Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator

Abstract

In this paper, we formulate the real-space Chern number in a supercell framework. In this framework, the overlap matrix between two corners of the Brillouin zone (BZ) is derived from diagonalizing the real-space Hamiltonian with periodic boundary conditions. The path-ordered product of overlap matrices around the BZ boundary forms a Wilson loop, and defines the Chern number in real space. It is analytically shown that the real-space Chern number is quantized at integers for large enough systems and coincides with the Bott index used in the previous studies. The formulation is greatly simplified for the former so that it makes numerical computations more efficient. The real-space formula is used to numerically elucidate topological phases in a disordered Chern insulator. The Chern insulator is modeled by dimensional extension of the Rice-Mele model consisting of two sublattices, and is disordered by including a random onsite potential. As disorder strength increases, the nontrivial-to-trivial phase transition takes place for normal disorder with no sublattice polarization. By contrast, the phase diagram is almost unaffected by polarized disorder, indicating that nontrivial topology persists against disorder. These observations are supported by the linear conductance and the density of bulk states.

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References in corpus (37)

• Topological Insulators
• Topological insulators and superconductors
• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological quantum matter with symmetries
• Experimental realisation of the topological Haldane model
• Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
• Mapping topological order in coordinate space
• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Disordered Topological Insulators via $C^{\ast }$-Algebras
• Topological Insulators in Random Lattices
• Scattering formula for the topological quantum number of a disordered multi-mode wire
• Scattering theory of topological insulators and superconductors
• Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice
• Charge Deficiency, Charge Transport and Comparison of Dimensions
• K-Theory and Pseudospectra for Topological Insulators
• Topological Insulators and C^*-Algebras: Theory and Numerical Practice
• Theory of spin Bott index for quantum spin Hall states in non-periodic systems
• Non-Commutative Tools for Topological Insulators
• Equality of bulk and edge Hall conductance revisited
• Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect
• Quasicrystalline Chern Insulators
• Locality of the anomalous Hall conductivity
• The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors
• On the Bott index of unitary matrices on a finite torus
• Orbital magnetization and Chern number in a supercell framework: Single k-point formula
• Estimating bulk and edge topological indices in finite open chiral chains
• Local Chern Marker for Periodic Systems
• Effective vacua for Floquet topological phases: A numerical perspective on switch-function formalism
• Topological charge pumping in quasiperiodic systems characterized by Bott index
• Single-point spin Chern number in a supercell framework
• Structural spillage: an efficient method to identify non-crystalline topological materials
• Theory of local ${\mathbb{Z}}_2$ topological markers for finite and periodic two-dimensional systems
• Energy polarization and energy pumping in Rice-Mele chains
• Topological Phases in Fractals: Local Spin Chern Marker in the Sierpinski carpet Kane-Mele-Rashba Model
• Adiabatic charge transport in extended SSH models
• Fluctuations and Correlations of Local Topological Order Parameters in Disordered Two-dimensional Topological Insulators
• Real-space formulation of topology for disordered Rice-Mele chains without chiral symmetry