Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models

We study the correspondence between boundary spectrum of non-chiral topological orders on an open manifold $\mathcal{M}$ with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed manifold. The closed manifold is bipartitioned into two subsystems, one of which has the same topology as $\mathcal{M}$. Specifically, we focus on the case of generalized string-net models and discuss the cases where $\mathcal{M}$ is a disk or a cylinde...

(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error correc...

Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC $\mathcal{A}$ representing the Witt class of an anomaly, the article [arXiv:2208.14018] gave a commuting projector model associated to an $\mathcal{A}$-enriched unitary fusion category $\mathcal{X}$ on a 2D boundary of the 3D Walker-Wang model associated to $\mathcal{A}$. That article claim...

Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group $G$, we consider a family of tensor network representations of its ground state subspace. This family is indexed by gapped boundary conditions encoded into module 2-categories over the input spherical fusion 2-category. Individual tensors are characterised by symmetry conditions with respect to non-local operators acting on entanglement degrees of freedom. In the case of Dirichlet and Neu...

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete $n$-categories which are mildly dualizable and have trivial centre. Dualizability encodes the word "topological," and we take it as the definition of "(separable) multifusion $n$-category"; triviality of the centre implements the physical principle of "remote detectability." We show that such $n$-categori...

$2+1$D bosonic topological orders can be characterized by the $S,T$ matrices that encode the statistics of topological excitations. In particular, the $S,T$ matrices can be used to systematically obtain the gapped boundaries of bosonic topological orders. Such an approach, however, does not naively apply to fermionic topological orders (FTOs). In this work, we propose a systematic approach to obtain the gapped boundaries of $2+1$D abelian FTOs. The main trick is to construct ...

We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for its ground state and low-lying quasiparticle excitations to be in the DFib topological phase. Using the string...

We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category $\mathrm{Rep}(H)$ of a semisimple weak Hopf algebra $H$, the fermionized TQFT has a superfusion category symmetry $\mathrm{SRep}(\mathcal{H}^u)$, which is the supercategory of super representations of a weak Hopf superalgebra $\mathcal{H}^u$. The weak Hopf superalgebra $\mathc...

We present three Lagrangian algebras in the modular 2-category associated to the 3+1D $\mathbb{Z}_2$ topological order and discuss their physical interpretations, connecting algebras with gapped boundary conditions, and interestingly, maps (braided autoequivalences) exchanging algebras with bulk domain walls. A Lagrangian algebra, together with its modules and local modules, encapsulates detailed physical data of strings condensing at a gapped boundary. In particular, the con...

To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable of creating anyonic excitations of particles and strings, well-defined in ...