Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase ...with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relation between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. A more general situation is also studied in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin–Wen types of lattice models and some chiral topological phases.
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We correct a mistake in arXiv:1307.8244, which was published in Nucl. Phys. B 886 (2014) 436–482. The main result of that paper, i.e. Theorembs 4.7 in arXiv:1307.8244, remains correct. We also take ...the opportunity to simplify the bootstrap analysis in arXiv:1307.8244.
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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.
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A
bstract
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.
Categories of quantum liquids I Kong, Liang; Zheng, Hao
The journal of high energy physics,
08/2022, Volume:
2022, Issue:
8
Journal Article
Peer reviewed
Open access
A
bstract
We develop a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd’s work on condensation completion. Based on this theory, we prove some fundamental results ...on
E
m
-multi-fusion higher categories and their higher centers. We also outline a theory of unitary higher categories based on a ∗-version of condensation completion. After these mathematical preparations, based on the idea of topological Wick rotation, we develop a unified mathematical theory of all quantum liquids, which include topological orders, SPT/SET orders, symmetry-breaking orders and CFT-like gapless phases. We explain that a quantum liquid consists of two parts, the topological skeleton and the local quantum symmetry, and show that all
n
D quantum liquids form a ∗-condensation complete higher category whose equivalence type can be computed explicitly from a simple coslice 1-category.
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.
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Boundary-bulk relation in topological orders Kong, Liang; Wen, Xiao-Gang; Zheng, Hao
Nuclear physics. B,
September 2017, 2017-09-00, 2017-09-01, Volume:
922, Issue:
C
Journal Article
Peer reviewed
Open access
In this paper, we study the relation between an anomaly-free n+1D topological order, which are often called n+1D topological order in physics literature, and its nD gapped boundary phases. We argue ...that the n+1D bulk anomaly-free topological order for a given nD gapped boundary phase is unique. This uniqueness defines the notion of the “bulk” for a given gapped boundary phase. In this paper, we show that the n+1D “bulk” phase is given by the “center” of the nD boundary phase. In other words, the geometric notion of the “bulk” corresponds precisely to the algebraic notion of the “center”. We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the “bulk” satisfies the same universal property as that of the “center” of an algebra in mathematics, i.e. “bulk = center”. The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.
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Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in1+1Dbosonic systems, there is ...no nontrivial topological order, while in2+1Dbosonic systems, the topological orders are classified by the following pair: a modular tensor category and a chiral central charge. In this paper, following a new line of thinking, we find that in3+1Dthe classification is much simpler than it was thought to be; we propose a partial classification of topological orders for3+1Dbosonic systems: If all the pointlike excitations are bosons, then such topological orders are classified by a simpler pair(G,ω4): a finite groupGand its group 4-cocycleω4∈H4G;U(1)(up to group automorphisms). Furthermore, all such3+1Dtopological orders can be realized by Dijkgraaf-Witten gauge theories.
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We performed a comprehensive review and meta-analysis to evaluate the diagnostic values of serum single and multiplex tumor-associated autoantibodies (TAAbs) in patients with lung cancer (LC).
We ...searched the MEDLINE and EMBASE databases for relevant studies investigating serum TAAbs for the diagnosis of LC. The primary outcomes included sensitivity, specificity and accuracy of the test.
The systematic review and meta-analysis included 31 articles with single autoantibody and 39 with multiplex autoantibodies. Enzyme-linked immunosorbent assay (ELISA) was the most common detection method. For the diagnosis of patients with all stages and early-stage LC, different single or combinations of TAAbs demonstrated different diagnostic values. Although individual TAAbs showed low diagnostic sensitivity, the combination of multiplex autoantibodies offered relatively high sensitivity. For the meta-analysis of a same panel of autoantibodies in patients at all stages of LC, the pooled results of the panel of 6 TAAbs (p53, NY-ESO-1, CAGE, GBU4-5, Annexin 1 and SOX2) were: sensitivity 38% (95% CI 0.35-0.40), specificity 89% (95% CI 0.86-0.91), diagnostic accuracy 65.9% (range 62.5-81.8%), AUC 0.52 (0.48-0.57), while the summary estimates of 7 TAAbs (p53, CAGE, NY-ESO-1, GBU4-5, SOX2, MAGE A4 and Hu-D) were: sensitivity 47% (95% CI 0.34-0.60), specificity 90% (95% CI 0.89-0.92), diagnostic accuracy 78.4% (range 67.5-88.8%), AUC 0.90 (0.87-0.93). For the meta-analysis of the same panel of autoantibodies in patients at early-stage of LC, the sensitivities of both panels of 7 TAAbs and 6 TAAbs were 40% and 29.7%, while their specificities were 91% and 87%, respectively.
Serum single or combinations of multiplex autoantibodies can be used as a tool for the diagnosis of LC patients at all stages or early-stage, but the combination of multiplex autoantibodies shows a higher detection capacity; the diagnostic value of the panel of 7 TAAbs is higher than the panel of 6 TAAbs, which may be used as potential biomarkers for the early detection of LC.
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