Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness

Understand

Two global symmetries are holo-equivalent if their algebras of local symmetric operators are isomorphic.

• Holo-equivalent classes of global symmetries are classified by gappable-boundary topological orders (TO) in one higher dimension (called symmetry TO), which leads to a symmetry/topological-order (Symm/TO) correspondence.
• We establish that: (1) For systems with a symmetry described by symmetry TO $M$, their gapped and gapless states are classified by condensable algebras $A$, formed by elementary excitations in $M$ with trivial self/mutual statistics.
• Such classified states (called $A$-states) can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc., in a unified way.