Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors

Abstract

Topological crystalline superconductors are known to have possible higher-order topology, which results in Majorana modes on $d-2$ or lower-dimensional boundaries. Given the rich possibilities of boundary signatures, it is desirable to have topological invariants that can predict the type of Majorana modes from band structures. Although symmetry indicators, a type of invariant that depends only on the band data at high-symmetry points, have been proposed for certain crystalline superconductors, there exist symmetry classes in which symmetry indicators fail to distinguish superconductors with different Majorana boundaries. Here, we systematically obtain topological invariants for an example of this kind, two-dimensional time-reversal symmetric superconductors with twofold rotational symmetry $C_2$. First, we show that the nontrivial topology is independent of band data on the high-symmetry points by conducting a momentum-space classification study. Then, from the resulting K groups, we derive calculable expressions for four ${\mathbb{Z}}_2$ invariants defined on high-symmetry lines or general points in the Brillouin zone. Finally, together with a real-space classification study, we establish the bulk-boundary correspondence and show that the four ${\mathbb{Z}}_2$ invariants can predict Majorana boundary types from band structures. Our proposed invariants can fuel practical material searches for $C_2$-symmetric topological superconductors featuring Majorana edge and corner modes.

arXiv comment27 pages, 5 figures, close to the published version

References in corpus (74)

• Topological Insulators with Inversion Symmetry
• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological quantum matter with symmetries
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Periodic table for topological insulators and superconductors
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Topological Crystalline Insulators in the SnTe Material Class
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Reflection symmetric second-order topological insulators and superconductors
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Odd-Parity Topological Superconductors: Theory and Application to Cu${}_x$Bi${}_2$Se${}_3$
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• The space group classification of topological band insulators
• Topological classification of crystalline insulators through band structure combinatorics
• Inversion Symmetric Topological Insulators
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Topology of crystalline insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• Classification of Topological Insulators and Superconductors
• Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors
• Stability of Fermi Surfaces and K-Theory
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Symmetry indicators and anomalous surface states of topological crystalline insulators
• Topological odd-parity superconductors
• Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
• Twisted equivariant matter
• Higher-order topological insulators protected by inversion and rotoinversion symmetries
• Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\text{ZrTe}$
• Topological phases protected by point group symmetry
• Building crystalline topological phases from lower-dimensional states
• Majorana Fermions and Disclinations in Topological Crystalline Superconductors
• Higher Order Topology and Nodal Topological Superconductivity in Fe(Se,Te) Heterostructures
• Geometric approach to fragile topology beyond symmetry indicators
• Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -
• Topological phases: Classification of topological insulators and superconductors of non-interacting Fermions, and beyond
• Symmetry protected topological phases in non-interacting fermion systems
• Inversion-protected higher order topological superconductivity in monolayer WTe${}_2$
• Fractional corner charges in spin-orbit coupled crystals
• Theory of interacting topological crystalline insulators
• Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials
• Refined symmetry indicators for topological superconductors in all space groups
• Topological states from topological crystals
• Symmetry representation approach to topological invariants in $C_{2z}T$-symmetric systems
• Unified understanding of symmetry indicators for all internal symmetry classes
• Symmetry-based indicators for topological Bogoliubov-de Gennes Hamiltonians
• Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock
• Chiral Dirac Superconductors: Second-order and Boundary-obstructed Topology
• Symmetry indicators of band topology
• Bott periodicity for $Z_2$ symmetric ground states of gapped free-fermion systems
• ${\mathbb{Z}}_2$-enriched symmetry indicators for topological superconductors in the 1651 magnetic space groups
• Atomic limit and inversion-symmetry indicators for topological superconductors
• Real-space recipes for general topological crystalline states
• Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology
• Symmetry indicators for topological superconductors
• Faithful derivation of symmetry indicators: A case study for topological superconductors with time-reversal and inversion symmetries
• High-$T_c$ Superconductor Fe(Se,Te) Monolayer: an Intrinsic, Scalable and Electrically-tunable Majorana Platform
• Topology invisible to eigenvalues in obstructed atomic insulators
• Time-reversal-invariant $C_2$-symmetric higher-order topological superconductors
• Bosonic Crystalline Symmetry Protected Topological Phases Beyond the Group Cohomology Proposal
• Lecture Notes on Topological Crystalline Insulators
• Variants of the symmetry-based indicator
• Topology in time-reversal symmetric crystals
• Effective field theories of topological crystalline insulators and topological crystals
• 4D beyond-cohomology topological phase protected by $C_2$ symmetry and its boundary theories
• Higher-Order Topological Superconductivity in Twisted Bilayer Graphene
• Takagi topological insulator with odd $\mathcal{P}\mathcal{T}$ pairs of corner states
• Classification of crystalline topological insulators through K-theory
• Homotopy invariant in time-reversal and twofold rotation symmetric systems

Cited by in corpus (7)

• Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups
• Intrinsic first and higher-order topological superconductivity in a doped topological insulator
• Interacting topological quantum chemistry of Mott atomic limits
• Higher-order topological superconductivity in a topological metal 1T${}^{\prime }$-MoTe${}_2$
• Classification of time-reversal symmetric topological superconducting phases for conventional pairing symmetries
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• Characterization of higher-order topological superconductors using Bott indices