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

Using quantum Monte Carlo simulations, we study spin-1/2 diagonal ladders coupled by ferromagnetic Heisenberg interactions. The model can also be viewed as usual ladders with ferromagnetic rung couplings coupled by antiferromagnetic diagonal couplings. We show that the model hosts a striped magnetic ordered phase and two topological nontrivial Haldane phases separated by two quantum critical points. We also show that the two quantum critical points are all in the three-dimensional O(3) universality class irrelevant to the topological properties of the Haldane phases. The properties of the surface formed by ladder ends in the two Haldane phases are studied, showing that the surface states are both gapless due to the symmetry-protected topological bulk states. We further demonstrate that extraordinary surface critical behaviors occur at both critical points on such gapless surfaces without enhancing the surface coupling. Notably, the surface is not expected to be ordered in the three-dimensional classical O(3) critical point, suggesting that the topological properties of the Haldane phases are responsible for this surface critical behavior. © 2023 American Physical Society.