In gauge theory, it is commonly stated that time-reversal symmetry only exists at $θ=0$ or $π$ for a $2π$-periodic $θ$-angle. In this paper, we point out that in both the free Maxwell theory and massive QED, there is a non-invertible time-reversal symmetry at every rational $θ$-angle, i.e., $θ= πp/N$. The non-invertible time-reversal symmetry is implemented by a conserved, anti-linear operator without an inverse. It is a composition of the naive time-reversal transformation and a fractional quantum Hall state. We also find similar non-invertible time-reversal symmetries in non-Abelian gauge theories, including the $\mathcal{N}=4$ $SU(2)$ super Yang-Mills theory along the locus $|τ|=1$ on the conformal manifold.
We investigate a family of invertible phases of matter with higher-dimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev's chain in 1+1d. The excitation has $\mathbb{Z}_2$ higher-form 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 $\mathbb{Z}_2$ one-form symmetry and the time-reversal symmetry, and has surface thermal Hall conductance not realized in conventional time-reversal symmetric o…
We study non-invertible global symmetries in $(3+1)$-dimensional axion electrodynamics with a massless axion and a massless photon. In addition to a previously known non-invertible 0-form shift symmetry of the axion, we find a non-invertible 1-form symmetry associated with the equation of motion for the photon. Correlation functions of non-invertible symmetry defects lead to invertible 1- and 2-form symmetry defects associated with Bianchi identities for the axion and photon. In terms of the correlation functions, we discuss several phenomena for extended objects, such as induced fractional el…
In this paper, using 1+1D models as examples, we study symmetries and anomalous symmetries via multi-component partition functions obtained through symmetry twists, and their transformations under the mapping class group of spacetime. This point of view allows us to treat symmetries and anomalous symmetries as non-invertible gravitational anomalies (which are also described by multi-component partition functions, transforming covariantly under the mapping group transformations). This allows us to directly see how symmetry and anomalous symmetry constraint the low energy dynamics of the systems…
Spontaneous breaking of non-invertible symmetries and duality to beyond-Landau transitions
Institute for Quantum Information and Matter, California Institute of Technology（加州理工学院量子信息物质研究所）；Department of Physics, California Institute of Technology（加州理工学院物理系）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；Department of Physics, Harvard University（哈佛大学物理系）；Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder（科罗拉多大学博尔德分校物理系与量子物质理论中心）；C.N. Yang Institute for Theoretical Physics, Stony Brook University（纽约州立大学石溪分校杨振宁理论物理研究所）
We elucidate the fate of classical symmetries which suffer from abelian Adler-Bell-Jackiw anomalies. Instead of being completely destroyed, these symmetries survive as non-invertible topological global symmetry defects with worldvolume anyon degrees of freedom that couple to the bulk through a magnetic one-form global symmetry as in the fractional hall effect. These non-invertible chiral symmetries imply selection rules on correlation functions and arise in familiar models of massless quantum electrodynamics and models of axions (as well as their non-abelian generalizations). When the associat…
We identify infinitely many non-invertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the $U(1)_\text{A}$ axial symmetry because of the ABJ anomaly, for every rational angle $2πp/N$, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic $U(1)$ gauge field. These conserved symmetry operators do not obey a group multiplication law, but a non-invertible fus…
It is known that the 't Hooft anomalies of invertible global symmetries can be characterized by an invertible TQFT in one higher dimension. The analogous statement remains to be understood for non-invertible symmetries. In this note we discuss how the linking invariants in a non-invertible TQFT known as the Symmetry TFT (SymTFT) can be used as a diagnostic for 't Hooft anomalies of non-invertible symmetries. When the non-invertible symmetry is non-intrinsically non-invertible, and hence the SymTFT is a Dijkgraaf-Witten model, the linking invariants can be computed explicitly. We illustrate thi…
Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Department of Physics and Astronomy, Rice University（莱斯大学物理与天文系）；Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Racah Institute of Physics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学拉卡物理研究所）；Rice Center for Quantum Materials and Advanced Materials Institute, Rice University（莱斯大学量子材料中心及先进材料研究所）；Division of Condensed Matter Physics and Materials Science, Brookhaven National Laboratory（布鲁克海文国家实验室凝聚态物理与材料科学部）；Department of Engineering and Physics, Karlstad University（卡尔斯特德大学工程与物理系）；Nordita, KTH Royal Institute of Technology and Stockholm University（诺尔迪塔，皇家理工学院与斯德哥尔摩大学）
