One dimensional $(1d)$ interacting systems with local Hamiltonians can be studied with various well-developed analytical methods. Recently novel $1d$ physics was found numerically in systems with either spatially nonlocal interactions, or at the $1d$ boundary of $2d$ quantum critical points, and the critical fluctuation in the bulk also yields effective nonlocal interactions at the boundary. This work studies the edge states at the $1d$ boundary of $2d$ strongly interacting symmetry protected topological (SPT) states, when the bulk is driven to a disorder-order phase transition. We will take the $2d$ Affleck-Kennedy-Lieb-Tasaki (AKLT) state as an example, which is a SPT state protected by the $SO(3)$ spin symmetry and spatial translation. We found that the original $(1+1)d$ boundary conformal field theory of the AKLT state is unstable due to coupling to the boundary avatar of the bulk quantum critical fluctuations. When the bulk is fixed at the quantum critical point, within the accuracy of our expansion method, we find that by tuning one parameter at the boundary, there is a generic direct transition between the long range antiferromagnetic N\'{e}el order and the valence bond solid (VBS) order. This transition is very similar to the N\'{e}el-VBS transition recently found in numerical simulation of a spin-1/2 chain with nonlocal spatial interactions. Connections between our analytical studies and recent numerical results concerning the edge states of the $2d$ AKLT-like state at a bulk quantum phase transition will also be discussed.

中文翻译：

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具有SO（3）对称性的非局部一维系统中的连续N \'{e} el-VBS量子相变

可以使用各种完善的分析方法来研究与当地哈密顿人的一维$（1d）$相互作用系统。最近，在具有空间非局部相互作用的系统中，或者在具有2d $个量子临界点的1d $边界的系统中，在数值上发现了新颖的1d $物理场，并且本体的临界波动还在边界处产生了有效的非局部相互作用。这项工作研究了当主体驱动到无序有序相变时，在具有强相互作用的对称保护拓扑（SPT）状态的$ 2d $边界处的边缘状态。我们以$ 2d $ Affleck-Kennedy-Lieb-Tasaki（AKLT）状态为例，这是受$ SO（3）$自旋对称性和空间平移保护的SPT状态。我们发现，AKLT状态的原始$（1 + 1）d $边界共形场理论由于与体量子临界涨落的边界化身耦合而不稳定。当将主体固定在量子临界点时，在我们扩展方法的精度范围内，我们发现通过在边界处调整一个参数，可以在长距离反铁磁N \'{e} el阶与价键固体（VBS）顺序。这种转变与最近在具有非局部空间相互作用的自旋1/2链的数值模拟中发现的N \'{e} el-VBS转变非常相似。我们还将讨论我们的分析研究与最近的数值结果之间的联系，该数值结果涉及在体量子相变时类似$ 2d $ AKLT态的边缘态。