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Using Quantum Monte Carlo simulations, we study spin-1/2 diagonal ladderscoupled by ferromagnetic Heisenberg interactions. The model can also be viewedas usual ladders with ferromagnetic rung couplings coupled by antiferromagneticdiagonal couplings. We find that the model hosts a striped magnetic orderedphase and two topological nontrivial Haldane phases, separated by two quantumcritical points. We show that the two quantum critical points are all in thethree-dimensional O(3) universality class irrelevant to the topologicalproperties of the Haldane phases. The properties of the surface formed byladder ends in the two Haldane phases are studied. We find that the surfacestates are both gapless due to the symmetry-protected topological bulk states.We further demonstrate that extraordinary surface critical behaviors arerealized at both critical points on such gapless surfaces without enhancing thesurface coupling. Notably, the surface is not expected to be ordered in thethree-dimensional classical O(3) critical point, suggesting that thetopological properties of the Haldane phases are responsible for such surfacecritical behavior.

Extraordinary surface critical behavior induced by symmetry-protected topological state of a two-dimensional quantum magnet