Abstract
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 Néel order and the valence bond solid (VBS) order. This transition is very similar to the Néel-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.
Highlights
One dimensional (1d) interacting systems with local Hamiltonians can be studied with various well-developed analytical methods
The interplay between the topological edge state and gapless quantum critical modes can lead to very nontrivial physics, which has been studied through numerical methods recently [4,5,6,7]
In this work we investigate the 2d symmetry protected topological (SPT) state protected by symmetry SO(3) × G, where SO(3) is the ordinary spin symmetry, while G is a discrete symmetry, which could be an onsite unitary Z2 symmetry, or an anti-unitary time-reversal Z2T
Summary
One dimensional (1d) interacting systems with local Hamiltonians can be studied with various well-developed analytical methods. Our main finding is that there is a generic continuous quantum phase transition between a long range antiferromagnetic Néel order which spontaneously breaks the SO(3) spin symmetry, and a valence bond solid state, at the 1d boundary of an AKLT state that couples to the bulk quantum critical modes. The bulk quantum critical modes effectively yield nonlocal interactions at the 1d boundary, which makes the long range Néel order possible.
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