A bstract We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading ℤ 2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.
A bstract We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading ℤ 2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.
We propose a general framework to characterize gapped infra-red (IR) phases of theories with non-invertible (or categorical) symmetries. In this paper we focus on (1+1)d gapped phases with fusion category symmetries. The approach that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key input: associated to a field theory in d d spacetime dimensions, the SymTFT lives in one dimension higher and admits a gapped boundary, which realizes the categorical symmetries. It also admits a second, physical, boundary, which is generically not gapped. Upon interval compactification of the SymTFT by colliding the gapped and physical boundaries, we regain the original theory. In this paper, we realize gapped symmetric phases by choosing the physical boundary to be a gapped boundary condition as well. This set-up provides computational power to determine the number of vacua, the symmetry breaking pattern, and the action of the symmetry on the vacua. The SymTFT also manifestly encodes the order parameters for these gapped phases, thus providing a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find that for non-invertible symmetries the order parameters involve multiplets containing both untwisted and twisted sector local operators, and hence can be interpreted as mixtures of conventional and string order parameters. We also observe that spontaneous breaking of non-invertible symmetries can lead to vacua that are physically distinguishable: unlike the standard symmetries described by groups, non-invertible symmetries can have different actions on different vacua of an irreducible gapped phase. This leads to the presence of relative Euler terms between physically distinct vacua. Along with the physical description of symmetric gapped phases, we also provide a mathematical one as pivotal 2-functors whose source 2-category is the delooping of the fusion category characterizing the symmetry and the target 2-category is the Euler completion of 2-vector spaces.
We study the classification of the gapped phases or massive renormalization group (RG) flows dual to the massless RG flows under changing the sign of the coupling constants. Whereas our formalism is based on combining Higgs- or Nambu-Goldstone-type arguments with Cardy's smeared boundary conformal field theories (SBCFTs), several puzzling structures arise. More specifically, the established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states, which are unphysical in boundary critical phenomena. Hence, the module of the gapped phases can be outside of that of boundary critical phenomena, whereas one can still calculate characterizing quantities by applying SBCFTs to the models. For example, we demonstrate that the massive RG flow dual to the massless RG flow from the tricritical Ising model to the Ising model, one of the simplest massless RG flows, has this unusual structure. This can be regarded as a quantum field-theoretic analogue of order-disorder coexistence in lattice models. More generally, the resultant gapped phases usually spontaneously break non-group-like symmetry (or noninvertible symmetry). Our work provides systematic quantum field theoretic descriptions of such unusual phases with spontaneous symmetry breaking of non-group-like (or noninvertible) symmetries.
Recent developments have revealed that symmetries need not form a group, but instead can be noninvertible. Here we use analytical arguments and numerical evidence to illuminate how spontaneous symmetry breaking of a noninvertible symmetry is similar yet distinct from ordinary, invertible, symmetry breaking. We consider one-dimensional chains of group-valued qudits, whose local Hilbert space is spanned by elements of a finite group G (reducing to ordinary qubits when G=Z_{2}). We construct Ising-type transverse-field Hamiltonians with Rep(G) symmetry whose generators multiply according to the tensor product of irreducible representations (irreps) of the group G. For non-Abelian G, the symmetry is noninvertible. In the symmetry broken phase there is one ground state per irrep on a closed chain. The symmetry breaking can be detected by local order parameters but, unlike the invertible case, different ground states have distinct entanglement patterns. We show that for each irrep of dimension greater than one the corresponding ground state exhibits string order, entanglement spectrum degeneracies, and has gapless edge modes on an open chain-features usually associated with symmetry-protected topological order. Consequently, domain wall excitations behave as one-dimensional non-Abelian anyons with nontrivial internal Hilbert spaces and fusion rules. Our Letter identifies properties of noninvertible symmetry breaking that existing quantum hardware can probe.
A bstract We initiate the S-matrix bootstrap analysis of theories with non-invertible symmetries in (1+1) dimensions. Our previous work [1] showed that crossing symmetry of S-matrices in such theories is modified, with modification characterized by the fusion category data. By imposing unitarity, symmetry and the modified crossing, we constrain the space of consistent S-matrices, identifying integrable theories with non-invertible symmetries at the cusps of allowed regions. We also extend the modified crossing rules to cases where vacua transform in non-regular representations of fusion category, utilizing a connection to a dual category CM∗ C M ∗ and Symmetry Topological Field Theory (SymTFT). This highlights the utility of SymTFT in the analysis of scattering amplitudes.
A bstract We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the ’t Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.
An anyon-chain-like lattice model with symmetry described by the Ising fusion category is studied. Combining numerical and analytical studies, we uncover a rich phase diagram that contains three phases: a symmetric critical phase and two categorical symmetry breaking phases. The symmetric phase lies in the same universality class as the usual critical Ising model. The first symmetry-breaking phase, dubbed the \emph{categorical ferromagnetic} phase, has the Ising fusion category fully broken and exhibits a threefold ground-state degeneracy, as expected from the generalized Landau paradigm. The other symmetry-breaking phase is analogous to a conventional antiferromagnet: it breaks lattice translation and part of the Ising fusion category, and therefore is termed the \emph{categorical antiferromagnetic} phase. Unlike ordinary antiferromagnetic states associated with finite invertible symmetry breaking, this phase itself is critical, being described by a fourfold degenerate Ising conformal field theory. We argue more generally that antiferromagnetic states associated with broken non-invertible symmetries have a large low-energy manifold that grows exponentially in system size, due to the greater-than-one quantum dimension of domain walls. We also numerically study the transitions between the three phases. The transition between the symmetric and categorical ferromagnetic phase is described by the c=7/10 tricritical Ising CFT, while the transition between the symmetric and categorical antiferromagnetic phases is less understood. Our numerical data suggest that the latter transition is continuous and described by a conformal field theory with central charge c=3/2.