SciPost Phys. 12, 202 (2022) ·
published 27 June 2022

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We study the gauging of a global U(1) symmetry in a gapped system in (2+1)d.
The gauging procedure has been wellunderstood for a finite global symmetry
group, which leads to a new gapped phase with emergent gauge structure and can
be described algebraically using the mathematical framework of modular tensor
category (MTC). We develop a categorical description of U(1) gauging in an MTC,
taking into account the dynamics of U(1) gauge field absent in the finite group
case. When the ungauged system has a nonzero Hall conductance, the gauged
theory remains gapped and we determine the complete set of anyon data for the
gauged theory. On the other hand, when the Hall conductance vanishes, we argue
that gauging has the same effect of condensing a special Abelian anyon
nucleated by inserting $2\pi$ U(1) flux. We apply our procedure to the
SU(2)$_k$ MTCs and derive the full MTC data for the $\mathbb{Z}_k$ parafermion
MTCs. We also discuss a dual U(1) symmetry that emerges after the original U(1)
symmetry of an MTC is gauged.
SciPost Phys. 12, 052 (2022) ·
published 3 February 2022

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We investigate a family of invertible phases of matter with higherdimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev's chain in 1+1d.
The excitation has $\mathbb{Z}_2$ higherform symmetry that mixes with the spacetime Lorentz symmetry to form a higher group spacetime symmetry.
We focus on the invertible exotic loop topological phase in 3+1d. This invertible phase is protected by the $\mathbb{Z}_2$ oneform symmetry and the timereversal symmetry, and has surface thermal Hall conductance not realized in conventional timereversal symmetric ordinary bosonic systems without local fermion particles and the exotic loops. We describe a UV realization of the invertible exotic loop topological order using the $SO(3)_$ gauge theory with unit discrete theta parameter, which enjoys the same spacetime twogroup symmetry. We discuss several applications including the analogue of ``fermionization'' for ordinary bosonic theories with $\mathbb{Z}_2$ nonanomalous internal higherform symmetry and timereversal symmetry.
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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