Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) $S^{\,}_3$ symmetry and the other with non-invertible $\mathsf{Rep}(S^{\,}_3)$ symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1D symmetry-topological-order (SymTO) of type $\mathrm{JK}^{\,}_4\boxtimes \overline{\mathrm{JK}}^{\,}_4$. The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual $S^{\,}_3$-symmetric and $\mathsf{Rep}(S^{\,}_3)$-symmetric models.
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）
Non-Invertible Symmetries and Boundaries for Two-Dimensional Fermions
INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Abdus Salam Centre for Theoretical Physics, Imperial College London（伦敦帝国理工学院阿卜杜斯·萨拉姆理论物理中心）；SISSA（的里雅斯特国际高等研究学院）
Non-invertible symmetries and mixed anomalies from equations of motion in (3+1)D twisted $BF$ topological quantum field theories
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；School of Physics, Sun Yat-sen University（中山大学物理学院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Additivity, Haag duality, and non-invertible symmetries
Center for Theoretical Physics - a Leinweber Institute, Massachusetts Institute of Technology（麻省理工学院莱因韦伯理论物理中心）；Yang Institute for Theoretical Physics, Stony Brook University（石溪大学杨理论物理研究所）
Non-invertible Symmetries in Weyl Fermions, and Applications to Fermion-Boundary Scattering Problem
Department of Physics, Nanjing University（南京大学物理系）；School of Quantum and Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学量子学院和理论科学研究所）
Generalized Wigner theorem for non-invertible symmetries
Indiana University（印第安纳大学）；Institute for Advanced Study, Princeton（普林斯顿高等研究院）；Rice University（莱斯大学）；Washington University in St. Louis（华盛顿大学圣路易斯分校）；Brookhaven National Laboratory（布鲁克海文国家实验室）；Technische Universität Chemnitz（开姆尼茨工业大学）；Indian Institute of Technology Madras（印度马德拉斯理工学院）
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）
