Holographic theory for continuous phase transitions: Emergence and symmetry protection of gaplessness

Abstract

Two global symmetries are holoequivalent if their algebras of local symmetric operators are isomorphic. A holoequivalent class of global symmetries is described by a topological order (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 $\mathcal{M}$, their gapped and gapless states are classified by condensable algebras $\mathcal{A}$, formed by elementary excitations in $\mathcal{M}$ with trivial self-/mutual statistics. These so-called $\mathcal{A}$ states can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc. in a unified way. (2) The local low-energy properties of an $\mathcal{A}$ state can be calculated from its reduced symmetry TO ${\mathcal{M}}_{/\mathcal{A}}$, using holographic modular bootstrap (holoMB) which takes ${\mathcal{M}}_{/\mathcal{A}}$ as an input. Here ${\mathcal{M}}_{/\mathcal{A}}$ is obtained from $\mathcal{M}$ by condensing excitations in $\mathcal{A}$. Notably, an $\mathcal{A}$ state must be gapless if ${\mathcal{M}}_{/\mathcal{A}}$ is nontrivial. This provides a unified understanding of the emergence and symmetry protection of gaplessness that applies to symmetries that are anomalous, higher-form, and/or noninvertible. (3) The relations between condensable algebras constrain the structure of the global phase diagram. We find that, for $1+1\mathrm{D} {\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{\ensuremath{'}}$ symmetry with mixed anomaly, there is a stable continuous transition (deconfined quantum critical point) between the ${\mathbb{Z}}_{2}$-breaking-${\mathbb{Z}}_{2}^{\ensuremath{'}}$-symmetric phase and the ${\mathbb{Z}}_{2}$-symmetric-${\mathbb{Z}}_{2}^{\ensuremath{'}}$-breaking phase. The critical point is the same as a ${\mathbb{Z}}_{4}$ symmetry breaking critical point. (4) $1+1\mathrm{D}$ bosonic systems with ${S}_{3}$ symmetry have four gapped phases with unbroken symmetries ${S}_{3}, {\mathbb{Z}}_{3}, {\mathbb{Z}}_{2}$, and ${\mathbb{Z}}_{1}$. We find a duality between two transitions ${S}_{3}\ensuremath{\leftrightarrow}{\mathbb{Z}}_{1}$ and ${\mathbb{Z}}_{3}\ensuremath{\leftrightarrow}{\mathbb{Z}}_{2}$: they are either both first order or both (stably) continuous, and in the latter case, they are described by the same conformal field theory (CFT). (5) The gapped and gapless states for $1+1\mathrm{D}$ bosonic systems with anomalous ${S}_{3}$ symmetries are obtained as well. For example, anomalous ${S}_{3}^{(1)}$ and ${S}_{3}^{(2)}$ symmetries can have symmetry protected chiral gapless states with only symmetric irrelevant and marginal operators.