Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase

We investigate (3+1)d topological orders in fermionic systems with an anomalous $\mathbb{Z}_{2N}^{\mathrm{F}}$ symmetry, where its $\mathbb{Z}_2^{\mathrm{F}}$ subgroup is the fermion parity. Such an anomalous symmetry arises as the discrete subgroup of the chiral U(1) symmetry of $\nu$ copies of Weyl fermions of the same chirality. Guided by the crystalline correspondence principle, we construct (3+1)d symmetry-preserving gapped states on the boundary of a closely related (4+...

We study bosonic symmetry protected topological (SPT) phases with $C_{2}$ rotational symmetry in four spatial dimensions which is not captured by the group cohomology classification. By using the topological crystal approach, we show that the topological crystalline state of this SPT phase is given by placing an $E_{8}$ state on the two dimensional rotational invariant plane, which provides a simple physical picture of this phase. Based on this understanding, we show that a v...

We analyze $2+1d$ and $3+1d$ Bosonic Symmetry Protected Topological (SPT) phases of matter protected by onsite symmetry group $G$ by using dual bulk and boundary approaches. In the bulk we study an effective field theory which upon coupling to a background flat $G$ gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Furthe...

We discuss physical constructions, and the boundary properties of various symmetry protected topological phases that involve 1-form symmetries, from one spatial dimension (1d) to four spatial dimensions (4d). For example, the prototype 3d boundary state of 4d SPT states involving 1-form symmetries can be either a gapless photon phase (quantum electrodynamics) or gapped topological order enriched by 1-form symmetries, namely the loop excitations of these topological orders car...

By definition, the physics of the $d-$dimensional (dim) boundary of a $(d+1)-$dim symmetry protected topological (SPT) state cannot be realized as itself on a $d-$dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a $d-$dim theory must be a boundary or not, is to couple this theory to a gauge field (or to "gauge" its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: can the boundary of...

Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ beco...

We discuss a strategy to construct gapped boundaries for a large class of symmetry-protected topological phases (SPT phases) beyond group cohomology. This is done by a generalization of the symmetry extension method previously used for cohomological SPT phases. We find that this method allows us to construct gapped boundaries for time-reversal-invariant bosonic SPT phases and for fermionic Gu-Wen SPT phases for arbitrary finite internal symmetry groups.

We study gapless phases in (3+1)d in the presence of 1-form and non-invertible duality symmetries. Using the Symmetry Topological Field Theory (SymTFT) approach, we classify the gapless symmetry-protected (gSPT) phases in these setups, with particular focus on intrinsically gSPTs (igSPTs). These are symmetry protected critical points which cannot be deformed to a trivially gapped phase without spontaneously breaking the symmetry. Although these are by now well-known in (1+1)d...

We propose a diagnostic tool for detecting non-trivial symmetry protected topological (SPT) phases protected by a symmetry group $G$ in 2+1 dimensions. Our method is based on directly studying the 1+1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests in the in-existence of boundary states that preserves both the con...

We use the Symmetry Topological Field Theory (SymTFT) to study and classify gapped phases in (2+1)d for a class of categorical symmetries, referred to as being of bosonic type. The SymTFTs for these symmetries are given by twisted and untwisted (3+1)d Dijkgraaf-Witten (DW) theories for finite groups G. A finite set of boundary conditions (BCs) of these DW theories is well-known: these simply involve imposing Dirichlet and Neumann conditions on the (3+1)d gauge fields. We refe...