A global symmetry (0-symmetry) in an n-dimensional space acts on the whole space.A higher symmetry acts on closed submanifolds (i.e., loops and membranes, etc.), and those transformations form a higher group.In this paper, we introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group.We show that an algebraic higher symmetry in a bosonic system in n-dimensional space is characterized and classified by a local fusion n-category.We find that, when restricted to its symmetric sub-Hilbert space, an algebraic higher symmetry can be fully characterized by a noninvertible gravitational anomaly (i.e., a topological order in one higher dimension).Thus, we also refer to a noninvertible gravitational anomaly as categorical symmetry to stress its connection to symmetry.This provides a holographic and entanglement view of symmetries.For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless.Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low-energy excitations, as long as they have the same categorical symmetry; and (4) the classification of gapped liquid phases for bosonic and/or fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry).This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.
For a zero-temperature Landau symmetry breaking transition in
n-dimensional space that completely breaks a finite symmetry
G, the critical point at the transition has the symmetry
G. In this paper, we show that the critical point also has a dual symmetry - a
(n−1)-symmetry described by a higher group when
G is Abelian or an algebraic
(n−1)-symmetry beyond higher group when
G is non-Abelian. In fact, any
G-symmetric system can be viewed as a boundary of
G-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk
G-gauge theory gives rise to the symmetry and the dual symmetry respectively. So any
G-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3+1D
Z2 gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a
Z2 0-symmetry and its dual - a
Z2 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a
Z2 1-symmetry and its dual - another
Z2 1-symmetry.
Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out that the algebra of local symmetric operators contains a special class of extended operators---transparent patch operators, which reveal the selection sectors and hence the corresponding symmetry. The algebra of those transparent patch operators in
n-dimensional space gives rise to a nondegenerate braided fusion
n-category, which happens to describe a topological order in one higher dimension (for finite symmetry). Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, noninvertible (higher) symmetries (also known as algebraic higher symmetries), and noninvertible gravitational anomalies. Thus, topological order in one higher dimension, replacing group, provides a unified and systematic description of the above generalized symmetries. This is referred to as symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: boundary uniquely determines the bulk, or more precisely, the algebra of local boundary operators uniquely determines the bulk topological order. As an application of the Symm/TO correspondence, we show the equivalence between ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry with mixed anomaly and
Z4 symmetry, as well as between many other symmetries, in 1-dimensional space.
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
M, their gapped and gapless states are classified by condensable algebras
A, formed by elementary excitations in
M with trivial self-/mutual statistics. These so-called
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
A state can be calculated from its reduced symmetry TO
M/A, using holographic modular bootstrap (holoMB) which takes
M/A as an input. Here
M/A is obtained from
M by condensing excitations in
A. Notably, an
A state must be gapless if
M/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
Z2-breaking-${\mathbb{Z}}_{2}^{\ensuremath{'}}$-symmetric phase and the
Z2-symmetric-${\mathbb{Z}}_{2}^{\ensuremath{'}}$-breaking phase. The critical point is the same as a
Z4 symmetry breaking critical point. (4)
1+1D bosonic systems with
S3 symmetry have four gapped phases with unbroken symmetries
S3,Z3,Z2, and
Z1. 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+1D bosonic systems with anomalous
S3 symmetries are obtained as well. For example, anomalous
S3(1) and
S3(2) symmetries can have symmetry protected chiral gapless states with only symmetric irrelevant and marginal operators.
Abstract In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and CFT-type gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category such that its monoidal center is trivial. We also study the categorical descriptions of the boundaries of these models. In the end, we obtain a classification of and the categorical descriptions of all 1-dimensional (spatial dimension) gapped quantum phases with a bosonic/fermionic finite onsite symmetry.
A bstract We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.