We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.
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）
Entropic order parameters and topological holography
Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Universidad Politécnica de Cartagena（卡塔赫纳理工大学）
SymTFT for Continuous Symmetries: Non-linear Realizations and Spontaneous Breaking
Universidad de Murcia（穆尔西亚大学）；Uppsala University（乌普萨拉大学）；Institut de Physique Théorique, Université Paris Saclay, CNRS, CEA（巴黎萨克雷大学理论物理研究所，法国国家科学研究中心，法国原子能和替代能源委员会）
