Topological zero modes and Dirac points protected by spatial symmetry and chiral symmetry

Abstract

We explore a new class of topologically stable zero energy modes which are protected by coexisting chiral and spatial symmetries. If a chiral symmetric Hamiltonian has an additional spatial symmetry such as reflection, inversion and rotation, the Hamiltonian can be separated into independent chiral-symmetric subsystems by the eigenvalue of the space symmetry operator. Each subsystem supports chiral zero energy modes when a topological index assigned to the block is nonzero. By applying the argument to Bloch electron systems, we detect band touching at symmetric points in the Brillouin zone. In particular, we show that Dirac nodes appearing in honeycomb lattice (e.g. graphene) and in half-flux square lattice are protected by three-fold and two-fold rotation symmetry, respectively. We also present several examples of Dirac semimetal with isolated band-touching points in three-dimensional $k$-space, which are protected by combined symmetry of rotation and reflection. The zero mode protection by spatial symmetry is distinct from that by the conventional winding number. We demonstrate that symmetry-protected band touching points emerge even though the winding number is zero. Finally, we identify relevant topological charges assigned to the gapless points.

12 pages, 6 figures

References in corpus (36)

• 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
• Discovery of a Three-dimensional Topological Dirac Semimetal, Na3Bi
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Weyl Semimetal in a Topological Insulator Multilayer
• Topological Crystalline Insulators
• Dirac semimetal and topological phase transitions in A3Bi (A=Na, K, Rb)
• Periodic table for topological insulators and superconductors
• Dirac semimetal in three dimensions
• Three Dimensional Dirac Semimetal and Quantum Transports in Cd3As2
• Topological Crystalline Insulators in the SnTe Material Class
• Observation of a topological 3D Dirac semimetal phase in high-mobility Cd3As2
• Experimental Realization of a Three-Dimensional Dirac Semimetal
• Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase
• Experimental realization of a topological crystalline insulator in SnTe
• Classification of stable three-dimensional Dirac semimetals with nontrivial topology
• Topological crystalline insulator states in Pb(1-x)Sn(x)Se
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Topological Origin of Zero-Energy Edge States in Particle-Hole Symmetric Systems
• A topological crystalline insulator (TCI) phase via topological phase transition and crystalline mirror symmetry
• The space group classification of topological band insulators
• Inversion Symmetric Topological Insulators
• Classification of topological insulators and superconductors in the presence of reflection symmetry
• Topology of Andreev Bound States with Flat Dispersion
• Stability of Fermi Surfaces and K-Theory
• Topological classification with additional symmetries from Clifford algebras
• Topological Mirror Superconductivity
• Topological Classification and Stability of Fermi Surfaces
• Majorana Fermions and Disclinations in Topological Crystalline Superconductors
• Symmetry-Protected Majorana Fermions in Topological Crystalline Superconductors: Theory and Application to Sr2RuO4
• Statistical Topological Insulators
• Topological Blount's theorem of odd-parity superconductors
• Weyl and Dirac semimetals with Z_2 topological charge
• Designing Dirac points in two-dimensional lattices

Cited by in corpus (49)

• Weyl and Dirac Semimetals in Three Dimensional Solids
• Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
• Topological Phases in Two-Dimensional Materials: A Brief Review
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
• Two-dimensional second-order topological insulator in graphdiyne
• Topological superconductivity in Dirac semimetals
• Topology of nonsymmorphic crystalline insulators and superconductors
• Unified theory of PT and CP invariant topological metals and nodal superconductors
• Magnetic Field Driven Nodal Topological Superconductivity in Monolayer Transition Metal Dichalcogenides
• Exchange rules for diradical π-conjugated hydrocarbons
• Topological Effects in Chiral Symmetric Driven Systems
• Interplay of topological phases in magnetic adatom-chains on top of a Rashba superconducting surface
• Edge states in graphene-like systems
• Topological Properties of Gapped Graphene Nanoribbons with Spatial Symmetries
• Spectroscopy of Twisted Bilayer Graphene Correlated Insulators
• Chiral spin-orbital liquids with nodal lines
• Higher-Order Topological Phase in a Honeycomb-Lattice Model with Anti-Kekulé Distortion
• Triggering a global density wave instability in graphene via local symmetry-breaking
• Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions
• Angular momentum and topology in semiconducting single-wall carbon nanotubes
• Non-Abelian Generalizations of the Hofstadter model: Spin-orbit-coupled Butterfly Pairs
• Magnetic-Field-Induced Topological Reorganization of a P-wave Superconductor
• Topology invisible to eigenvalues in obstructed atomic insulators
• Novel ${\mathbb{Z}}_2$-topological metals and semimetals
• Majorana fermion fingerprints in spin-polarised scanning tunneling microscopy
• Optoelectronic Fingerprints of Interference between Different Charge Carriers in Graphene Superlattices and Analogies to Twisted Graphene Bilayers
• Type-III Dirac Cones from Degenerate Directionally Flat Bands: Viewpoint from Molecular-Orbital Representation
• Magnetic manipulation of topological states in p-wave superconductors
• Topological junction states and their crystalline network in chiral-symmetric systems: application to graphene nanoribbons
• Nodal S-wave Superconductivity in Antiferromagnetic Semimetals
• Magnetic frustration and fractionalization in oligo(indenoindenes)
• Helical metals and insulators and sheet singularity of inflated Berry monopole
• Topologically invisible defects in chiral mirror lattices
• Molecular-orbital representation with random U(1) variables
• Topological Modes Protected by Chiral and Two-Fold Rotational Symmetry in a Spring-Mass Model with a Lieb Lattice Structure
• Conductance zeros in complex molecules and lattices from the interference set method
• Emergent chiral symmetry in non-bipartite kagome and pyrochlore lattices with spin-orbit coupling
• Emergence of Topological Superconductivity in Doped Topological Dirac Semimetals under Symmetry-Lowering Lattice Distortions
• Discriminant indicator with generalized rotational symmetry
• Universal Distribution of Would-be Topological Zero Modes in Coupled Chiral Systems
• Stability against contact interactions of a topological superconductor in two-dimensional space protected by time-reversal and reflection symmetries
• General response theory of topologically stable Fermi points and its implications for disordered cases
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- part II: Higher-dimensional spectral invariants
• Boundary states in the chiral symmetric systems with a spatial symmetry
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Floquet composite Dirac semimetals
• Engineering Topological Nodal Line Semimetals in Rashba Spin-Orbit Coupled Atomic Chains
• Single $\pi$-flux hosting topological defect modes in bilayer acoustic metamaterials
• From Cyclic Nanorings to Single-Walled Carbon Nanotubes: Disclosing the Evolution of their Electronic Structure with the Help of Theoretical Methods