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 …

Quantum spin liquids may be considered'quantum disordered'ground states of spin systems,
in which zero-point fluctuations are so strong that they prevent conventional magnetic long …

We discuss invertible and non-invertible topological condensation defects arising from
gauging a discrete higher-form symmetry on a higher codimensional manifold in spacetime …

The discovery of the quantum spin Hall effect and topological insulators more than a decade
ago has revolutionized modern condensed matter physics. Today, the field of topological …

We examine the interplay of symmetry and topological order in 2+ 1-dimensional topological
quantum phases of matter. We present a precise definition of the topological symmetry …

A global symmetry (0-symmetry) in an n-dimensional space acts on the whole space. A
higher symmetry acts on closed submanifolds (ie, loops and membranes, etc.), and those …

The recent discovery of the Mott insulating and superconducting phases in twisted bilayer
graphene has generated tremendous research interest. Here, we develop a weak coupling …

In a periodically driven (Floquet) system, there is the possibility for new phases of matter, not
present in stationary systems, protected by discrete time-translation symmetry. This includes …

We consider topological phases in periodically driven (Floquet) systems exhibiting many-
body localization, protected by a symmetry G. We argue for a general correspondence …

We put the theory of interacting topological crystalline phases on a systematic footing. These
are topological phases protected by space-group symmetries. Our central tool is an …