Ground-state degeneracy for Abelian anyons in the presence of gapped boundaries

Kapustin, Anton

doi:10.1103/physrevb.89.125307

Cited by 50 publications

(56 citation statements)

References 22 publications

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“…Our analysis, on GSD with gapped domain walls, closely follows [30] (See also a related work [68]). The periodicity for φ 0 ∼ φ 0 + 2π imposes the quantization of its conjugate variable P φ ∈ Z.…”

Section: D2 Zero Modes and Gsd Counting For Percolating Pf|apf Domaimentioning

confidence: 81%

Theory of the disordered ν=52 quantum thermal Hall state: Emergent symmetry and phase diagram

Lian

Wang

2018

Phys. Rev. B

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Fractional quantum Hall (FQH) system at Landau level filling fraction ν = 5/2 has long been suggested to be non-Abelian, either Pfaffian (Pf) or antiPfaffian (APf) states by numerical studies, both with quantized Hall conductance σ xy = 5e 2 /2h. Thermal Hall conductances of the Pf and APf states are quantized at κ xy = 7/2 and κ xy = 3/2 respectively in a proper unit. However, a recent experiment shows the thermal Hall conductance of ν = 5/2 FQH state is κ xy = 5/2. It has been speculated that the system contains random Pf and APf domains driven by disorders, and the neutral chiral Majorana modes on the domain walls may undergo a percolation transition to a κ xy = 5/2 phase. In this work, we do perturbative and nonperturbative analyses on the domain walls between Pf and APf. We show the domain wall theory possesses an emergent SO(4) symmetry at energy scales below a threshold Λ 1 , which is lowered to an emergent U(1)×U(1) symmetry at energy scales between Λ 1 and a higher value Λ 2 , and is finally lowered to the composite fermion parity symmetry Z F 2 above Λ 2 . Based on the emergent symmetries, we propose a phase diagram of the disordered ν = 5/2 FQH system, and show that a κ xy = 5/2 phase arises at disorder energy scales Λ > Λ 1 . Furthermore, we show the gapped double-semion sector of N D compact domain walls contributes non-local topological degeneracy 2 N D −1 , causing a low-temperature peak in the heat capacity. We implement a nonperturbative method to bootstrap generic topological 1+1D domain walls (2-surface defects) applicable to any 2+1D non-Abelian topological order. We also identify potentially relevant spin topological quantum field theories (TQFTs) for various ν = 5/2 FQH states in terms of fermionic version of U(1) ±8 Chern-Simons theory×Z 8 -class TQFTs.

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Section: D2 Zero Modes and Gsd Counting For Percolating Pf|apf Domaimentioning

confidence: 81%

Theory of the disordered ν=52 quantum thermal Hall state: Emergent symmetry and phase diagram

Lian

Wang

2018

Phys. Rev. B

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“…(3.12), (3.13) and (3.15) in 4d). Such a boundary's defect Verlinde formula may be helpful to constrain other physical observables of systems of gapped boundary with defects, such as the boundary topological degeneracy [58][59][60] [57]. Moreover, by "the bulk-boundary 3d-2d correspondence," we see that ♣ The 2d boundaries/interfaces of 3d TQFT systems can be regarded as the 2d defects surfaces in 3d TQFT; via "dimensional reductions (lowering one dimension)" thus the above discussion intimately relates to ♣ The 1d boundaries/interfaces of 2d CFT, or the 1d defect lines in 2d CFT.…”

Section: Physics and Laboratory Realization And Future Directionsmentioning

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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“…The phase associated with braiding particles of types a and a is e 2πib(a,a ) , where b is a finite bilinear form b : D × D → Q/Z. The different gapped boundary conditions [3][4][5][6][7][9][10][11][12][13]15,16 for the edge between this topological phase and the vacuum are given by the different Lagrangian subgroups L ⊂ D. These are subgroups satisfying the following two conditions: (1) for any x, y ∈ L, b(x, y) = 0; and (2) for any a / ∈ L, there is some x ∈ L such that b(x, a) = 0. In other words, all of the particles in the Lagrangian subgroup braid trivially with each other while every particle that is outside the Lagrangian subgroup braids non-trivially with at least one particle in the Lagrangian subgroup.…”

Section: Appendix B: Dimensional Reduction: Understanding Loop Excitamentioning

confidence: 99%

Cheshire charge in (3+1)-dimensional topological phases

Else

Nayak

2017

Phys. Rev. B

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We show that (3 + 1)-dimensional topological phases of matter generically support loop excitations with topological degeneracy. The loops carry "Cheshire charge": topological charge that is not the integral of a locally-defined topological charge density. Cheshire charge has previously been discussed in non-Abelian gauge theories, but we show that it is a generic feature of all (3+1)-D topological phases (even those constructed from an Abelian gauge group). Indeed, Cheshire charge is closely related to non-trivial three-loop braiding. We use a dimensional reduction argument to compute the topological degeneracy of loop excitations in the (3 + 1)-dimensional topological phases associated with Dijkgraaf-Witten gauge theories. We explicitly construct membrane operators associated with such excitations in soluble microscopic lattice models in Z2 × Z2 Dijkgraaf-Witten phases and generalize this construction to arbitrary membrane-net models. We explain why these loop excitations are the objects in the braided fusion 2-category Z(2Vect ω G ), thereby supporting the hypothesis that 2-categories are the correct mathematical framework for (3 + 1)-dimensional topological phases. arXiv:1702.02148v2 [cond-mat.str-el]

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Ground-state degeneracy for Abelian anyons in the presence of gapped boundaries

Cited by 50 publications

References 22 publications

Theory of the disordered ν=52 quantum thermal Hall state: Emergent symmetry and phase diagram

Theory of the disordered ν=52 quantum thermal Hall state: Emergent symmetry and phase diagram

Quantum statistics and spacetime topology: Quantum surgery formulas

Cheshire charge in (3+1)-dimensional topological phases

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