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We construct a set of lattice models of non-interacting topologicalinsulators with chiral symmetry in three dimensions. We build a model of thetopological insulators in the class AIII by coupling lower dimensional modelsof $\mathbb{Z}$ classes. By coupling the two AIII models related bytime-reversal symmetry we construct other chiral symmetric topologicalinsulators that may also possess additional symmetries (the time-reversaland/or particle-hole).  There are two different chiral symmetry operators for the coupled model, thatcorrespond to two distinct ways of defining the sublattices. The integertopological invariant (the winding number) in case of weak coupling can beeither the sum or difference of indices of the basic building blocks, dependenton the preserved chiral symmetry operator. The value of the topological indexin case of weak coupling is determined by the chiral symmetry only and does notdepend on the presence of other symmetries. For $\mathbb{Z}$ topologicalclasses AIII, DIII, and CI with chiral symmetry are topologically equivalent,it implies that a smooth transition between the classes can be achieved if itconnects the topological sectors with the same winding number. We demonstratethis explicitly by proving that the gapless surface states remain robust in$\mathbb{Z}$ classes as long as the chiral symmetry is preserved, and thecoupling 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 aredistinct, and can not be adiabatically connected.

Elementary models of 3D topological insulators with chiral symmetry