  For conventional topological phases, the boundary gapless modes are
determined by bulk topological invariants. Based on developing an analytic
method to solve higher-order boundary modes, we present $PT$-invariant $2$D
topological insulators and $3$D topological semimetals that go beyond this
bulk-boundary correspondence framework. With unchanged bulk topological
invariant, their first-order boundaries undergo transitions separating
different phases with second-order-boundary zero-modes. For the $2$D
topological insulator, the helical edge modes appear at the transition point
for two second-order topological insulator phases with diagonal and
off-diagonal corner zero-modes, respectively. Accordingly, for the $3$D
topological semimetal, the criticality corresponds to surface helical Fermi
arcs of a Dirac semimetal phase. Interestingly, we find that the $3$D system
generically belongs to a novel second-order nodal-line semimetal phase,
possessing gapped surfaces but a pair of diagonal or off-diagonal hinge Fermi
arcs.
