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While strong topological insulators (STI) have been experimentally realizedsoon after their theoretical predictions, a weak topological insulator (WTI)has yet to be unambiguously confirmed. A major obstacle is the lack of distinctnatural cleavage surfaces to test the surface selective hallmark of WTI. With anew scheme, we discover that Bi$_{4}$X$_{4}$ (X=Br, I), stable or synthesizedbefore, can be WTI with two natural cleavage surfaces, where two anisotropicDirac cones stabilize and annihilate, respectively. We further find foursurface state Lifshitz transitions under charge doping and two bulk topologicalphase transitions under uniaxial strain. Near the WTI-STI transition, thereemerges a novel Weyl semimetal phase, in which the Fermi arcs genericallyappear at both cleavage surfaces whereas the Fermi circle only appears at oneselected surface.

Weak Topological Insulators and Composite Weyl Semimetals: $\beta$-Bi$_{4}$X$_{4}$ (X=Br, I)