Theory of local ${\mathbb{Z}}_2$ topological markers for finite and periodic two-dimensional systems

Abstract

The topological phases of two-dimensional time-reversal symmetric insulators are classified by a ${\mathbb{Z}}_2$ topological invariant. Usually, the invariant is introduced and calculated by exploiting the way time-reversal symmetry acts in reciprocal space, hence implicitly assuming periodicity and homogeneity. Here, we introduce two space-resolved ${\mathbb{Z}}_2$ topological markers that are able to probe the local topology of the ground-state electronic structure also in the case of inhomogeneous and finite systems. The first approach leads to a generalized local spin-Chern marker, that usually remains well-defined also when the perpendicular component of the spin, $S_z$, is not conserved. The second marker is solely based on time-reversal symmetry, hence being more general. We validate our markers on the Kane-Mele model both in periodic and open boundary conditions, also in presence of disorder and including topological/trivial heterojunctions.

13 pages, 7 figures

References in corpus (30)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Observation of phononic helical edge states in a mechanical 'topological insulator'
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Computing topological invariants without inversion symmetry
• Wannier representation of Z_2 topological insulators
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Exponential localization of Wannier functions in insulators
• Robustness of the Spin-Chern number
• The insulator/Chern-insulator transition in the Haldane model
• Topological Weaire-Thorpe models of amorphous matter
• Toward Realistic Amorphous Topological Insulators
• Spin-Resolved Topology and Partial Axion Angles in Three-Dimensional Insulators
• Universal topological marker
• Amorphous topological matter: theory and experiment
• Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter
• Local markers for crystalline topology
• Kibble-Zurek behavior in disordered Chern insulators
• Orbital magnetization and Chern number in a supercell framework: Single k-point formula
• Quantised Bulk Conductivity as a Local Chern Marker
• Probing Chern number by opacity and topological phase transition by a nonlocal Chern marker
• Phase transitions and scale invariance in topological Anderson insulators
• Local Chern Marker for Periodic Systems
• Single-point spin Chern number in a supercell framework
• Optical absorption measurement of spin Berry curvature and spin Chern marker
• Structural spillage: an efficient method to identify non-crystalline topological materials
• Replica Higher-Order Topology of Hofstadter Butterflies in Twisted Bilayer Graphene
• Local spin Hall conductivity
• Mirror Chern numbers in the hybrid Wannier representation

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