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).

arXiv comment13 pages, 3 figures; V2: Added supplemental material

References in corpus (42)

• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Anyons in an exactly solved model and beyond
• Topological Insulators with Inversion Symmetry
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Topological Field Theory of Time-Reversal Invariant Insulators
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Periodic table for topological insulators and superconductors
• Magnetoelectric polarizability and axion electrodynamics in crystalline insulators
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Riemannian symmetric superspaces and their origin in random matrix theory
• Amorphous topological insulators constructed from random point sets
• Mapping topological order in coordinate space
• Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder
• Many-body localization characterized from a one-particle perspective
• Disordered Topological Insulators via $C^{\ast }$-Algebras
• Topological Insulators in Random Lattices
• Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice
• Disordered Topological Insulators: A Non-Commutative Geometry Perspective
• Theory of orbital magnetoelectric response
• Higher-order topological insulators in amorphous solids
• K-Theory and Pseudospectra for Topological Insulators
• Witten effect in a crystalline topological insulator
• Theory of spin Bott index for quantum spin Hall states in non-periodic systems
• Chern-Simons orbital magnetoelectric coupling in generic insulators
• Non-Commutative Tools for Topological Insulators
• Topological Amorphous Metals
• Topological Weaire-Thorpe models of amorphous matter
• Robust Edge States in Amorphous Gyromagnetic Photonic Lattices
• Amorphous topological superconductivity in a Shiba glass
• Surface theorem for the Chern-Simons axion coupling
• Topological phase transitions in glassy quantum matter
• Local Topological Markers in Odd Dimensions
• Geometric and nongeometric contributions to the surface anomalous Hall conductivity
• Amorphous topological phases protected by continuous rotation symmetry
• Topological Crystalline Phases in a Disordered Inversion-Symmetric Chain
• Topological and conventional phases of a three dimensional electron glass
• Amorphous Bi${}_2$Se${}_3$ structural, electronic, and topological nature by first-principles
• Microscopic characteristics and tomography scheme of the local Chern marker
• 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