3,643 research outputs found
Boundary Degeneracy of Topological Order
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
allows us to derive the ground state degeneracy formula in terms of boundary
gapping conditions, which encodes more than the fusion algebra of
fractionalized quasiparticles. We apply our theory to Kitaev's toric code and
Levin-Wen string-net models. We predict that the toric code and
double-semion model (more generally, the gauge theory and the non-chiral fractional quantum Hall state at even integer )
can be numerically and experimentally distinguished, by measuring their
boundary degeneracy on an annulus or a cylinder.Comment: 15 pages, 4 figures. v3: the expanded version, add new tables for
clarification, with some new correction
Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D 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
and a 4-cocycle twist of 's cohomology group
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. We express
the SL generators and in
terms of the gauge group and the 4-cocycle . As we compactify one
of the spatial directions into a compact circle with a gauge flux
inserted, we can use the generators and of
an SL subgroup to study the dimensional reduction of the 3D
topological order to a direct sum of degenerate
states of 2D topological orders in different flux
sectors: .
The 2D topological orders are described by 2D gauge
theories of the group twisted by the 3-cocycles ,
dimensionally reduced from the 4-cocycle . We show that the
SL generators, and , 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.Comment: 36 pages, many figures, 17 tables. v3: Accepted by Phys. Rev. B. Add
acknowledgements to Louis H. Kauffma
Weak Decays of Doubly-Heavy Tetraquarks
We study the weak decays of exotic tetraquark states
with two heavy quarks. Under the SU(3) symmetry for light quarks, these
tetraquarks can be classified into an octet plus a singlet: . We will concentrate on the octet tetraquarks with
, and study their weak decays, both semileptonic and nonleptonic.
Hadron-level effective Hamiltonian is constructed according to the irreducible
representations of the SU(3) group. Expanding the Hamiltonian, we obtain the
decay amplitudes parameterized in terms of a few irreducible quantities. Based
on these amplitudes, relations for decay widths are derived, which can be
tested in future. We also give a list of golden channels that can be used to
look for these states at various colliders.Comment: 14 pages,3 figure
Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
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 , whose entries are the fusion-space dimensions
, to label different types of gapped domain walls. By studying
many examples, we find evidence that the tunneling matrices are powerful
quantities to classify different types of gapped domain walls. Since a gapped
boundary is a gapped domain wall between a bulk topological order and the
vacuum, regarded as the trivial topological order, our theory of gapped domain
walls inclusively contains the theory of gapped boundaries. In addition, we
derive a topological ground state degeneracy formula, applied to arbitrary
orientable spatial 2-manifolds with gapped domain walls, including closed
2-manifolds and open 2-manifolds with gapped boundaries.Comment: 5+9 pages, 3 figures, updated references, fixed typos and
refinements, added proof for equivalence to Lagrangian subgroups in Abelian
case
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