Iris Cong, Meng Cheng, Zhenghan Wang
We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.
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This Letter discusses topological quantum computation with gapped boundaries of two-dimensional topological phases. Systematic methods are presented to encode quantum information topologically using gapped boundaries, and to perform topologically protected operations on this encoding. In particular, we introduce a new and general computational primitive of topological charge measurement and present a symmetry-protected implementation of this primitive. Throughout the Letter, a concrete physical example, the $\mathbb{Z}_3$ toric code ($\mathfrak{D}(\mathbb{Z}_3)$), is discussed. For this exampl…
This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protec…
We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory o…
Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix $\mathcal W$, whose entries are the fusion-space dimensions $\mathcal W_{ia}$, to label different types of gapped domain …
Defects between gapped boundaries provide a possible physical realization of projective non-abelian braid statistics. A notable example is the projective Majorana/parafermion braid statistics of boundary defects in fractional quantum Hall/topological insulator and superconductor heterostructures. In this paper, we develop general theories to analyze the topological properties and projective braiding of boundary defects of topological phases of matter in two spatial dimensions. We present commuting Hamiltonians to realize defects between gapped boundaries in any $(2+1)D$ untwisted Dijkgraaf-Wit…
We study gapped boundaries of Abelian type-I fracton systems in three spatial dimensions. Using the X-cube model as our motivating example, we give a conjecture, with partial proof, of the conditions for a boundary to be gapped. In order to state our conjecture, we use a precise definition of fracton braiding and show that bulk braiding of fractons has several features that make it \textit{insufficient} to classify gapped boundaries. Most notable among these is that bulk braiding is sensitive to geometry and is "nonreciprocal," that is, braiding an excitation $a$ around $b$ need not yield the …
Gapped Boundaries of Kitaev's Quantum Double Models: A Lattice Realization of Anyon Condensation from Lagrangian Algebras
本文提出通过拉格朗日代数数据构建Kitaev量子双模型间隙边界的系统框架，揭示了任意子凝聚的微观动态。
Gapped phases with long-range entanglement may admit gapped boundaries. If the boundary is gapped, the ground-state degeneracy is well-defined and can be computed using methods of Topological Quantum Field Theory. We derive a general formula for the ground-state degeneracy for abelian Fractional Quantum Hall phases, including the cases when connected components of the boundary are subdivided into an arbitrary number of segments, with a different boundary condition on each segment, and in the presence of an arbitrary number of boundary domain walls.
We present a detailed microscopic investigation of fractional quantum Hall states with gapped boundaries in a coupled bilayer lattice model featuring holes whose counterpropagating chiral edge states are hybridized and gapped out. We focus on a lattice limit for cold-atom experiments, in which each hole just consists of a single removed site. Although the holes distort the original band structure and lead to ingap remnants of the continuum edge modes, we find that the lowest nearly flat band representing a higher-genus system may naturally form by controlling the local hopping terms that gap o…
An Algebraic Theory of Gapped Domain Wall Partons
本文提出了一种针对二维拓扑相之间带隙畴壁上parton量子数的范畴描述，揭示了其复合扇区结构，并与广义对称性和SymTFT思想建立了联系。
Non-anomalous non-invertible symmetries in 1+1D from gapped boundaries of SymTFTs
本文借助 SymTFT 的有能隙边界与 Lagrangian 代数刻画 1+1 维 QFT 中不可逆对称性的非反常条件，并提出跨融合范畴传输线算子的方法。
On gapped boundaries for SPT phases beyond group cohomology
本文通过推广对称扩张方法，为超越群上同调的一大类SPT相构造带隙边界，涵盖时间反演不变的玻色子SPT相和任意有限内部对称群的费米子Gu-Wen SPT相。
This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the $d$-dimensional quantum double model by using $d$-cocycles functions ($d\geq 2$). We point out that the system supports $m$-dimensional excitations ($m<d$), which we shall call fluctuating charges, that are superpositions of point-like electric charges characterized by $m$-…
We generalize the Electric-magnetic (EM) duality in the quantum double (QD) models to the case of topological orders with gapped boundaries. We also map the QD models with boundaries to the Levin-Wen (LW) models with boundaries. To this end, we Fourier transform and rewrite the extended QD model with a finite gauge group $G$ on a trivalent lattice with a boundary. Gapped boundary conditions of the model before the transformation are known to be characterized by the subgroups $K \subseteq G$. We find that after the transformation, the boundary conditions are then characterized by the Frobenius …
We study gapped boundaries characterized by "fermionic condensates" in 2+1 d topological order. Mathematically, each of these condensates can be described by a super commutative Frobenius algebra. We systematically obtain the species of excitations at the gapped boundary/ junctions, and study their endomorphisms (ability to trap a Majorana fermion) and fusion rules, and generalized the defect Verlinde formula to a twisted version. We illustrate these results with explicit examples. We also connect these results with topological defects in super modular invariant CFTs. To render our discussion …
We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category $\calC$ as in the Levin-Wen model, whereas the boundary is associated with a module category over $\calC$. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictio…
We show that the gapless boundary signatures - namely, chiral/helical hinge modes or localized zero modes - of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on surface is `anomalous' in the sense that it can be made consistent with symmetry only on the surface of a three dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-or…
The boundary of a fractionalized topological phase can be gapped by condensing a proper set of bosonic quasiparticles. Interestingly, in the presence of a global symmetry, such a boundary can have different symmetry transformation properties. Here we present an explicit example of this kind, in the double semion state with time reversal symmetry. We find two distinct cases where the semionic excitations on the boundary can transform either as time reversal singlets or as time reversal (Kramers) doublets, depending on the coherent phase factor of the Bose condensate. The existence of these two …
Gapped boundaries of (3+1)d topological orders
本文研究(3+1)维拓扑序带隙边界的分类问题，提出按弦状激发能否终止于边界来定义边界类，并以玻色环面码为例构造了超出已知三种边界的大量新边界，揭示其与有限群子群的对应关系。
The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. In this paper, we use those representations to determine the possible gapped boundaries of a 2+1D topological order, as well as the domain walls between two topological orders. We find that mapping-class-group representations for both genus-1 and genus-2 surfaces are needed to determine the gapped domain walls and boundaries. Our systematic theory is based on the fixed-point partition functions for the walls (or the boundaries), which completely characterize…
Whether two boundary conditions of a two-dimensional topological order can be continuously connected without a phase transition in between remains a challenging question. We tackle this challenge by constructing an effective Hamiltonian, describing anyon interaction, that realizes such a continuous deformation. At any point along the deformation, the model remains a fixed point model describing a gapped topological order with gapped boundaries. That the deformation retains the gap is due to the anomaly cancelation between the boundary and bulk. Such anomaly inflow is quantitatively studied usi…
Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part II
本文利用对称拓扑场论系统研究 2+1 维中具有非阿贝尔范畴对称性的带隙相，通过分析极小与非极小带隙边界条件及其对称性结构，揭示了广义电荷凝聚产生的超越 Landau 的拓扑相变。
Anyon exclusions statistics on surfaces with gapped boundaries
本文提出带隙边界曲面上的扩展任意子排他统计，引入任意子与边界分量间的相互排他统计，并通过系统基构造揭示非阿贝尔任意子系统中伪物种的物理含义。
Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries
本文将傅里叶变换与基底重写方法应用于三维拓扑序的规范理论模型，发现带隙边界条件由Frobenius代数刻画，并将该模型映射为Walker-Wang模型，从而建立了扩展Dijkgraaf-Witten与Crane-Yetter拓扑场论之间的对应关系。
Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
本研究针对带阿贝尔在位对称性的1+1维有能隙相，用富集融合范畴描述其时空可观测量，统一了对称相分类与格点对偶性，给出非平凡SPT相等实例，通过全息对偶关联二维拓扑序与一维有能隙量子液体，为量子相统一定义提供思路。
In this work we consider general fermion systems in two spatial dimensions, both with and without charge conservation symmetry, which realize a nontrivial fermionic topological order with only Abelian anyons. We address the question of precisely how these quantum phases differ from their bosonic counterparts, both in terms of their edge physics and in the way one would identify them in numerics. As in previous works, we answer these questions by studying the theory obtained after gauging the global fermion parity symmetry, which turns out to have a special and simple structure. Using this stru…
Characterizing the ambiguity in topological entanglement entropy
本文提出模型无关的划分图像，揭示拓扑纠缠熵在纠缠边界接触有能隙边界时因边界条件与相干性选择而产生歧义，并用两个参数完整刻画Z2环面码中的歧义性。
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 cylinder. When $\mathcal{M}$ has the topology of a cylinder, different combinations of boundary conditions of the cylinder will…
Exotic massive fermionic systems with huge vacuum degeneracy at boundaries
本研究探究有质量非相对论费米子系统，发现引入边界后系统仍有能隙但存在巨大真空简并度，剩余熵与边界面积成正比，揭示了边界对这类对称破缺费米系统的奇特影响。
Gapped boundaries of fermionic topological orders and higher central charges
本研究提出一种仅基于超模张量范畴模数据的费米子拓扑序高阶中心荷零值检验方法，无需显式计算模扩张，可用于判断有隙边界存在的必要条件，是迈向Gauss-Milgram公式费米子版本的第一步。
$2+1$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 a bosonic extension in which the fermionic excitation is "condensed" to form the associated FTOs. Here we choose the par…
Gapped Interfaces in Fracton Models and Foliated Fields
本文利用分层规范场论和晶格模型系统研究3+1维分形子相的有能隙界面，在X-cube模型及其与toric code模型之间发现多种新型有能隙边界和界面，包括线性子转换、Kramers-Wannier对偶型界面和电磁对偶界面。
We extend the twisted gauge theory model of topological orders in three spatial dimensions to the case where the three spaces have two dimensional boundaries. We achieve this by systematically constructing the boundary Hamiltonians that are compatible with the bulk Hamoltonian. Given the bulk Hamiltonian defined by a gauge group $G$ and a four-cocycle $ω$ in the fourth cohomology group of $G$ over $U(1)$, a boundary Hamiltonian can be defined by a subgroup $K$ of $G$ and a three-cochain $α$ in the third cochain group of $K$ over $U(1)$. The boundary Hamiltonian to be constructed must be gapped…
The structure of extrinsic defects in topologically ordered states of matter is host to a rich set of universal physics. Extrinsic defects in 2+1 dimensional topological states include line-like defects, such as boundaries between topologically distinct states, and point-like defects, such as junctions between different line defects. Gapped boundaries in particular can themselves be \it topologically \rm distinct, and the junctions between them can localize topologically protected zero modes, giving rise to topological ground state degeneracies and projective non-Abelian statistics. In this pa…
We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the mathematical community as state-sum constructions or lattice topological quantum field theories. All of the established ansatzes for fixed-point models imply the existence of a gapped boundary as well as a commuting-projector Hamiltonian. Thus, they fail to capture topological phases without a gapped boundary or commuting-projector Hamiltonian, most notably ch…
Pristine and Pseudo-gapped Boundaries of the Deconfined Quantum Critical Points
本文研究去禁闭量子临界点（DQCP）的边界效应，发现体序参量涨落与边界条件可产生赝能隙边缘态，并探讨了弱测量对奇异关联子的影响。
Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes
本文提出算法构造二维Pauli稳定子码的间隙边界和缺陷，利用算子代数建立拓扑数据与一维边界异常子码的对应关系，通过Laurent多项式和矩阵运算实现自动生成。
Categorical Landau Paradigm for Gapped Phases
提出基于对称拓扑场论（SymTFT）的统一框架，实现对任意维度下具有范畴对称性的带隙红外相的分类，建立了广义的范畴化朗道范式。
We show that boundaries of 3D weak topological insulators can become gapped by strong interactions while preserving all symmetries, leading to Abelian surface topological order. The anomalous nature of the weak topological insulators manifests itself in a non-trivial action of symmetries on the quasiparticles; most strikingly, translations change the anyon types in a manner impossible in strictly 2D systems with the same symmetry. As a further consequence, screw dislocations form non-Abelian defects that trap $\mathbb{Z}_4$ parafermion zero modes.
We propose an order parameter for a general one-dimensional gapped system with an open boundary condition. The order parameter can be computed from the ground state entanglement entropy of some regions near one of the boundaries. Hence, it is well-defined even in the presence of arbitrary interaction and disorder. We also show that it is invariant under a finite-depth local quantum circuit, suggesting its stability against an arbitrary local perturbation that does not close the energy gap. Further, it can unambiguously distinguish Majorana chain from a trivial chain under a global fermion pari…
Topological Holography for 2+1-D Gapped and Gapless Phases with Generalized Symmetries
本文利用高阶线性代数与高阶凝聚理论，通过SymTFT三明治构造和极小可凝聚代数，统一刻画2+1维有能隙相及含2-群、不可逆对称性的无能隙gSPT。
We present a binary classifier based on neural networks to detect gapped quantum phases. By considering the errors on top of a suitable reference state describing the gapped phase, we show that a neural network trained on the errors can capture the correlation between the errors and can be used to detect the phase boundaries of the gapped quantum phase. We demonstrate the application of the method for matrix product state calculations for different quantum phases exhibiting local symmetry-breaking order, symmetry-protected topological order, and intrinsic topological order.
Gapped non-liquid state (also known as fracton state) is a very special gapped quantum state of matter that is characterized by a microscopic cellular structure. Such microscopic cellular structure has a macroscopic effect at arbitrary long distances and cannot be removed by renormalization group flow, which makes gapped non-liquid state beyond the description of topological quantum field theory with a finite number of fields. Using Abelian and non-Abelian topological orders in 2-dimensional (2d) space and the different ways to glue them together via their gapped boundaries, we propose a syste…
In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these m…
String Condensations in 3+1D and Lagrangian Algebras
本研究在3+1维$\mathbb{Z}_2$拓扑序的模2范畴中提出三个拉格朗日代数，揭示其与有能隙边界和域壁的对应，首次系统研究弦凝聚的数学与物理，提供分层构造和格点实现。
We study the conductivities and entanglement structures of two different holographic gapped systems at zero density in the presence of boundaries within AdS/BCFT. The first gapped system is described by the Einstein-scalar gravity and the second one is the dual of AdS soliton geometry. We show that in both these two systems the bulk and boundary conductivities along the spatial direction of the boundary of BCFT are trivial. For the first system, when we increase the size of the subsystem the renormalized entanglement entropy is always non-negative and monotonically decreasing with discontinuou…
We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category. We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial. In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e. an topological order in one higher dimension). Thus we also ref…
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 characteriz…
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 $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric …
We compute the entanglement entropy in a 2+1 dimensional topological order in the presence of gapped boundaries. Specifically, we consider entanglement cuts that cut through the boundaries. We argue that based on general considerations of the bulk-boundary correspondence, the "twisted characters" feature in the Renyi entropy, and the topological entanglement entropy is controlled by a "half-linking number" in direct analogy to the role played by the S-modular matrix in the absence of boundaries. We also construct a class of boundary states based on the half-linking numbers that provides a "clo…
Recently we conjectured that a certain set of universal topological quantities characterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the ground state wave functions. For systems with gapped boundaries, these quantities are representations of the mapping class group $MCG(\mathcal M)$ of the space manifold $\mathcal M$ on which the systems lives. We will here consider simple examples in three dimensions and give physical interpretation of these quantities, related to fusion algebra and statistics of particle and string excitations. In pa…
Local observable errors from truncating interaction tails in gapped quantum lattice systems
本文针对有能隙量子格点系统，推导了截断长程相互作用尾部产生的局域可观测量误差界，明确了误差与截断范围的关系，证明了所得界的最优性，为有限范围截断的适用性提供了量化依据。
We introduce the concept of boundary degeneracy of topologically ordered states on a compact orientable spatial manifold with boundaries, and emphasize that the boundary degeneracy provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions. By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries. By Chern-Simons theory, this …
Fracton phases exhibit striking behavior which appears to render them beyond the standard topological quantum field theory (TQFT) paradigm for classifying gapped quantum matter. Here, we explore fracton phases from the perspective of defect TQFTs and show that topological defect networks---networks of topological defects embedded in stratified 3+1D TQFTs---provide a unified framework for describing various types of gapped fracton phases. In this picture, the sub-dimensional excitations characteristic of fractonic matter are a consequence of mobility restrictions imposed by the defect network. …
Hall conductivity pump
本文将Thouless电荷泵概念推广到三维，构造了一个拓扑不变量来度量霍尔电导在三维材料二维边界间的泵送。
Boundary symmetries of (2+1)D topological orders
本文提出代数框架描述(2+1)维拓扑序带隙边界的内禀拓扑对称性，并给出范畴论障碍，阻碍体态群对称性与边界对称性的相干性。
A topological insulator and its spin analogue as a gapped spin liquid have characteristic low energy excitations (edge states) within the gap when the systems have boundaries. This is the bulk-edge correspondence, which implies that the edge states themselves characterize the gapped bulk spin liquid. Based on the general principle, we analyzed the vector chirality phase of the spin-1/2 ladder with ring exchange by using the edge states and the entanglement entropy.
We study $(d-1)$-dimensional excitations in the $d$-dimensional color code that are created by transversal application of the $R_{d}$ phase operators on connected subregions of qubits. We find that such excitations are superpositions of electric charges and can be characterized by fixed-point wavefunctions of $(d-1)$-dimensional bosonic SPT phases with $(\mathbb{Z}_{2})^{\otimes d}$ symmetry. While these SPT excitations are localized on $(d-1)$-dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the non-triviality of emerging SPT wa…
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This …
Theory of Topologically Ordered Superfluids in 2+1 Dimensions
本研究针对2+1维自发破缺电荷U(1)c对称性的玻色子系统，建立拓扑有序超流体的表征框架，通过双向规范映射编码其拓扑数据，还给出含非阿贝尔涡旋的实例，为相关研究提供理论基础。
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