Braiding with Borromean Rings in (
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)-Dimensional Spacetime

Chan, AtMa P. O.; Ye, Peng; Ryu, Shinsei

doi:10.1103/physrevlett.121.061601

Cited by 37 publications

(18 citation statements)

References 79 publications

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“…Our present results combined together with Ref. [7] positively support the previous attempts based on continuum TQFTs [12,36,[38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Various data derived from continuum TQFTs can be checked and compared through the discrete cocycle and lattice formulations [8-10, 29, 30, 37, 52-57].…”

Section: The Plan Of the Article And A Short Summarysupporting

confidence: 87%

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Wang

Ohmori

Putrov

et al. 2018

Progress of Theoretical and Experimental Physics

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Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain Topological Quantum Field Theories (TQFT), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFT. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry protected topological states (with fermion parity Z f 2 ) of symmetry group Z 4 × Z 2 and (Z 4 ) 2 in 3+1D, also Z 2 and (Z 2 ) 2 in 2+1D. Gauging the last three cases begets non-Abelian spin TQFT (fermionic topological order). We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with boundary, such that the bulk and boundary are fully-gapped and short or long-range entangled (SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condensed, but only fuzzy-composite objects of extended operators can end (e.g. a stringlike composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g. 0-form/higher-form/"composite" breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some of such LRE systems. 1 We denote the spacetime dimensions as d + 1D 2 One can also consider a generalization of this relation by turning on a background flat connection A (G) for a global symmetry G. First, non-trivial holonomies along 1-cycles of M d will result in replacement H M d by the corresponding twisted Hilbert space H tw M d . Second, a non-trivial holonomy g ∈ G along the time S 1 will result in insertion of ρ(g) into the trace, where ρ is the representation of G on the Hilbert space:( 1.2) In condensed matter, this is related to the symmetry twist inserted on M d to probe the Symmetry Protected/Enriched Topological states (SPTs/SETs) [2,12]. In this work, instead we mainly focus on eqn. (1.1).

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Section: The Plan Of the Article And A Short Summarysupporting

confidence: 87%

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Wang

Ohmori

Putrov

et al. 2018

Progress of Theoretical and Experimental Physics

View full text Add to dashboard Cite

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“…Refs. [79][80][81]. However, L + L does not appear to fit into the framework of these works since e.g.…”

Section: Dynamical Foliationsmentioning

confidence: 99%

Foliated field theory and string-membrane-net condensation picture of fracton order

2019

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Foliated fracton order is a qualitatively new kind of phase of matter. It is similar to topological order, but with the fundamental difference that a layered structure, referred to as a foliation, plays an essential role and determines the mobility restrictions of the topological excitations. In this work, we introduce a new kind of field theory to describe these phases: a foliated field theory. We also introduce a new lattice model and string-membrane-net condensation picture of these phases, which is analogous to the string-net condensation picture of topological order.Contents arXiv:1812.01613v3 [cond-mat.str-el]

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“…Fermionic version of TQFT gauge theory: Fermionic finite-group gauge theory and spin-TQFT, and their braiding statistics or topological link invariants [35,45,46] and 3. Higher-gauge theory as TQFTs, see some selective examples in [47,48], [37], [33], [49][50][51][52][53].…”

Section: Examples Of Topological Orders Tqfts and Topological Invarimentioning

confidence: 99%

Quantum statistics and spacetime topology: Quantum surgery formulas

2019

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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 gapped states of matter with intrinsic topological orders. Second, we introduce the braiding statistics data of particles and strings, such as the geometric Berry matrices for particle-string Aharonov-Bohm, 3-string, 4-string, or multi-string adiabatic loop braiding process, encoded by submanifold linkings, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new "quantum surgery" formulas and constraints, analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, 3d and 4d TQFTs and potentially correlated to bootstrap boundary physics such as gapless modes, extended defects, 2d and 3d conformal field theories or quantum anomalies.This article is meant to be an extended and further detailed elaboration of our previous work [1] and Chapter 6 of Ref. [2]. Our theory applies to general quantum theories and quantum mechanical systems, also applicable to, but not necessarily requiring the quantum field theory description.

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Braiding with Borromean Rings in ( 3+1 )-Dimensional Spacetime

Cited by 37 publications

References 79 publications

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Foliated field theory and string-membrane-net condensation picture of fracton order

Quantum statistics and spacetime topology: Quantum surgery formulas

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