2D Topological Orders Research Articles

Boundary and domain wall theories of 2d generalized quantum double model

The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.

The boundary phase transitions of the 2+1D ℤN topological order via topological Wick rotation

In this work, we show that a critical point of a 1d self-dual boundary phase transition between two gapped boundaries of the ℤN topological order can be described by a mathematical structure called an enriched fusion category. The critical point of a boundary phase transition can be viewed as a gappable non-chiral gapless boundary of the ℤN topological order. A mathematical theory of the gapless boundaries of 2d topological orders developed by Kong and Zheng (arXiv:1905.04924 and arXiv:1912.01760) tells us that all macroscopic observables on the gapless boundary form an enriched unitary fusion category, which can be obtained by a holographic principle called the “topological Wick rotation.” Using this method, we obtain the enriched fusion category that describes a critical point of the phase transition between the e-condensed boundary and the m-condensed boundary of the ℤN topological order. To verify this idea, we also construct a lattice model to realize the critical point and recover the mathematical data of this enriched fusion category. The construction further shows that the categorical symmetry of the boundary is determined by the topological defects in the bulk, which indicates the holographic principle indirectly. This work shows, as a concrete example, that the mathematical theory of the gapless boundaries of 2+1D topological orders is a powerful tool to study general phase transitions.

A Characterization of a Finite-Dimensional Commuting Square Producing a Subfactor of Finite Depth

Abstract We give a characterization of a finite-dimensional commuting square of $C^*$-algebras with a normalized trace that produces a hyperfinite type II$_1$ subfactor of finite index and finite depth in terms of Morita equivalent unitary fusion categories. This type of commuting squares was studied by N. Sato and we show that a slight generalization of his construction covers the fully general case of such commuting squares. We also give a characterization of such a commuting square that produces a given hyperfinite type II$_1$ subfactor of finite index and finite depth. These results also give a characterization of certain 4-tensors that appear in recent studies of matrix product operators in 2D topological order.

A mathematical theory of gapless edges of 2d topological orders. Part II

This is the second part of a two-part work on the unified mathematical theory of gapped and gapless edges of 2+1D topological orders. In Part I, we have developed the mathematical theory of chiral gapless edges. In Part II, we study boundary-bulk relation and non-chiral gapless edges. In particular, we explain how the notion of the center of an enriched monoidal category naturally emerges from the boundary-bulk relation. After the study of 0+1D gapless walls, we give the complete boundary-bulk relation for 2+1D topological orders with chiral gapless edges (including gapped edges) and 0d walls between edges. This relation is stated precisely and proved rigorously as a monoidal equivalence, which generalizes the functoriality of the usual Drinfeld center to an enriched setting. We also develop the mathematical theory of non-chiral gapless edges and 0+1D walls, and explain how to gap out certain non-chiral 1+1D gapless edges and 0+1D gapless walls categorically. In the end, we show that all anomaly-free 1+1D boundary-bulk rational CFT's can be recovered from 2d topological orders with chiral gapless edges via a dimensional reduction process. This provides physical meanings to some mysterious connections between mathematical results in fusion categories and those in rational CFT's.

Topological phase transition on the edge of two-dimensional Z2 topological order

The unified mathematical theory of gapped and gapless edges of 2d topological orders was developed by two of the authors. It provides a powerful tool to study pure edge topological phase transitions on the edges of 2d topological orders (without altering the bulks). In particular, it implies that the critical points are described by enriched fusion categories. In this work, we illustrate this idea in a concrete example: the 2d $\mathbb{Z}_2$ topological order. In particular, we construct an enriched fusion category, which describes a gappable non-chiral gapless edge of the 2d $\mathbb{Z}_2$ topological order; then use an explicit lattice model construction to realize the critical point and, at the same time, all the ingredients of this enriched fusion category.

A mathematical theory of gapless edges of 2d topological orders. Part I

This is the first part of a two-part work on a unified mathematical theory of gapped and gapless edges of 2d topological orders. We analyze all the possible observables on the 1+1D world sheet of a chiral gapless edge of a 2d topological order, and show that these observables form an enriched unitary fusion category, the Drinfeld center of which is precisely the unitary modular tensor category associated to the bulk. This mathematical description of a chiral gapless edge automatically includes that of a gapped edge (i.e. a unitary fusion category) as a special case. Therefore, we obtain a unified mathematical description and a classification of both gapped and chiral gapless edges of a given 2d topological order. In the process of our analysis, we encounter an interesting and reoccurring phenomenon: spatial fusion anomaly, which leads us to propose the Principle of Universality at RG fixed points for all quantum field theories. Our theory also implies that all chiral gapless edges can be obtained from a so-called topological Wick rotations. This fact leads us to propose, at the end of this work, a surprising correspondence between gapped and gapless phases in all dimensions.

Characterizing topological order by the information convex

Motivated by previous efforts in detecting topological orders from the ground state(s) wave function, we introduce a new quantum information tool, coined the information convex, to capture the bulk and boundary topological excitations of a 2D topological order. Defined as a set of reduced density matrices that minimizes the energy in a subsystem, the information convex encodes not only the bulk anyons but also the gapped boundaries of 2D topological orders. Using untwisted gapped boundaries of non-Abelian quantum doubles as an example, we show how the information convex reveals and characterizes deconfined bulk and boundary topological excitations, and the condensation rule relating them. Interference experiments in cold atoms provide potential measurements for the invariant structure of information convex.

Gapless edges of 2d topological orders and enriched monoidal categories

In this work, we give a mathematical description of a chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological edge excitations, boundary CFT's and walls between boundary CFT's. These observables can be described by a chiral algebra and an enriched monoidal category. This mathematical description automatically includes that of gapped edges as special cases. Therefore, it gives a unified framework to study both gapped and gapless edges. Moreover, the boundary-bulk duality also holds for gapless edges. More precisely, the unitary modular tensor category that describes the 2d bulk phase is exactly the Drinfeld center of the enriched monoidal category that describes the gapless/gapped edge. We propose a classification of all gapped and chiral gapless edges of a given bulk phase. In the end, we explain how modular-invariant bulk rational conformal field theories naturally emerge on certain gapless walls between two trivial phases.

Spin TQFTs and fermionic phases of matter

We study lattice constructions of gapped fermionic phases of matter. We show that the construction of fermionic Symmetry Protected Topological orders by Gu and Wen has a hidden dependence on a discrete spin structure on the Euclidean space-time. The spin structure is needed to resolve ambiguities which are otherwise present. An identical ambiguity is shown to arise in the fermionic analog of the string-net construction of 2D topological orders. We argue that the need for a spin structure is a general feature of lattice models with local fermionic degrees of freedom and is a lattice analog of the spin-statistics relation.

Symmetry-respecting topologically ordered surface phase of three-dimensional electron topological insulators

A 3d electron topological insulator (ETI) is a phase of matter protected by particle-number conservation and time-reversal symmetry. It was previously believed that the surface of an ETI must be gapless unless one of these symmetries is broken. A well-known symmetry-preserving, gapless surface termination of an ETI supports an odd number of Dirac cones. In this paper we deduce a symmetry-respecting, gapped surface termination of an ETI, which carries an intrinsic 2d topological order, Moore-Read x anti-semion. The Moore-Read sector supports non-Abelian charge 1/4 anyons, while the Abelian, anti-semion sector is electrically neutral. Time-reversal symmetry is implemented in this surface phase in a highly non-trivial way. Moreover, it is impossible to realize this phase strictly in 2d, simultaneously preserving its implementation of both the particle number and time-reversal symmetries. A 1d edge on the ETI surface between the topologically-ordered phase and the topologically trivial time-reversal-broken phase with a Hall conductivity sigma_H = 1/2, carries a right-moving neutral Majorana mode, a right-moving bosonic charge mode and a left-moving bosonic neutral mode. The topologically-ordered phase is separated from the surface superconductor by a direct second order phase transition in the XY* universality class, which is driven by the condensation of a charge 1/2 boson, when approached from the topologically-ordered side, and proliferation of a flux 4\pi-vortex, when approached from the superconducting side. In addition, we prove that time-reversal invariant (interacting) electron insulators with no intrinsic topological order and electromagnetic response characterized by a \theta-angle, \theta = \pi, do not exist if the electrons transform as Kramers singlets under time-reversal.

Non-Abelian string and particle braiding in topological order: ModularSL(3,Z)representation and(3+1)-dimensional twisted gauge theory

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist $\omega_4$ of $G$'s cohomology group $\mathcal{H}^4(G,\mathbb{R}/\mathbb{Z})$ in 3 dimensional space and 1 dimensional time (3+1D). We establish the topological spin and the spin-statistics relation for the closed strings, and their multi-string braiding statistics. The 3+1D twisted gauge theory can be characterized by a representation of a modular transformation group SL$(3,\mathbb{Z})$. We express the SL$(3,\mathbb{Z})$ generators $\mathsf{S}^{xyz}$ and $\mathsf{T}^{xy}$ in terms of the gauge group $G$ and the 4-cocycle $\omega_4$. As we compactify one of the spatial directions $z$ into a compact circle with a gauge flux $b$ inserted, we can use the generators $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$ of an SL$(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3D topological order $\mathcal{C}^{3\text{D}}$ to a direct sum of degenerate states of 2D topological orders $\mathcal{C}_b^{2\text{D}}$ in different flux $b$ sectors: $\mathcal{C}^{3\text{D}} = \oplus_b \mathcal{C}_b^{2\text{D}}$. The 2D topological orders $\mathcal{C}_b^{2\text{D}}$ are described by 2D gauge theories of the group $G$ twisted by the 3-cocycles $\omega_{3(b)}$, dimensionally reduced from the 4-cocycle $\omega_4$. We show that the SL$(2,\mathbb{Z})$ generators, $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.