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Momentum polarization: an entanglement measure of topological spin and chiral central charge
Topologically ordered states are quantum states of matter with topological
ground state degeneracy and quasi-particles carrying fractional quantum numbers
and fractional statistics. The topological spin is an
important property of a topological quasi-particle, which is the Berry phase
obtained in the adiabatic self-rotation of the quasi-particle by . For
chiral topological states with robust chiral edge states, another fundamental
topological property is the edge state chiral central charge . In this paper
we propose a new approach to compute the topological spin and chiral central
charge in lattice models by defining a new quantity named as the momentum
polarization. Momentum polarization is defined on the cylinder geometry as a
universal subleading term in the average value of a "partial translation
operator". We show that the momentum polarization is a quantum entanglement
property which can be computed from the reduced density matrix, and our
analytic derivation based on edge conformal field theory shows that the
momentum polarization measures the combination of
topological spin and central charge. Numerical results are obtained for two
example systems, the non-Abelian phase of the honeycomb lattice Kitaev model,
and the Laughlin state of a fractional Chern insulator described by a
variational Monte Carlo wavefunction. The numerical results verifies the
analytic formula with high accuracy, and further suggests that this result
remains robust even when the edge states cannot be described by a conformal
field theory. Our result provides a new efficient approach to characterize and
identify topological states of matter from finite size numerics.Comment: 13 pages, 8 figure
Enhancing Hydrogen Generation Through Nanoconfinement of Sensitizers and Catalysts in a Homogeneous Supramolecular Organic Framework.
Enrichment of molecular photosensitizers and catalysts in a confined nanospace is conducive for photocatalytic reactions due to improved photoexcited electron transfer from photosensitizers to catalysts. Herein, the self-assembly of a highly stable 3D supramolecular organic framework from a rigid bipyridine-derived tetrahedral monomer and cucurbit[8]uril in water, and its efficient and simultaneous intake of both [Ru(bpy)3 ]2+ -based photosensitizers and various polyoxometalates, that can take place at very low loading, are reported. The enrichment substantially increases the apparent concentration of both photosensitizer and catalyst in the interior of the framework, which leads to a recyclable, homogeneous, visible light-driven photocatalytic system with 110-fold increase of the turnover number for the hydrogen evolution reaction
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