SciPost Phys. 11, 033 (2021) ·
published 18 August 2021

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We investigate the behavior of higherform symmetries at various quantum phase transitions. We consider discrete 1form symmetries, which can be either part of the generalized concept ``categorical symmetry" (labelled as $\tilde{Z}_N^{(1)}$) introduced recently, or an explicit $Z_N^{(1)}$ 1form symmetry. We demonstrate that for many quantum phase transitions involving a $Z_N^{(1)}$ or $\tilde{Z}_N^{(1)}$ symmetry, the following expectation value $ \langle \left( \log
O_\mathcal{C} \right)^2 \rangle$ takes the form $\langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim  \frac{A}{\epsilon} P+ b \log P $, where $O_\mathcal{C} $ is an operator defined associated with loop $\mathcal{C} $ (or its interior $\mathcal{A} $), which reduces to the Wilson loop operator for cases with an explicit $Z_N^{(1)}$ 1form symmetry. $P$ is the perimeter of $\mathcal{C} $, and the $b \log P$ term arises from the sharp corners of the loop $\mathcal{C} $, which is consistent with recent numerics on a particular example. $b$ is a universal microscopicindependent number, which in $(2+1)d$ is related to the universal conductivity at the quantum phase transition. $b$ can be computed exactly for certain transitions using the dualities between $(2+1)d$ conformal field theories
developed in recent years. We also compute the ``strange correlator" of $O_\mathcal{C} $: $S_{\mathcal{C} } = \langle 0  O_\mathcal{C}  1 \rangle / \langle 0  1 \rangle$ where $0\rangle$ and $1\rangle$ are manybody states with different topological nature.
SciPost Phys. 10, 033 (2021) ·
published 12 February 2021

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One dimensional $(1d)$ interacting systems with local Hamiltonians
can be studied with various welldeveloped 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 disorderorder phase transition. We
will take the $2d$ AffleckKennedyLiebTasaki (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}elVBS
transition recently found in numerical simulation of a spin1/2
chain with nonlocal spatial interactions. Connections between our
analytical studies and recent numerical results concerning the
edge states of the $2d$ AKLTlike state at a bulk quantum phase
transition will also be discussed.
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in Submissions  report on Continuous N\'{e}elVBS Quantum Phase Transition in NonLocal onedimensional systems with SO(3) Symmetry