Symmetry Protected Topological Charge in Symmetry Broken Phase: Spin-Chern, Spin-Valley-Chern and Mirror-Chern Numbers

Abstract

The Chern number is a genuine topological number. On the other hand, a symmetry protected topological (SPT) charge is a topological number only when a symmetry exists. We propose a formula for the SPT charge as a derivative of the Chern number in terms of the Green function in such a way that it is valid and related to the associated Hall current even when the symmetry is broken. We estimate the amount of deviation from the quantized value as a function of the strength of the broken symmetry. We present two examples. First, we consider Dirac electrons with the spin-orbit coupling on honeycomb lattice, where the SPT charges are given by the spin-Chern, valley-Chern and spin-valley-Chern numbers. Though the spin-Chern charge is not quantized in the presence of the Rashba coupling, the deviation is estimated to be $10^{-7}$ in the case of silicene, a silicon cousin of graphene. Second, we analyze the effect of the mirror-symmetry breaking of the mirror-Chern number in a thin-film of topological crystalline insulator.

6 pages, 1 figure

References in corpus (14)

• Topological Field Theory of Time-Reversal Invariant Insulators
• Topological Crystalline Insulators
• Valley-Polarized Metals and Quantum Anomalous Hall Effect in Silicene
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Quantum Spin Hall Effect and Topologically Invariant Chern Numbers
• Photo-Induced Topological Phase Transition and a Single Dirac-Cone State in Silicene
• Coupling the valley degree of freedom to antiferromagnetic order
• Spin-filtered Edge States with an Electrically Tunable Gap in a Two-Dimensional Topological Crystalline Insulator
• Robustness of the Spin-Chern number
• Spin-Valleytronics in Silicene: Quantum-Spin-Quantum-Anomalous Hall Insulators and Single-Valley Semimetals
• Topological Kirchhoff Law and Bulk-Edge Correspondence for Valley-Chern and Spin-Valley-Chern Numbers
• Topological Phase Transition without Gap Closing
• Magnetic ordering phenomena of interacting quantum spin Hall models
• Marginality of bulk-edge correspondence for single-valley Hamiltonians

Cited by in corpus (17)

• Topologically-protected refraction of robust kink states in valley photonic crystals
• Monolayer Topological Insulators: Silicene, Germanene and Stanene
• Giant magnetoresistance and perfect spin filter in silicene, germanene, and stanene
• Canted Spin Texture and Quantum Spin Hall Effect in WTe2
• Electrically Tunable Conductance and Edge Modes in Topological Crystalline Insulator Thin Films: Tight-Binding Model Analysis
• Valleytronics on the Surface of Topological Crystalline Insulator: Elliptic Dichroism and Valley-Selective Optical Pumping
• Spin-Orientation Dependent Topological States in Two-Dimensional Antiferromagnetic NiTl${}_2$S${}_4$ Monolayers
• Topological phases in Iridium oxide superlattices: quantized anomalous charge or valley Hall insulators
• Zitterbewegung in monolayer silicene in a magnetic field
• Phase transitions and scale invariance in topological Anderson insulators
• Magnetic-Field Induced Semimetal in Topological Crystalline Insulator Thin Films
• Relation Between the Spin Hall Conductivity and the Spin Chern Number for Dirac-like Systems
• Two-dimensional topological superconducting phases emerged from d-wave superconductors in proximity to antiferromagnets
• Spin Currents of Charged Dirac Particles in Rotating Coordinates
• A Quantized Interband Topological Index in Two-Dimensional Systems
• Topological invariant for holographic Weyl-${\mathrm{Z}}_2$ semimetal
• Phase structure of topological insulators by lattice strong-coupling expansion