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

What are topological phases of matter? First, they are phases of matter at zero temperature.
Second, they have a nonzero energy gap for the excitations above the ground state. Third …

A bstract 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 …

We show that the standard 1+ 1 DZ 2× Z 2 cluster model has a noninvertible global
symmetry, described by the fusion category Rep (D 8). Therefore, the cluster state is not only …

Long-range entanglement—the backbone of topologically ordered states—cannot be
created in finite time using local unitary circuits, or, equivalently, adiabatic state preparation …

A fundamental distinction between many-body quantum states are those with short-and long-
range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits …

Symmetry-protected topological (SPT) phases are many-body quantum states that are
topologically nontrivial as long as the relevant symmetries are unbroken. In this work we …

We present a general approach for detecting when a fusion category symmetry is
anomalous, based on the existence of a special kind of Lagrangian algebra of the …

We study fourfold rotation-invariant gapped topological systems with time-reversal symmetry
in two and three dimensions (d= 2, 3). We show that in both cases nontrivial topology is …

We analyze lattice Hamiltonian systems whose global symmetries have't Hooft anomalies.
As is common in the study of anomalies, they are probed by coupling the system to classical …