Surface phase transitions in a (1+1)-dimensional $SU(2)_1$ conformal field theory boundary coupled to a (2+1)-dimensional $Z_2$ bulk

TL;DR

This paper investigates surface phase transitions in a (2+1)D quantum spin model, revealing novel extraordinary surface critical behaviors at the bulk critical point, including a transition involving an $SU(2)_1$ CFT boundary coupled to a $Z_2$ bulk.

Contribution

It uncovers a new type of surface phase transition involving an $SU(2)_1$ conformal boundary coupled to a $Z_2$ bulk, challenging previous theoretical expectations.

Findings

Identified different surface critical behaviors, including ordinary and extraordinary SCBs.

Discovered a novel surface phase transition not fitting previous models.

Showed the transition involves an $SU(2)_1$ CFT boundary, not a $Z_2$ or KT transition.

Abstract

We design a (2+1))-dimensional [(2+1)D] quantum spin model in which spin-1/2 ladders are coupled through antiferromagnetic Ising interactions. The model hosts a quantum phase transition in the (2+1)D $Z_2$ universality class from the Haldane phase to the antiferromagnetic Ising ordered phase. We focus on studying the surface properties of three different surface configurations when the Ising couplings are tuned. Different behaviors are found on different surfaces. We find ordinary and two different extraordinary surface critical behaviors (SCBs) at the bulk critical point. The ordinary SCBs belong to the surface universality class of the classical 3D Ising bulk transition. One extraordinary SCBs is induced by the topological properties of the Haldane phase. Another extraordinary SCBs at the bulk critical point is induced by an unconventional surface phase transition where the surface develops an Ising order before the bulk. This surface transition is realized by coupling a (1+1)-dimensional [(1+1)D] $SU(2)_1$ CFT boundary to a (2+1)D bulk with $Z_2$ symmetry. We find that the transition is neither a (1+1)D $Z_2$ transition, expected based on symmetry consideration, nor a Kosterlitz-Thouless-like transition, violating the previous theoretical prediction. This new surface phase transition and related extraordinary SCBs deserve further analytical and numerical exploration.