Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter

Abstract

Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker -- the Chern number expressed in terms of the Fourier transformed Chern character -- is an easily applicable local marker in even dimensions, but there are no analogous expressions for odd dimensions. We provide general analytic expressions for local markers for free-fermion topological states in odd dimensions protected by local symmetries: a Chiral marker, a local $\mathbb{Z}$ marker which in case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local ${\mathbb{Z}}_2$ marker characterizing all nonchiral phases in odd dimensions. We achieve this by introducing a one-parameter family $P_{\vartheta }$ of single-particle density matrices interpolating between a trivial state and the state of interest. By interpreting the parameter $\vartheta$ as an additional dimension, we calculate the Chern marker for the family $P_{\vartheta }$. We demonstrate the practical use of these markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions: a topological superconductor ($\mathbb{Z}$ classification) and a topological insulator (${\mathbb{Z}}_2$ classification).

13 pages, 3 figures; V2: Added supplemental material

References in corpus (12)

• Topological Insulators with Inversion Symmetry
• Topological Field Theory of Time-Reversal Invariant Insulators
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Theory of orbital magnetoelectric response
• Robust Edge States in Amorphous Gyromagnetic Photonic Lattices
• Surface theorem for the Chern-Simons axion coupling
• Local Topological Markers in Odd Dimensions
• Amorphous topological phases protected by continuous rotation symmetry
• Amorphous Bi${}_2$Se${}_3$ structural, electronic, and topological nature by first-principles
• Estimating bulk and edge topological indices in finite open chiral chains
• Topological transitions and Anderson localization of light in disordered atomic arrays
• Quantized electromagnetic response of three-dimensional chiral topological insulators

Cited by in corpus (26)

• Universal topological marker
• Amorphous topological matter: theory and experiment
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Robustness of topological order against disorder
• Topological marker approach to an interacting Su-Schrieffer-Heeger model
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Topological superconductivity in Fibonacci quasicrystals
• 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
• Scaling of the Integrated Quantum Metric in Disordered Topological Phases
• Interacting Local Topological Markers: A one-particle density matrix approach for characterizing the topology of interacting and disordered states
• Fractal Surface States in Three-Dimensional Topological Quasicrystals
• Orbital Magnetization under an Electric Field and Orbital Magnetoelectric Polarizabilty for a Bilayer Chern System
• Floquet Amorphous Topological Orders in a One-dimensional Rydberg Glass
• Topological zero-modes of the spectral localizer of trivial metals
• Quantum Hall criticality in an amorphous Chern insulator
• Evaluation of real-space second Chern number using the kernel polynomial method
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Aharanov-Bohm oscillations and perfectly transmitted mode in amorphous topological insulator nanowires
• Three-Dimensional Quantum Hall Effect in Topological Amorphous Metals
• Symmetry and Topology of Monitored Quantum Dynamics
• Regularized universal topological markers for Dirac systems
• Topological transition induced by selective random defects on a honeycomb lattice
• Dimensional crossover of class D real-space topological invariants
• Coarse geometric approach to topological phases: Invariants from real-space representations
• Real-space chirality from crystalline topological defects in the Kitaev spin liquid