# Elementary models of 3D topological insulators with chiral symmetry

## TL;DR

This paper constructs lattice models of 3D topological insulators with chiral symmetry, analyzing their topological invariants, surface states, and class transitions, revealing robustness and distinctions among different topological classes.

## Contribution

It introduces new lattice models for 3D chiral symmetric topological insulators and analyzes their topological invariants and class relationships.

## Key findings

- Weak coupling topological index depends only on chiral symmetry.
- Surface states remain robust as long as chiral symmetry is preserved.
- Class CII and AII are topologically distinct and cannot be connected adiabatically.

## Abstract

We construct a set of lattice models of non-interacting topological insulators with chiral symmetry in three dimensions. We build a model of the topological insulators in the class AIII by coupling lower dimensional models of $\mathbb{Z}$ classes. By coupling the two AIII models related by time-reversal symmetry we construct other chiral symmetric topological insulators that may also possess additional symmetries (the time-reversal and/or particle-hole). There are two different chiral symmetry operators for the coupled model, that correspond to two distinct ways of defining the sublattices. The integer topological invariant (the winding number) in case of weak coupling can be either the sum or difference of indices of the basic building blocks, dependent on the preserved chiral symmetry operator. The value of the topological index in case of weak coupling is determined by the chiral symmetry only and does not depend on the presence of other symmetries. For $\mathbb{Z}$ topological classes AIII, DIII, and CI with chiral symmetry are topologically equivalent, it implies that a smooth transition between the classes can be achieved if it connects the topological sectors with the same winding number. We demonstrate this explicitly by proving that the gapless surface states remain robust in $\mathbb{Z}$ classes as long as the chiral symmetry is preserved, and the coupling does not close the gap in the bulk. By studying the surface states in $\mathbb{Z}_2$ topological classes, we show that class CII and AII are distinct, and can not be adiabatically connected.