Surface critical behaviors of coupled Haldane chains

TL;DR

This study investigates how the surface critical behaviors differ in a 2D system of coupled Haldane chains at quantum criticality, revealing that surface orientation influences whether the transition is ordinary or nonordinary due to SPT phase effects.

Contribution

It demonstrates that the gapless surface states in SPT phases can induce nonordinary surface critical behaviors at quantum critical points, a phenomenon previously observed mainly in dimerized models.

Findings

Different surface orientations lead to distinct critical behaviors.

Surface along the chain direction shows ordinary critical behavior.

Perpendicular surfaces exhibit nonordinary, special transition behavior.

Abstract

The special surface transition at (2+1)-dimensional quantum critical point is precluded in corresponding classical critical point. The mechanism of such behavior which is only found in dimerized Heisenberg models so far is still under debate. To illuminate the role of symmetry protected topological (SPT) phase in inducing such nonordinary behaviors, we study a system on a two-dimensional square lattice consisted by interacting spin-1 Haldane chains, which has a genuine SPT phase--the Haldane phase--at weak interchain interactions and a quantum critical point belonging to the classical 3D O(3) universality class to the N\'eel phase. Different from models studied previously, there is no dimerization in the current model. Cutting the system along the chain direction or perpendicular to the chain direction exposes two different surfaces. Using unbiased quantum Monte Carlo simulations, we find that the two different types of surface show completely different surface critical behaviors at the bulk critical point, resulted from different surface states in the SPT phase. For the system with surfaces along the chain direction, the surface critical behavior is of ordinary type of the bulk 3D O(3) critical point, while for the surfaces perpendicular to the chain direction, the surface critical behavior is nonordinary, consistent with special transitions found in dimerized Heisenberg models. Our numerical results demonstrate that the gapless surface state in the gapped SPT phase together with the gapless mode of critical point is a pure quantum scenario that leads to the nonordinary transition.