Three-dimensional spinless Euler insulators with rotational symmetry

Abstract

The Euler class is a $\mathbb{Z}$-valued topological invariant that characterizes a pair of real bands in a two-dimensional Brillouin zone. One of the symmetries that permits its definition is $C_{2z}T$, where $C_{2z}$ denotes a twofold rotation about the $z$ axis and $T$ denotes time-reversal symmetry. Here, we study three-dimensional spinless insulators characterized by the Euler class, focusing on the case where additional $C_{4z}$ or $C_{6z}$ rotational symmetry is present, and investigate the relationship between the Euler class of the occupied bands and their rotation eigenvalues. We first consider two-dimensional systems and clarify the transformation rules for the real Berry connection and curvature under point group operations, using the corresponding sewing matrices. Applying these rules to $C_{4z}$ and $C_{6z}$ operations, we obtain explicit formulas that relate the Euler class to the rotation eigenvalues at high-symmetry points. We then extend our analysis to three-dimensional systems, focusing on the difference in the Euler class between the two $C_{2z}T$-invariant planes. We derive analytic expressions that relate the difference in the Euler class to two types of representation-protected invariants and analyze their phase transitions. We further construct tight-binding models and perform numerical calculations to support our analysis and elucidate the bulk-boundary correspondence.

21 pages, 12 figures

References in corpus (63)

• Topological Insulators
• Topological insulators and superconductors
• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• WannierTools: An open-source software package for novel topological materials
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Topological Crystalline Insulators
• Periodic table for topological insulators and superconductors
• Topological quantum chemistry
• The (High Quality) Topological Materials In The World
• Catalogue of Topological Electronic Materials
• An equivalent expression of Z2 Topological Invariant for band insulators using Non-Abelian Berry's connection
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Topological classification of crystalline insulators through band structure combinatorics
• Topology of crystalline insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• All "Magic Angles" Are "Stable" Topological
• Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle
• Symmetry indicators and anomalous surface states of topological crystalline insulators
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Non-Abelian band topology in noninteracting metals
• New classes of topological crystalline insulators with unpinned surface Dirac cones
• All Topological Bands of All Nonmagnetic Stoichiometric Materials
• Efficient Topological Materials Discovery Using Symmetry Indicators
• Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\text{ZrTe}$
• Topological Euler class as a dynamical observable in optical lattices
• Experimental observation of non-Abelian topological charges and bulk-edge correspondence
• Observation of non-Abelian topological semimetals and their phase transitions
• Dispersion of Fermi arcs in Weyl semimetals and their evolutions to Dirac cones
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Geometric approach to fragile topology beyond symmetry indicators
• Topology of Disconnected Elementary Band Representations
• Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice
• Spinless Topological Insulators without Time-Reversal Symmetry
• Topological Insulators from Group Cohomology
• Band Representations and Topological Quantum Chemistry
• Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates
• Berry-phase description of Topological Crystalline Insulators
• Stiefel-Whitney classes and topological phases in band theory
• Symmetry representation approach to topological invariants in $C_{2z}T$-symmetric systems
• Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
• Fragile Phases As Affine Monoids: Classification and Material Examples
• Crystallographic splitting theorem for band representations and fragile topological photonic crystals
• Topological Materials Discovery By Large-order symmetry indicators
• Unconventional topological phase transition in two-dimensional systems with space-time inversion symmetry
• Four-band non-Abelian topological insulator and its experimental realization
• Multi-gap topology and non-Abelian braiding of phonons from first principles
• Minimal non-abelian nodal braiding in ideal metamaterials
• Differential topology of semimetals
• Observation of topological Euler insulators with a trapped-ion quantum simulator
• Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions
• Two-dimensional topological materials discovery by symmetry-indicator method
• Topological Euler insulators
• Non-Abelian charged nodal links in dielectric photonic crystal
• Phase transitions of non-Abelian charged nodal links in a spring-mass system
• Chirality flip of Weyl nodes and its manifestation in strained MoTe${}_2$
• Interferometry of non-Abelian band singularities and Euler class topology
• Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Ideal Spin-Orbit-Free Dirac Semimetal and Diverse Topological Transitions in Pr${}_8$CoGa${}_3$ Family
• Representation-protected topology of spin-singlet $s$-wave superconductors