1,421 research outputs found
Non-Abelian Fusion, Shrinking and Quantum Dimensions of Abelian Gauge Fluxes
Braiding and fusion rules of topological excitations are indispensable
topological invariants in topological quantum computation and topological
orders. While excitations in 2D are always particle-like anyons, those in 3D
incorporate not only particles but also loops -- spatially nonlocal objects --
making it novel and challenging to study topological invariants in higher
dimensions. While 2D fusion rules have been well understood from bulk
Chern-Simons field theory and edge conformal field theory, it is yet to be
thoroughly explored for 3D fusion rules from higher dimensional bulk
topological field theory. Here, we perform a field-theoretical study on (i) how
loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into
particles in the path integral, which generates fusion rules, loop-shrinking
rules, and descendent invariants, e.g., quantum dimensions. We first assign a
gauge-invariant Wilson operator to each excitation and determine the number of
distinct excitations through equivalence classes of Wilson operators. Then, we
adiabatically shift two Wilson operators together to observe how they fuse and
are split in the path integral; despite the Abelian nature of the gauge fluxes
carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we
adiabatically deform world-sheets of unknotted loops into world-lines and
examine the shrinking outcomes; we find that the resulting loop-shrinking rules
are algebraically consistent to fusion rules. Interestingly, fusing a pair of
loop and anti-loop may generate multiple vacua, but fusing a pair of anyon and
anti-anyon in 2D has one vacuum only. By establishing a field-theoretical
ground for fusion and shrinking in 3D, this work leaves intriguing directions,
e.g., symmetry enrichment, quantum gates, and physics of braided monoidal
2-category of 2-group.Comment: Title adjusted. Abstract, Intro and Discussions revised. about 30
pages, 5 figures. 9 table
Continuum field theory of 3D topological orders with emergent fermions and braiding statistics
Universal topological data of topologically ordered phases can be captured by
topological quantum field theory in continuous space time by taking the limit
of low energies and long wavelengths. While previous continuum
field-theoretical studies of topological orders in D real space focus on
either self-statistics, braiding statistics, shrinking rules, fusion rules or
quantum dimensions, it is yet to systematically put all topological data
together in a unified continuum field-theoretical framework. Here, we construct
the topological field theory with twisted terms (e.g., and )
as well as a -matrix term, in order to simultaneously explore all such
topological data and reach anomaly-free topological orders. Following the
spirit of the famous -matrix Chern-Simons theory of D topological orders,
we present general formulas and systematically show how the -matrix
term confines topological excitations, and how self-statistics of particles is
transmuted between bosonic one and fermionic one. In order to reach
anomaly-free topological orders, we explore, within the present continuum
field-theoretical framework, how the principle of gauge invariance
fundamentally influences possible realizations of topological data. More
concretely, we present the topological actions of (i) particle-loop braidings
with emergent fermions, (ii) multiloop braidings with emergent fermions, and
(iii) Borromean-Rings braidings with emergent fermions, and calculate their
universal topological data. Together with the previous efforts, our work paves
the way toward a more systematic and complete continuum field-theoretical
analysis of exotic topological properties of D topological orders. Several
interesting future directions are also discussed
Research on Self-Calibration of HF Ground Wave Radar Antenna Arrays
Since the performance of high-resolution direction finding algorithm for HF Ground Wave Radar (GWR) is severely degraded by sensor phase and amplitude errors, the radar system's phase calibration is the prerequisite of keeping the radar working in order. According to the characteristic of HF GWR's sea echo, this paper, based on an arbitrary triangular array, presents that Space -Time DOA(direction of arrival) Matrix Method, which is used to estimate 2D DOA under ideal conditions, can be used to estimate planar wave's DOA and sensor phase and amplitude errors simultaneously so as to achieve self-calibration. Its validity is verified not only by computer simulation, but also by comparing treatment results of measured data before and after calibration with the GPS-measured result
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