SciPost Phys. 15, 004 (2023) ·
published 10 July 2023

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Symmetry acting on a (2+1)$D$ topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)$D$ onsite symmetry. In this paper, we develop a (3+1)$D$ topological quantum field theory to calculate the anomaly indicators of a (2+1)$D$ topological order with a general symmetry group $G$, which may be discrete or continuous, Abelian or nonAbelian, contain antiunitary elements or not, and permute anyons or not. These anomaly indicators are partition functions of the (3+1)$D$ topological quantum field theory on a specific manifold equipped with some $G$bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Our framework is applied to derive the anomaly indicators for various symmetry groups, including $\mathbb{Z}_2\times\mathbb{Z}_2$, $\mathbb{Z}_2^T\times\mathbb{Z}_2^T$, $SO(N)$, $O(N)^T$, $SO(N)\times \mathbb{Z}_2^T$, etc, where $\mathbb{Z}_2$ and $\mathbb{Z}_2^T$ denote a unitary and antiunitary order2 group, respectively, and $O(N)^T$ denotes a symmetry group $O(N)$ such that elements in $O(N)$ with determinant $1$ are antiunitary. In particular, we demonstrate that some anomaly of $O(N)^T$ and $SO(N)\times \mathbb{Z}_2^T$ exhibit symmetryenforced gaplessness, i.e., they cannot be realized by any symmetryenriched topological order. As a byproduct, for $SO(N)$ symmetric topological orders, we derive their $SO(N)$ Hall conductance.
Weicheng Ye, Meng Guo, YinChen He, Chong Wang, Liujun Zou
SciPost Phys. 13, 066 (2022) ·
published 26 September 2022

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LiebSchultzMattis (LSM) theorems provide powerful constraints on the emergibility problem, i.e. whether a quantum phase or phase transition can emerge in a manybody system. We derive the topological partition functions that characterize the LSM constraints in spin systems with $G_s\times G_{int}$ symmetry, where $G_s$ is an arbitrary space group in one or two spatial dimensions, and $G_{int}$ is any internal symmetry whose projective representations are classified by $\mathbb{Z}_2^k$ with $k$ an integer. We then apply these results to study the emergibility of a class of exotic quantum critical states, including the wellknown deconfined quantum critical point (DQCP), $U(1)$ Dirac spin liquid (DSL), and the recently proposed nonLagrangian Stiefel liquid. These states can emerge as a consequence of the competition between a magnetic state and a nonmagnetic state. We identify all possible realizations of these states on systems with $SO(3)\times \mathbb{Z}_2^T$ internal symmetry and either $p6m$ or $p4m$ lattice symmetry. Many interesting examples are discovered, including a DQCP adjacent to a ferromagnet, stable DSLs on square and honeycomb lattices, and a class of quantum critical spinquadrupolar liquids of which the most relevant spinful fluctuations carry spin$2$. In particular, there is a realization of spinquadrupolar DSL that is beyond the usual parton construction. We further use our formalism to analyze the stability of these states under symmetrybreaking perturbations, such as spinorbit coupling. As a concrete example, we find that a DSL can be stable in a recently proposed candidate material, NaYbO$_2$.
Mr Ye: "We thank the referee for think..."
in Submissions  report on Anomaly of $(2+1)$Dimensional SymmetryEnriched Topological Order from $(3+1)$Dimensional Topological Quantum Field Theory