Abstract It was well known that there are e-particles and m-strings in the 3-dimensional (spatial dimension) toric code model, which realizes the 3-dimensional ℤ2 topological order. Recent mathematical result, however, shows that there are additional string-like topological defects in the 3-dimensional ℤ2 topological order. In this work, we construct all topological defects of codimension 2 and higher, and show that they form a braided fusion 2-category satisfying a braiding non-degeneracy condition.
Gravitational anomalies can be realized on the boundary of topologically ordered states in one higher dimension and are described by topological orders in one higher dimension. In this paper, we try to develop a general theory for both topological order and gravitational anomaly in any dimensions. (1) We introduce the notion of BF category to describe the braiding and fusion properties of topological excitations that can be point-like, string-like, etc. A subset of BF categories -- closed BF categories -- classify topological orders in any dimensions, while generic BF categories classify (potentially) anomalous topological orders that can appear at a boundary of a gapped quantum liquid in one higher dimension. (2) We introduce topological path integral based on tensor network to realize those topological orders. (3) Bosonic topological orders have an important topological invariant: the vector bundles of the degenerate ground states over the moduli spaces of closed spaces with different metrics. They may fully characterize topological orders. (4) We conjecture that a topological order has a gappable boundary iff the above mentioned vector bundles are flat. (5) We find a holographic phenomenon that every topological order with a gappable boundary can be uniquely determined by the knowledge of the boundary. As a consequence, BF categories in different dimensions form a (monoid) cochain complex, that reveals the structure and relation of topological orders and gravitational anomalies in different dimensions. We also studied the simplest kind of bosonic topological orders that have no non-trivial topological excitations. We find that this kind of topological orders form a
Z class in 2+1D (with gapless edge), a
Z2 class in 4+1D (with gappable boundary), and a
Z⊕Z class in 6+1D (with gapless boundary).
We review the condensation completion of a modular tensor category
C, which yields a fusion 2-category
ΣC of separable algebras, bimodules over algebras and bimodule maps in
C. Physically,
ΣC is the fusion 2-category of codimension-1 defects, codimension-2 defects and instantons in the
2+1D topological order
C. We realize the rough-rough wall and
e-
m exchange wall in Toric Code model on the lattice by deforming the Hamiltonian based on the corresponding algebraic data. We apply condensation completion to Toric Code,
3F, two-laryer semion and
Z4 topological orders, and explicitly enumerate their
1d and
0d defects along with fusion rules. We also mention other applications of condensation completion: alternative interpretations of condensation completion of a braided fusion category; condensation completion of the category of symmetry charges and its correspondence to gapped phases with symmetry; for a topological order
C, one can find all gapped boundaries of the stacking of
C with its time-reversal conjugate through computing the condensation completion of
C.
We construct large families of mixed-state topological orders in (3+1)D, including intrinsically mixed-state topological orders. We realize these theories by relating the data parameterizing braided fusion 2-categories to the strong higher-form symmetries and their anomalies that characterize mixed-state topological order. We focus here on bosonic theories, which include theories which admit an emergent fermion in the spectrum of line operators. Our general framework is exemplified on the lattice, where we present concrete models and identify how the data of braided fusion 2-categories are physically encoded. We then formalize this data and show that every braided fusion 2-category arises from a general construction (decoherence followed by strong and weak gauging) as the category of topological operators of a (3+1)D mixed state, with intrinsically mixed-state topological orders distinguished by the presence of strong higher-form symmetries whose anomalies cannot be realized in any (3+1)D gapped ground state. Notably, we show that (3+1)D admits a new mechanism with no (2+1)D counterpart: intrinsically mixed-state order whose topological operators braid non-degenerately, but which carries a gravitational anomaly.
(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-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of \mathbb{Z}_2 ℤ 2 gauge theory with fermionic charges, in \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ( D_n D n ) and alternating ( A_6 A 6 ) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H^4 H 4 cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A_6 A 6 gauge theory.
On top of a D D -dimensional gapped bulk state, Low Entanglement Excitations (LEE) on d(<D) d ( < D ) -dimensional sub-manifolds can have extensive energy but preserves the entanglement area law of the ground state. Due to their multi-dimensional nature, the LEEs embody a higher-category structure in quantum systems. They are the ground state of a modified Hamiltonian and hence capture the notions of ‘defects’ of generalized symmetries. In previous works, we studied the low-entanglement excitations in a trivial phase as well as those in invertible phases. We find that LEEs in these phases have the same structure as lower-dimensional gapped phases and their defects within. In this paper, we study the LEEs inside non-invertible topological phases. We focus on the simple example of \mathbb{Z}_2 ℤ 2 toric code and discuss how the fusion result of 1d LEEs with 0d morphisms can depend on both the choice of fusion circuit and the ordering of the fused defects.
A hallmark feature of topologically ordered states of matter is the dependence of ground-state degeneracy (GSD) on the topology of the manifold determined by the global shape of the system. Although the topology of a physical system is practically hard to manipulate, recently, it was shown that in certain topologically ordered phases, topological defects can introduce extra topological GSD. Here the topological defects can be viewed as effectively changing the topology of the physical system. Previous studies have been focusing on two spatial dimensions with pointlike topological defects. In three dimensions, linelike topological defects can appear. They are closed loops in the bulk that can be linked and knotted, effectively leading to complex three-dimensional manifolds in certain topologically ordered states. This paper studies the properties of such line defects in a particular context: the lattice dislocations. We give an analytical construction, together with support from exact numerical calculations, for the dependence of the GSD on dislocations of certain doubled versions of the exactly solvable Kitaev's toric code models in both two and three dimensions. We find that the GSD of the 3D model depends only on the total number of dislocation loops, no matter how they are linked or knotted. The results are extended to
Zn generalizations of the model. Additionally, we consider the phases in which the crystalline orders are destroyed through proliferation of double dislocations. The resulting phases are shown to host topological orders described by non-Abelian gauge theories.