In this lecture note, we give a basic introduction to the rapidly developing concepts of
generalized symmetries, from the perspectives of both high energy physics and condensed …

The most well-known mechanism for fermions to acquire a mass is the Nambu–Goldstone–
Anderson–Higgs mechanism, ie, after a spontaneous symmetry breaking, a bosonic field …

Symmetry-protected topological (SPT) states have boundary't Hooft anomalies that obstruct
the effective boundary theory realized in its own dimension with UV completion and with an …

Fracton phases exhibit striking behavior which appears to render them beyond the standard
topological quantum field theory (TQFT) paradigm for classifying gapped quantum matter …

The classification and construction of symmetry-protected topological (SPT) phases in
interacting boson and fermion systems have become a fascinating theoretical direction in …

The classification and lattice model construction of symmetry-protected topological (SPT)
phases in interacting fermion systems are very interesting but challenging. In this paper, we …

Abstract Topological Quantum Field Theories (TQFTs) pertinent to some emergent low
energy phenomena of condensed matter lattice models in 2+ 1 and 3+ 1 dimensions are …

Dynamical systems may host a number of remarkable symmetry-protected phases that are
qualitatively different from their static analogs. In this work, we consider the phase space of …

A bstract We develop a mathematical theory of symmetry protected trivial (SPT) orders and
anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different …

The Lieb-Schultz-Mattis (LSM) theorem and its extensions forbid trivial phases from arising
in certain quantum magnets. Constraining infrared behavior with the ultraviolet data …