One-dimensional non-interacting topological insulators with chiral symmetry

Abstract

We construct microscopical models of one-dimensional non-interacting topological insulators in all of the chiral universality classes. Specifically, we start with a deformation of the Su-Schrieffer-Heeger (SSH) model that breaks time-reversal symmetry, which is in the AIII class. We then couple this model to its time-reversal counterpart in order to build models in the classes BDI, CII, DIII and CI. We find that the $\mathbb{Z}$ topological index (the winding number) in individual chains is defined only up to a sign. This comes from noticing that changing the sign of the chiral symmetry operator changes the sign of the winding number. The freedom to choose the sign of the chiral symmetry operator on each chain independently allows us to construct two distinct possible chiral symmetry operators when the chains are weakly coupled -- in one case, the total winding number is given by the sum of the winding number of individual chains while in the second case, the difference is taken. We find that the chiral models that belong to $\mathbb{Z}$ classes, AIII, BDI and CII are topologically equivalent, so they can be adiabatically deformed into one another so long as the chiral symmetry is preserved. We study the properties of the edge states in the constructed models and prove that topologically protected edge states must all be localised on the same sublattice (on any given edge). We also discuss the role of particle-hole symmetry on the protection of edge states and explain how it manages to protect edge states in ${\mathbb{Z}}_2$ classes, where the integer invariant vanishes and chiral symmetry alone does not protect the edge states anymore. We discuss applications of our results to the case of an arbitrary number of coupled chains, construct possible chiral symmetry operators for the multiple chain case, and briefly discuss the generalisation to any odd number of dimensions.

arXiv comment21 page

References in corpus (42)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Non-Abelian Anyons and Topological Quantum Computation
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Unpaired Majorana fermions in quantum wires
• Topological Insulators in Three Dimensions
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• 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
• Quantum Spin Hall Effect
• A spin-orbit coupled Bose-Einstein condensate
• First observation of Spin-Momentum Helical Locking in Bi2Se3 and Bi2Te3, demonstration of Topological-Order at 300K and a realization of topological-transport-regime
• Periodic table for topological insulators and superconductors
• Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices
• First direct observation of Spin-textures in Topological Insulators : Spin-resolved ARPES as a probe of topological quantum spin Hall effect and Berry's phase
• Direct Measurement of the Zak phase in Topological Bloch Bands
• Spin-Orbit Coupled Degenerate Fermi Gases
• Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas
• Nonlocal edge state transport in the quantum spin Hall state
• Topological phases of fermions in one dimension
• The Quantum Spin Hall Effect: Theory and Experiment
• Riemannian symmetric superspaces and their origin in random matrix theory
• The effects of interactions on the topological classification of free fermion systems
• Topology of Andreev Bound States with Flat Dispersion
• Topological invariants for spin-orbit coupled superconductor nanowires
• Realistic Rashba and Dressehaus spin-orbit coupling for neutral atoms
• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• Topological Order Parameters for Interacting Topological Insulators
• Entanglement entropy and the Berry phase in solid states
• Topological phases: Classification of topological insulators and superconductors of non-interacting Fermions, and beyond
• Search for non-Abelian Majorana braiding statistics in superconductors
• Real-space imaging of a topological protected edge state with ultracold atoms in an amplitude-chirped optical lattice
• Quantum criticality of quasi one-dimensional topological Anderson insulators
• Topology vs. Anderson localization: non-perturbative solutions in one dimension
• Anderson localization and the topology of classifying spaces
• Topological invariant for generic 1D time reversal symmetric superconductors in class DIII
• Realizing and detecting a topological insulator in the AIII symmetry class
• Orbital embedding and topology of one-dimensional two-band insulators

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• Weakly interacting one-dimensional topological insulators: a bosonization approach
• Adiabatic charge transport in extended SSH models
• A non-Hermitian Su-Schrieffer-Heeger model with the energy levels of free parafermions
• Emergent topological phases in an extended Su-Schrieffer-Heeger model with Rashba spin-orbit interaction, higher order hopping and domain wall
• Revealing inverted chirality of hidden domain wall states in multiband systems without topological transition
• One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems