Gapless States with Generalized Symmetries and Large Conserved Quantities

We investigate the behavior of generalized symmetries in various gapless systems. To unambiguously characterize whether symmetry is spontaneously broken or not, we introduce the concept of "order diagnosis operator" (ODO). We first consider discrete 1-form symmetries at quantum phase transitions, which can be either part of "categorical symmetry" introduced recently or an explicit 1-form symmetry. We discuss the universal term in ODOs for Z_N 1-form symmetries. Then we move to systems with a large number of conserved quantities, such as fermi-surface states and systems with subsystem symmetries. The special features of ODOs in some examples will be mentioned.