Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions

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 a 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 + 1-dimensional Z 2 gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a Z 2 0-symmetry and its dual, a Z 2 2-symmetry, while the critical point of the conﬁnement transition has a categorical symmetry formed by a Z 2 1-symmetry and its dual, another Z 2 1-symmetry.

Consider a Landau symmetry breaking transition [1,2] in a quantum system in d-dimensional space at zero temperature that completely breaks a finite on-site symmetry G.The critical point at the transition is a gapless state with G-symmetry.When d = 1, it is well known that the 1+1D gapless critical point has two decoupled sectors at low energies: right-movers and left-movers [3,4].Thus the critical point has a low energy emergent symmetry G × G.In this paper, we like to show that a similar symmetry "doubling" phenomenon also appears for critical points of Landau symmetry breaking phase transitions in all other dimensions.
Regardless dimensions, the quantum critical point always connects two phases: a symmetric phase with no ground state degeneracy and a symmetry breaking phase with |G| degenerate ground states.The symmetric phase has a G-symmetry, which is characterized by its pointlike excitations that form an irreducible representations of G.The collection of those point-like excitations, plus their trivial braiding and non-trivial fusion properties, give us a local fusion d-category denoted by Rep(G) [5][6][7].The conservation of the those point-like excitations is encoded by the non-trivial fusion of the irreducible representations R q 's which reflects the symmetry.In fact, due to Tannaka theorem, the local fusion d-category Rep(G) completely characterize the symmetry group G. Since the critical point touch the symmetric phase, the critical point also has the symmetry G.
The symmetry breaking phase has ground states labeled by the group elements g ∈ G, |Ψ g , that is not invariant under the symmetry transformation: U h |Ψ g = |Ψ hg .We can always consider a symmetrized ground state |Ψ 0 = g |Ψ g that is invariant under the symmetry transformation: U h |Ψ 0 = |Ψ 0 .The symmetry breaking phase has domain wall excitations that have a codimension 1. (We also only consider the symmetrized states with domain walls.)Thus domain walls can also fuse in a non-trivial way, which form a local fusion dcategory R w (see Ref. 7 and Appendix A).The nontrivial fusion also leads to a "conservation" of domainwall excitations.Ref. 7 conjectured that there is a generalization of Tannaka theorem: local fusion category R w completely characterizes an algebraic higher symmetry [7] (in the present case, an algebraic (d−1)-symmetry).When the symmetry group is Abelian, such an algebraic (d − 1)-symmetry is a (d − 1)-symmetry described by a higher group.We see that the symmetry breaking phase has such an algebraic (d−1)-symmetry.Since the critical point touches the symmetry breaking phase, the critical point also has the algebraic (d − 1)-symmetry.
To contrast higher symmetry (described by higher group) and algebraic higher symmetry (beyond higher group), we note that, for example, both group-like higher symmetry and algebraic higher symmetry can be generated by closed string operators W i (S 1 ) that commute with lattice Hamiltonian for all the closed loops S 1 .For group-like higher symmetry, the string operators satisfy a group-like algebra while for algebraic higher symmetry, the string operators satisfy a more general algebra 1 Higher form symmetry and higher symmetry are similar.They have only a small difference: The action of higher form symmetry becomes identity when acts on contractable closed subspaces, while the action of higher symmetry may not be identity when acts on contractable closed subspaces.Higher symmetry is a symmetry in a lattice model.Higher symmetry reduces to higher form symmetry in gapped ground state subspaces (i.e. in low energy effective topological quantum field theory).
One way to see the appearance of algebraic higher symmetry in any G-symmetric system is to note that when restricted to the symmetric sub Hilbert space, a d + 1D G-symmetric system can be viewed as a system that has a non-invertible gravitational anomaly, i.e. can be viewed as the boundary of G-gauge theory in one higher dimension [24,25].The conservation of the gauge charge and gauge flux in G-gauge theory in the bulk leads to the G 0-symmetry and algebraic (d − 1)-symmetry respectively, in our G-symmetric system that corresponds to the boundary.More precisely, the G-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry (from the conservation of gauge charge) and the algebraic (d − 1)symmetry (from the conservation of gauge flux), with non-trivial mutual statistics between gauge charge and gauge flux.Such a categorical symmetry is fully characterized by the G-gauge theory in one higher dimension.
Since the gapped boundaries of G-gauge theory always come from the condensation of the gauge charges and/or the gauge flux, the gapped ground state of G-symmetric system always breaks the categorical symmetry partially, either the G-symmetry, or the algebraic (d−1)-symmetry, or some other mixtures of the two symmetries.A state with the full categorical symmetry (i.e. both the Gsymmetry and the algebraic (d − 1)-symmetry) must be gapless.We show that such a gapless state describes the critical point of Landau symmetry breaking transition.
More generally, all possible gapless states in a Gsymmetric system are classified by gapless boundaries of G-gauge theory in one higher dimension.This universal emergence of categorical symmetry at critical point, and its origin from non-invertible gravitational anomaly (i.e.topological order in one higher dimension), may help us to systematically understand gapless states of matter.
In the following, we will study a few concrete examples, to show the appearance of categorical symmetries in Landau symmetry breaking transitions and in topological phase transitions.

A. Duality transformation in 1+1D Ising model
A common scenario happening at the critical point between symmetric phase and symmetry breaking phase is the emergent symmetry.The example we know the best is the Ising transition in one dimension.The paramagnetic phase is Z 2 symmetric, and the ferromagnetic phase spontaneously breaks the Z 2 symmetry.The critical point is Z 2 symmetric.More than that, it also has an additional Z 2 symmetry [26][27][28][29][30].
To see both Z 2 symmetries, we consider Ising model on a ring of L sites, where on each site i there are spin-up and spin-down state in the Pauli-Z i basis.So the Hilbert space is Each state in the Hilbert space can be also labeled in an alternative way, that is via the absence or presence of a domain wall (DW) in the dual lattice of L links.On each link i + 1 2 , a domain wall means the spins on i and i + 1 are anti-parallel.It follows that each basis state |ψ of V and its Z 2 partner |ψ ′ are labeled by the same kink variable on the dual lattice.So the Z 2 symmetric state in V is labeled uniquely by the DW variable.Moreover, a configuration of odd number of domain walls on links cannot be mapped to any configuration of spins on sites.Thus each DW variable with an even number of DW's labels a unique Z 2 symmetric state.Therefore, we say that the Z 2 symmetric Hilbert space of spins on the sites is in one-to-one correspondence with the Hilbert space of even number of DWs on the links.Each of the Hilbert space is of dimension 2 L−1 .
Next, we demonstrate an isomorphism between a set of operators on sites and that on links.
Here, we use the following notation, There is a unitary transformation performing the map, It is a linear depth operator that scales with the system size.Physically, the spin-flipping X i is the same as creating two DW's on i − 1 2 and i + 1 2 links, represented by X i− 1 2 X i+ 1 2 .The Ising coupling term Z i Z i+1 also measures the energy cost of a domain wall, which is represented by Z i+ 1  2 .Formally, the two sets of operators {X i , Z i Z i+1 } and { X i− 1 2 X i+ 1 2 , Z i+ 1 2 } are two ways (representations) to write the same set of operators {W i , W i+ 1  2 } defined by the operator algebra, for i We also have a further global condition, We have two 2 L−1 dimensional representations of W i 's, satisfying these relations (8) and (9).In particular, the following Hamiltonian, has the same spectrum independent of the representation.In the "spin representation", the Hamiltonian reduces to In the "DW representation", the Hamiltonian reduces to The unitary transformation (5) between the "spin representation" and "DW representation" is also known as Z 2 gauging.In the current case, it is also the same as Kramers-Wannier duality.
The Ising model has two exact Z 2 symmetries.However, in our spin and DW representations, we do not see them simultaneously.In the spin representation, we see one Z 2 symmetry generated by which is denoted as Z 2 .In the DW representation, we see the other Z 2 symmetry generated by We have shown that the Ising model always has the Z 2 × Z 2 categorical symmetry.It is interesting to note that, in its ground state, either Z 2 × Z 2 categorical symmetry is spontaneously broken partially (for example, one of the Z 2 is spontaneously broken) or the ground state is gapless [31].Since a symmetric gapped ground state is not allowed, one may wonder if the Z 2 × Z 2 categorical symmetry has a 't Hooft anomaly.In fact, we will show that this is actually an effect of non-invertible gravitational anomaly [25], or more precisely, the non-trivial "mutual statistics" between the Z 2 and Z 2 symmetries.
To see the non-invertible gravitational anomaly in the 1+1D Ising model, we concentrate on the so called symmetric sub Hilbert space V symm that is invariant under the U Z2 transformation.The space V symm of a L-site system does not have tensor product expansion of form Let us construct the boundary effective theory for the m condensed boundary of Z 2 topological order.Such a boundary contains a gapped excitation that corresponds to the e-type particle.One might expect a second boundary excitation corresponding to the f -type particle.However, since m is condensed on the boundary, the e-type particle and the f -type particle are actual equivalent on the boundary.The simplest boundary effective lattice Hamiltonian that describes the gapped e-type particles has a form (on a ring) Here a spin X i = 1 corresponds to an empty site and an spin X i = −1 corresponds to a site occupied with an e-type particle (with 2h as its energy gap).However, the boundary Hilbert space does not have a direct product decomposition since the number of the e-type particles on the boundary must be even (assume the bulk has no topological excitations).This is a reflection of non-invertible gravitational anomaly.A more general boundary effective theory may have a form where Z L+1 ≡ Z 1 and Z i Z i+1 creates a pair of the etype particles, or move an e-type particle from one site to another.Also, ǫ 0 is chosen to cancel the L-linear term in the ground state energy.As a model with the non-invertible gravitational anomaly characterized by 2+1D Z 2 topological order, its partition functions has four-components.For such mcondensed boundary (with |J| < h), the four-component partition functions are given by [25] Here where H Is A is the model eqn.( 19) with an "anti-periodic boundary condition": i.e. the e-type particle moving around the ring see a πflux.
From boundary effective theory, H Is P and H Is A , we can also obtain a gapless boundary by adjusting J = h [25, [32][33][34]: This corresponds to the critical point at the Z 2 symmetry breaking transition.When J > h, we obtain the second gapped boundary: This corresponds to the Z 2 symmetry breaking phase of the Ising model.We see that, indeed, the critical point of Ising transition, plus its two neighboring gapped states, can be described by a gapless edge, and its neighbors, of 2 + 1D Z 2 topological order.What is impressive is that requiring the categorical symmetry characerized by 2 + 1D Z 2 topological order allows us to determine a gapless state described by eqn.(25).This points to a direction that gapless states are determined by categorical symmetries, i.e. by topological order in one higher dimension.
The Ising model (19), when restricted to the symmetric sector V symm , can also be viewed as the low energy effective theory of the following model with h, J > 0 and U = +∞, where the Z 2 × Z 2 symmetry becomes explicit: The above Hamiltonian is conventionally known as describing Z 2 matter field coupled to Z 2 gauge field in 1+1D.The Z 2 × Z 2 symmetry (or more precisely, the Z 2 × Z 2 categorical symmetry) is explicit in the above model, which is generated by U Z2 and U Z2 as an on-site symmetry of the model.The U -terms tie a domain wall in Z i to U Z2 charge Z ı+ 1 2 , and a domain wall in X i to U Z2 charge X i .Even though the Z 2 × Z 2 symmetry is on site in the model (27), it is in some sense anomalous when restricted in the low energy sector.This is because in U = +∞ limit, the model cannot have gapped ground state that respects the full Z 2 × Z 2 symmetry.(Certainly, when U < J, h, the model can have a gapped ground state that respects the full Z 2 × Z 2 symmetry, such as when We note that the low energy sector of the model eqn.(27) describes the boundary of the 2+1D Z 2 gauge theory with Z 2 -charge e and Z 2 -vortex m, where e and m particle has low energies on the boundary.An e particle on the boundary corresponds to X i = −1 and a m particle corresponds to Z i+ 1 2 = −1 in eqn.(27).The Z 2 × Z 2 symmetry is the mod-2 conservation of e and m particles.This explains the Z 2 × Z 2 symmetry in the symmetric sector of the Ising model.Note that, on the boundary, we may have a e or m condensation.The condensations may spontaneously break the Z 2 × Z 2 symmetry in the ground state.However, the model itself always has Z 2 × Z 2 symmetry.
It is more precise to refer the Z 2 × Z 2 symmetry as Z 2 × Z 2 categorical symmetry.This is because the Z 2 and Z 2 symmetries are not independent.The Z 2 -charge (the e particle) and the Z 2 -charge (the m particle) have a π mutual statistics, when viewed as particles in one higher dimension.The term Z 2 × Z 2 categorical symmetry includes such non-trivial "mutual statistics" between the Z 2 and Z 2 symmetry.We see that the Z 2 × Z 2 categorical symmetry is characterized by Z 2 topological order in one higher dimension, that include the above non-trivial mutual statistics.
The mutual π-statistics between e and m in the 2+1D bulk is encoded at boundary by requiring the Z 2 domain wall to carry Z 2 charge and the Z 2 domain wall to carry Z 2 charge.This non-trivial mutual statistics has a highly non-trivial effect: in a gapped ground state, one and only one of Z 2 and Z 2 symmetries must be spontaneously broken [31].Thus, a symmetric state that does not break the Z 2 × Z 2 categorical symmetry must be gapless.This is a consequence of 1+1D non-invertible gravitational anomaly [25] characterized by 2+1D Z 2 topological order (i.e.Z 2 gauge theory).
In general, a theory with non-invertible gravitational anomaly has emergent symmetries (i.e. the categorical symmetry), which come from the conservation of topological excitations in one-higher-dimension bulk.Thus the categorical symmetry is fully characterized by the bulk topological order.Part of the categorical symmetry must be broken in any gapped phase, and the symmetric phase must be gapless.There is a gapless phase that respects the full categorical symmetry.

1-SYMMETRY IN 2+1D ISING MODEL
In two dimensions, it is well known that the critical point for Z 2 symmetry breaking transition has the Z 2 symmetry.In this section, we show that the critical point also has a 1-symmetry.
Let us consider the following two models: Z 2 Ising model and Z 2 gauge model on the square lattice.We will demonstrate that the Z 2 Ising model restricted to Z 2 even (chargeless) sector is exactly dual to the lattice Z 2 gauge model in the limit where the Z 2 vortex has infinity gap.
The Z 2 Ising model is given by where l sums over all links, i sums over all vertices, i 1 (l) and i 2 (l) are two vertices connected by the link l.
The lattice Z 2 gauge model is given by where i sums over all sites, s sums over all squares, l⊃i is a product over all the four links that contain vertex i, and l∈s is a product over all the four link on the boundary of square s.We consider the limit g = +∞.
The two models can be mapped into each other via the map that preserves the operator algebra We will refer such a map as "gauging".
The Ising model H has a global Z 2 symmetry generated by After gauging, it is mapped into an identity operator 1.The lattice Z 2 gauge model has a Z (1) 2 1-symmetry generated by the Wilson-line operators along any closed path It is mapped to identity operator in the ungauged Ising model.
Assume the space to be a torus with N = L × L vertices.The Hilbert space of the Ising model has a dimension 2 N .The subspace of Z 2 symmetric states, V symm Ising , has a dimension 2 N −1 .The Hilbert space of the Z 2 gauge model has a dimension 2 2N .In g = +∞ limit, the low energy subspace has a dimension 2 N ×2, The extra factor 2 is due to the operator identity s l∈s Z l = 1.In the low energy Hilbert space, we have l∈s Z l = 1 and if C can be deformed into C ′ .Now we consider the subspace, V symm gauge , of the low energy Hilbert space where (S 1 y ) = 1, where S 1 x and S 1 y are the two non-contractible loops wrapping around the system in x-and y-directions.V symm gauge has a dimension 2 Ising and H in V symm gauge are equivalent via an unitary transformation.In this sense, the Ising model eqn.(28) is exactly dual to the Z 2 gauge model eqn.(67).Now, let us assume J, U > 0. It is interesting to note that, for U ≫ J, the trivial phase of the Ising model is mapped to the topologically ordered phase (the Z (1) 2 1symmetry breaking phase) of the Z 2 gauge model, while for U ≪ J, the Z 2 symmetry breaking phase of the Ising model is mapped to the trivial phase (the symmetric phase of the Z symmetry (or more precisely, the Z 2 × Z (1) 2 categorical symmetry) is the mod-2 conservation of e particles and s strings.The mutual π statistics between e and s in the 3+1D bulk has a highly non-trivial effect on the boundary: in a gapped ground state of the 2+1D model (28), one and only one of the Z 2 and Z (1) 2 symmetries must be spontaneously broken.Thus, a ground state of model (28) with the full Z 2 × Z (1) 2 categorical symmetry must be gapless.This is a consequence of 2+1D non-invertible gravitational anomaly [25] and the Z 2 × Z (1) 2 categorical symmetry characterized by 3+1D Z 2 topological order (i.e.Z 2 gauge theory).

IV. THE MODEL WITH ANOMALOUS SYMMETRY AS THE BOUNDARY OF TOPOLOGICAL ORDER IN ONE HIGHER DIMENSION
In the above, we show that, through a few examples, a model with a finite symmetry G, when restricted in the symmetric sector, can be viewed as the boundary of G-gauge theory in one higher dimension.The conservation's (the fusion rules) of the point-like gauge charge, and codimension-2 gauge flux give rise to the symmetry and algebraic higher symmetry (whose combination becomes the so-called categorical symmetry) of the Gsymmetric model, The categorical symmetry is not spontaneously broken at the critical point of the symmetry breaking transition (see Section V for more details).
We know that a G-gauge theory can be twisted and becomes Dijkgraaf-Witten theory [35].We will show that boundary of such twisted G-gauge theory has an anomalous G-symmetry.This implies that a 1+1D system with anomalous G-symmetry also has an algebraic higher symmetry.The combination of the two symmetries corresponds to the categorical symmetry described by the twisted G-gauge theory in one higher dimension.Such a system has a gapless state, where the categorical symmetry is not spontaneously broken.Also, a state with the unbroken categorical symmetry must be gapless.And the gapped states of the system must spontaneously break the anomalous G-symmetry.

A. Boundary of double-semion model
In this section, we will study the boundary of doublesemion (DS) model (i.e. the twisted Z 2 gauge theory in 2+1D) to illustrate the above result.
A 2+1D double-semion (DS) topological order has four types of excitations 1, s, s * , b.Here s, s * , f are topological excitations s, s * are semions with statistics ± i, and b is a boson.They satisfy the following fusion relation s and s have a mutual π statistics and s and s * have a mutual boson statistics.As a result, s and b have a mutual π statistics.
We consider a gapped boundary from condensing b excitations.Since b ⊗ b = 1 and b particles have mod-2 conservation, we assume the b condensation gives rise to two degenerate ground states, one with Z i = 1 and the other with Z i = −1.The domain wall between Z i = 1 and Z i = −1 regions corresponds a s particle.
We like to point out that, on the boundary, although stype particle and e-type particle (in the Z 2 gauge theory discussed before) have the same fusion rule s ⊗ s = 1 and e ⊗ e = 1, their fusion F -tensor are different [36,37].In particular, fusing three s-type particles into one s-type particle in two different ways differ by a phase −1: In contrast, fusing three e-type particles (described by into one e-type particle in two different ways have the same phase: For the boundary of Z 2 topological order, the above two processes of fusing e particles are induced, respectively, by a pair-annihilation operator X + i X + i+1 and a hopping operator , where Indeed, we have The pair-annihilation operator Z + i Z + i+1 and hopping operator i+1 are allowed local operations, and we can use them to construct effective boundary Hamiltonian which describes the boundary of 2+1D Z 2 topological order. For the boundary of DS topological order, the two processes for fusing s particles eqn.(35) are also induced by a pair-annihilation operator and a hopping operator.Here we choose the hopping operator to be X The pair-annihilation or pair-creation operator is given by Z i−1 (X i + Z i−1 X i Z i+1 ), which creates or annihilates a pair of domain walls at i + 1 2 and i − 1 2 .For three s-type particles (the domain walls) at i − where |sss = | ↑ i−1 ↓ i ↑ i+1 ↓ i+2 .Now we can construct the boundary effective theory for the b condensed boundary of DS topological order.We note that such a boundary contains a gapped excitation that corresponds to the s-type particle.One might expect a second boundary excitation corresponding to the s * -type particle.However, since b is condensed on the boundary, the s-type particle and the s * -type particle are actual equivalent on the boundary.The simplest boundary effective lattice Hamiltonian that describes the gapped s particles has a form which has two degenerate ground states and the s particles correspond to domain walls.
Using the above allowed local operations ), we can construct a more general boundary effective theory where site-i and site-(i + L) are identified.Here ǫ 0 is chosen to cancel the L-linear term in the ground state energy.
We note that the above Hamiltonian is not invariant under the spin-flip transformation i X i .In fact, it is invariant under a non-on-site transformation [38]: where s ij acts on two spins as The transformation has a simple picture: it flips all the spins and include a (−) N ↑→↓ phase, where N ↑→↓ is the number of ↑→↓ domain wall.We see that the transformation is a Z 2 transformation (i.e.square to 1).From Appendix B, the Z 2 transformation has the following action, We find that eqn.( 42) is invariant under the Z 2 transformation.
From the above discussion, we see that the different fusion properties lead to different local operators.The boundary effective theories for Z 2 topological order and for the double semion topological order are different.In particular, the boundary effective theory for Z 2 topological order has an on-site Z 2 symmetry, while the boundary effective theory for the double semion topological order has a non-on-site Z 2 symmetry.The non-on-site Z 2 symmetry U Z2 implies that the model ( 42) cannot have a gapped Z 2 symmetric ground state [38].

B. Z2 dual symmetry
We have seen that a 1+1D lattice model (42) with an anomalous Z 2 symmetry (non-on-site symmetry [38,39]) can be viewed as a boundary of twisted 2+1D Z 2 gauge theory (i.e.DS topological order).The anomalous Z 2 symmetry comes from the mod-2 conserved b particles.The mod-2 conserved s particles will give rise to another symmetry, which will be referred as dual Z 2 symmetry.In other words, we claim that the model ( 42) has both the Z 2 symmetry and the Z 2 symmetry.
To see the Z 2 symmetry explicitly, we do a dual transformation on the model ( 42): We find The duality transformation changes the Hamiltonian (42) into: We see that the dual Z 2 symmetry is generated by This way, we obtain the explicit expression of the dual Z 2 symmetry.The on-site Z 2 symmetry U Z2 implies that the model ( 42) can have a gapped Z 2 symmetric ground state, which correspond to a Z 2 symmetry breaking state.
In the dual model, Z i+ 1 2 = 1 describes a site with no semion s, while Z i+ 1 2 = −1 describes a site occupied with a semion s.The term is the hopping term for the s particle, while the term ) creates a pair of s particles.

V. APPEARANCE OF ALGEBRAIC HIGHER SYMMETRY IN SYMMETRY BREAK TRANSITION FOR GENERAL FINITE SYMMETRY
After the above simple examples in 1+1D and 2+1D, in this section, we are going to describe the emergence of categorical symmetry at and off the critical point of Landau symmetry breaking transition for a general finite symmetry in any dimensions.

A. A duality point of view
We consider two lattice model defined on the triangulation of d-dimensional space.The vertices of the triangulation are labeled by i, the links labeled by ij, etc .
In the first model, the physical degrees of freedom lives on the vertices and are labeled by group elements g of a finite group G.The many-body Hilbert space the following local basis The Hamiltonian is given by where J(g) is a real function of g which is maximized when g = 1.Also the operator T h (i) is given by The Hamiltonian H 1 has an on-site G 0-symmetry We see that when J ≫ h, H 1 is in the symmetry breaking phase, and when J ≪ h, H 1 is in the symmetric phase.
Our second lattice bosonic model has degrees of freedom living on the links.On an oriented link ij, the degrees of freedom are labeled by g ij ∈ G. g ij 's on links with opposite orientations satisfy The many-body Hilbert space has the following local basis The Hamiltonian of the second model is given by where Q h (i) acts on all the links that connect to the vertex i: and The second model has an algebraic (d − 1)-symmetry [7] W q (S 1 )H = HW q (S 1 ), W q (S 1 ) = Tr for all loops S 1 formed by links, where R q is an irreducible representation of G.This is because Q h (i) can be viewed as a "gauge" transformation and the Wilson loop operator W q (S 1 ) is gauge invariant, and hence W q (S 1 ) commute with other terms in H 2 since they are all diagonal in the |{g ij } basis.The ground state of H 2 is a trivial product state |{g ij = 1} in the limit h ≪ J ≪ U .The ground state of H 2 is a topologically ordered state (described by the G-gauge theory) in the limit J ≪ h ≪ U .
When U → +∞ and when the space is a d-dimensional sphere S d , the low energy part of H 2 can be mapped to H 1 via the following map where i 0 is a fixed base point.We note that to map a configuration g i to a configuration g ij , we need to pick a base point i 0 and a value g i0 .Therefore, the above map is a |G|-to-one map.It maps the following H 2 have the same G-symmetric low energy dynamics.In particular they have the same phase transition and critical point.The G-symmetry breaking phase of H 1 corresponds to the trivial phase of H 2 (which is the symmetric phase of the algebraic (d − 1)-symmetry) and the G-symmetric phase of H 1 corresponds to the topologically ordered phase of H 2 (which is the symmetry breaking phase of the algebraic (d − 1)-symmetry) [7,9].Now we see that the critical point at the symmetry breaking transition touch a phase with G 0-symmetry and another phase with algebraic (d − 1)-symmetry.Thus the critical point has both the G 0-symmetry and the algebraic (d − 1)-symmetry.In other words, the critical point has a categorical symmetry which is the combination the G 0-symmetry and the algebraic (d − 1)-symmetry.The categorical symmetry is characterized by G-gauge theory in one higher dimension.

B. An example of Rep(S3) in 2+1D theories
The simplest example where the algebraic symmetry is beyond a higher symmetry is in the 2+1D S 3 lattice gauge theory (also called S 3 quantum double model).
Here, S 3 = s, r|s 3 = r 2 = 1, rsr = s 2 is the permutation group on 3 elements.The Hamiltonian is shown in (57) with G = S 3 .There are 8 types of anyonic excitations in the model.Their fusion rules are shown in Table I.
In the D(S 3 ) model, there is a Rep(S 3 ) algebraic 1symmetry.The generators are two Wilson lines labeled by the anyons a 1 and a 2 (see Table I).
The fusion of two W a 2 (S 1 )'s reveals that the symmetry is an algebraic 1-symmetry.
Let us examine the relation between S 3 lattice gauge model and a S 3 model with symmetry breaking in more details.A model with S 3 symmetry breaking can be realized by a Z 3 Ising model in 2 + 1D, which has three states labeled by g ∈ Z 3 = x|x 3 = 1 on each sites. where are the generalized Pauli matrices acting on the three states.The model in fact has S 3 global symmetry, generated by In particular, Γ r i exchange the state g and g −1 on the i th site.
It corresponds to S 3 gauge model by first gauging the Z 3 symmetry, and then the Z 2 symmetry.The first step is straightforward.We get the Z 3 gauge theory with the Hamiltonian where i sums over all sites, p sums over all plaquettes, l⊃i is a product over all the four links that contain vertex i, and l∈p is a product over all the four link on the boundary of plaquette p.We consider the limit g = +∞.Indeed, gauging Z 3 symmetry, we get the 3 2 anyons e a m b , a, b = 0, 1, 2, generated by the charge e and flux m, where e 3 = 1, m 3 = 1.The remaining Z 2 = U r S3 symmetry now acts on anyons as On the lattice, under Z 2 , the Wilson lines W q (S 1 ) and Q h (i) as defined in (60) and ( 58) are transformed into W −q (S 1 ) and Q h −1 (i) for q = 0, 1, 2 labeling the three irreducible representation of Z 3 and h ∈ Z 3 .Gauging this Z 2 symmetry, we get the anyons in D(S 3 ).First, the vacuum anyon 1 becomes vacuum anyon 1 = ([1], 1), a 1 = ([1], −), where "−" represents the sign representation of S 3 , as well as c 0 = ([r], 1), c 1 = ([r], −), two sectors with Z 2 flux.This is similar to the case when gauging the Z 2 symmetry in Z 2 Ising model, the only superselection sector is the vacuum sector, it becomes the four anyon sectors in the D(Z 2 ) model.Here, the sign representation of S 3 is the same as the sign representation of the quotient group Z 2 = S 3 /Z 3 .In summary, the D(Z 3 ) anyons and D(S 3 ) anyons are related via gauging the Z 2 in (68) as follows, We see that the model with S 3 symmetry breaking and S 3 lattice gauge theory can be related via the twostep gauging.The S 3 symmetry breaking model when restricted within S 3 symmetric Hilbert space is dual to the S 3 lattice gauge theory with infinite energy for the S 3 flux excitations.

C. A categorical point of view
Let us start with the lattice model H 1 (52) with a finite 0-symmetry G.If we restrict to the symmetric sub Hilbert space, the G-symmetry transformation (54) will be trivial.But we can see the G-symmetry via pointlike excitations which carry symmetry charge described by representations of G.The non-trivial fusion of Grepresentations give rise to the G 0-symmetry.
The lattice model H 1 (52), when restricted to the symmetric sub Hilbert space, can be viewed as a boundary of G-gauge theory in one higher dimension [25].The gauge charges in the G-gauge theory also carry representations of G.The non-trivial fusion of G-representations give rise to the G 0-symmetry both in the bulk and at the boundary.This is how the finite 0-symmetry G in model H 1 (52) is recovered via the G-gauge theory in one higher dimension.
But the G-gauge theory also has other excitations (such as the gauge flux -codimension-2 excitations), which also fuse in a non-trivial way and give rise to additional symmetry to the lattice model H 1 (52).So the complete symmetry of the lattice model H 1 (52) is given by the non-trivial fusion of all the excitations.Such a complete symmetry is the so-called categorical symmetry of the lattice model H 1 (52) (when restricted to the symmetric sub Hilbert space).Clearly, the categorical symmetry is fully characterized by the G-gauge theory in one higher dimension, and include the G 0-symmetry.This is the categorical understanding of the categorical symmetry.
Using a categorical language (for details see Ref. 7 and Appendix A), the braiding and fusion of the particles carrying G representation is described by fusion d-category Rep(G).Every fusion higher category can be mapped into a braided fusion higher category by Drinfeld center Z.The Drinfeld center of d-category Rep(G) gives rise to a braided fusion d-category D(G) = Z(Rep(G)), which is the G-gauge theory in one higher dimension.Therefore, for a system with symmetry described by fusion d-category Rep(G) (which is nothing but the G 0-symmetry), the categorical symmetry is given by the Drinfeld center of Rep(G).This is the categorical point of view of the categorical symmetry.
We stress that the lattice model H 1 (52) (when restricted to the symmetric sub Hilbert space) has the full categorical symmetry, but its ground states may spontaneously break part of the categorical symmetry.Those different ground states correspond to different boundaries of the G-gauge theory.Since a gapped boundary of Ggauge theory always comes from condensation of gauge charges, or gauge flux, or some combinations of them, a gapped boundary always spontaneously breaks some part of the categorical symmetry.Therefore, the gapped ground states of H 1 always spontaneously break some part of the categorical symmetry.Because the gauge charge and gauge flux have non-trivial mutual statistics between them, we cannot condense all gauge charges and gauge fluxes simultaneously.Therefore, any ground states of the lattice model H 1 (52) cannot break the categorical symmetry completely.
The G-gauge theory has a gapless boundary without the condensation of gauge charges and gauge flux.Such a boundary does not break the categorical symmetry.Thus the lattice model H 1 (52) has a gapless ground state where the categorical symmetry is not spontaneously broken.This gapless state should correspond to the critical point of Landau G-symmetry breaking transition.

VI. APPEARANCE OF CATEGORICAL SYMMETRY AT AND OFF CRITICAL POINT OF TOPOLOGICAL TRANSITIONS FOR ANY TOPOLOGICAL STATES A. A general categorical discussion
In the last section V C, we show that for a system with particle described by fusion d-category Rep(G), its categorical symmetry is characterized by Z(Rep(G)).The critical point for condensing the particles in Rep(G) has the full categorical symmetry Z(Rep(G)).
This result can be generalized.Consider a topological order described by a fusion category F .Assume its low energy excitations are described by a subcategory R which is also a local fusion category (see Ref. 7 and Appendix A).Ref. 7 points out that the local fusion category R characterizes an algebraic higher symmetry, which is a generalization of Rep(G).Since R describes all the low energy excitations, we can ignore other excitations and view the system as having an emergent algebraic higher symmetry characterized by R. Such a system also has a larger categorical symmetry characterized by Z(R).The critical point for condensing the particles in R has the full categorical symmetry Z(R).In the next section, we will demonstrate the above ideas in some lattice models.
B. An example: Higgs and confinement transition in 3+1D Z2 gauge theory Let us discuss a simple example of 3+1D Z 2 topological order to illustrate the above general results.In the first case, we choose the local energy subcategory R p to be the one formed by all the point-like excitations, i.e. the Z 2 charges.We assume all other excitations (such as gauge flux) to have infinite energy.In this case, we can focus only on excitations described by R p = Rep(Z 2 ), and our system can be viewed as a system with Z 2 0-symmetry.Our previous discussions on G-symmetric system will apply.In particular, we have an (emergent) categorical symmetry characterized by Z(R p ) = D(Z 2 ), which is a Z 2 -gauge theory in 4+1D.The categorical symmetry contains Z 2 0-symmetry from the fusion of the point-like Z 2 charges in the 4+1D Z 2 -gauge theory.The categorical symmetry also contains Z categorical symmetry.In fact, such a critical point is the same as the Z 2 symmetry breaking critical point discussed before.
In the second case, we choose the low energy subcategory R s to be the one formed by the pure string-like excitations, i.e. the Z 2 flux lines.We assume all other excitations (such as gauge charges) to have infinite energy.After ignore other excitations, the only excitations categorical symmetry.Thus categorical symmetries deepen our understanding of Higgs transition and confinement transition, as well as their relation.
We would like to thank Liang Kong, Tian Lan, Shuheng Shao, Hao Zheng for many helpful discussions.This research was partially supported by NSF DMS-1664412.This work was also partially supported by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440).
Appendix A: List of terminologies Here, we explain some mathematical terminologies used in this paper, at the level of physical rigorousness.An ordinary category has objects and morphisms (also called 1-morphisms).A 2-category generalizes this by also including 2-morphisms between the 1-morphisms.Continuing this up to n-morphisms between (n − 1)morphisms gives an n-category.
In condensed matter physics, an D-category corresponds to a collection of topological orders in Ddimensional spacetime.The topological orders in the collection correspond to the simple objects in the category.The composite objects correspond to degenerate states where several different topological orders happen to have the same energy.The gapped domain walls between different topological orders correspond to 1morphisms between different simple objects.The gapped domain walls within the same topological order correspond to 1-morphisms between the same simple object.1-morphisms can also have domain walls between them, which correspond to 2-morphisms, etc .In general, a mmorphism that connects the trivial m − 1 morphism to itself corresponds to a codimension-m excitation.A mmorphism that connects a non-trivial m − 1 morphism to itself corresponds to a domain wall on the codimensionm − 1 excitation.A m-morphism that connect two m − 1 morphisms corresponds to a domain wall between the two codimension-m − 1 excitations.A D-morphism corresponds to an "instanton" in spacetime (i.e. an insertion of a local operator in spacetime).A (D − 1)-morphism corresponds a point-like excitation.
Two topological orders can be stacked to form the third topological order.Under the stacking operation, topological orders form a monoid (which is similar to a group, but without inverse operation) [24].If we include the stacking operation in the D-category, the D-category will become a monoidal D-category.Thus a collection of topological orders in D-dimensional spacetime is actually described by a monoidal D-category.
A single topological order in D-dimensional spacetime is described by a D-category with only one simple object.We will call such a D-category as a fusion (D − 1)category.We drop the simple object in the D-category.The 1-morphisms in the D-category correspond to the objects in the fusion (D − 1)-category.The composition of the 1-morphisms in the D-category becomes the fusion of the objects in the fusion (D − 1) category.The 2-morphisms in the D-category correspond to the 1-morphisms in the fusion (D − 1)-category, etc .
An example of fusion 1-category is Vec.The objects in Vec are the vector spaces.The simple objects are the (equivalent classes of) 1-dimensional vector spaces and composite objects are multi-dimensional vector spaces.There is only one simple object.We see that composite objects are direct sums of simple objects.The morphisms (the 1-morphisms) correspond to the linear operators acting on the vector spaces.The tensor product of the vector spaces defines the fusion of the objects.We see that 1-dimensional vector space is the unit of the tensor product, and hence the simple object is the unit of the fusion.We also call such a fusion unit as the trivial object.The 1-morphisms that connect the simple object to itself is proportional to the one-by-one identity operator, and thus are also trivial.So we refer fusion 1-category Vec as trivial category.We also have trivial higher category, which has only one simple morphism at every level.We denote the trivial higher category as nVec.
Another We also need a notion of local fusion d-category: A fusion d-category F is local if we can add morphisms in a consistent way, such that all the resulting simple morphisms are isomorphic to the trivial one.Physically, the process of "adding morphisms" corresponds to explicit breaking of the (algebraic higher) symmetry.This is because, F only has morphisms that correspond to symmetric operators.Adding morphisms means include morphisms that correspond to symmetry breaking operators.If after breaking all the symmetry, F becomes a trivial product phase of bosons or fermions, then F is a local fusion d-category.forms X i in the following way: In other words ) of the Z 2 gauge model.At the gapless critical point of the 2+1D Z 2 symmetry breaking transition (also the Z breaking transition), we have theZ 2 × Z (1)2 symmetry which is not spontaneously broken.This way, we show the appearance of 1-symmetry in the 2+1D Z 2 symmetry breaking transition.Again the Z 2 symmetric sector of the 2+1D Ising model eqn.(28) can be viewed as the boundary of the 3+1D Z 2 gauge theory with Z 2 -charge e and Z 2 -vortex string s, where e particles and s strings has low energies at the boundary.An e particle at the boundary corresponds to X i = −1 in the Ising model eqn.(28), and a s string corresponds to Z l = −1 along a loop in the dual lattice in the Z 2 gauge model eqn.(67).The Z 2 × Z (1) 2 The point-like excitations and their fusion rule in 2+1D S3 topological order.Here b and c correspond to pure S3 flux excitations, a 1 and a 2 pure S3 charge excitations, 1 the trivial excitation, while b 1 , b 2 , and c 1 are charge-flux composites.
symmetry from the fusion of the membrane-like Z 2 flux in the 4+1D Z 2 -gauge theory.The condensation of the Z 2 charge induces a Higgs transition from the 3 + 1D Z 2 topological order to trivial order.The critical point of the Higgs transition has the Z 2 × Z (2) 2 example of fusion d-category is Rep(G).The (d − 1)-morphisms in Rep(G) are the representation of the group G which correspond to point-like excitations in d-dimensional space.The tensor product of the G-representations defines the fusion of the (d − 1)morphisms.The simple (d−1)-morphisms are the (equivalent classes of) irreducible representations and composite objects are the reducible representations.The reducible representations are direct sums of irreducible representations, and thus composite morphisms are direct sums of simple morphisms.The d-morphisms correspond the symmetric operators acting one on group representations.If a fusion d-category has trivial 1-morphisms, then it is a braided fusion (d − 1)-category.It also describes a topological order in d-dimensional space.