
needed to generate a higher-order topological insulating
phase. This work suggests experimental routes, for example in a
cold atomic system, to obtain different topological phases of
fermions to study both the properties of these phases and the
phase transitions between them.   

Type-II quadrupole topological insulators

Recently, the formulation of polarization based on the Berry phase has been extended to
higher multipole moments, through the discovery of the so-called quadupole topological insulator.
It has been established by a classical electromagnetic theory that in a two-dimensional material
the quantized properties for the quadupole topological insulator should satisfy a basic relation.
Here we discover a new type of quadupole topological insulator (dubbed as type-II) that violates
this relation due to the breakdown of a previously established theory that a Wannier band and
an edge energy spectrum are topologically equivalent in a closed quantum system. We find that,
similar to the previously discovered (referred to as type-I) quadrupole topological insulator, the
type-II hosts topologically protected corner states carrying fractional corner charges. However, the
edge polarizations only occur at a pair of boundaries in the type-II insulating phase, leading to
the violation of the classical constraint. We propose an experimental scheme to realize such a new
topological phase of matter. The existence of the new topological insulating phase means that
new multipole topological insulators with distinct properties can exist in broader contexts beyond
classical constraints.   

Layer Construction of Topological Crystalline Insulator LaSbTe

Topological crystalline insulator (TCI) is one of the symmetry-protected topological states. Any TCI can be looked as a simple product state of several decoupled two-dimensional (2D) topologically nontrivial layers in its lattice respecting its crystalline symmetries, so called layer construction (LC) scheme. In this work, based on first-principles calculations we have revealed that both the tetragonal LaSbTe (t-LaSbTe) and the orthorhombic LaSbTe (o-LaSbTe) can be looked as a stacking of 2D topological insulators in each lattice space. The structural phase transition from t-LaSbTe to o-LaSbTe due to soft phonon modes demonstrates how the real space change can lead to the modification of topological states. Their symmetry-based indicators and topological invariants have been analyzed based on LC. We propose that LaSbTe is an ideal paradigm perfectly demonstrating the LC scheme, which bridges the crystal structures in real space to the band topology in momentum space.   

Filling-enforced Dirac loops and their evolutions under various perturbations

Based on symmetry analysis, we find that filling-enforced Dirac loops (FEDLs) in non-magnetic systems exist in only five space groups. We further explore all possible configurations of these FEDLs in these space groups, and classify them accordingly. We study the evolutions of the FEDLs under various types of perturbations such as applied strain or field. It is interesting that the FEDL-materials can serve as both parent materials of nodal semimetals having Dirac points or nodal-loops, and topological insulators/topological crystalline insulators. Many the FEDL-materials are predicted in DFT calculations.