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NMR-active nuclei have energy levels called 'spin states' that are associated with the orientations of their nuclear magnetic moments. In the absence of a magnetic field, the nuclear magnetic moments are randomly oriented, and the spin states are degenerate. When an external magnetic field is applied, the spin states have only 2 + 1 orientations available to them. A proton with = ½ has two available orientations. Similarly, for a quadrupolar nucleus with a nuclear spin value of one, the...
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Two NMR-active nuclei bonded to a central atom can be involved in geminal or two-bond coupling. Geminal coupling is commonly seen between diastereotopic protons in chiral molecules and unsymmetrical alkenes, among others.
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Spin–Spin Coupling: One-Bond Coupling01:17

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Coupling interactions are strongest between NMR-active nuclei bonded to each other, where spin information can be transmitted directly through the pair of bonding electrons. While nuclei polarize their electrons to the opposite spins, the bonding electron pair has opposite spins. Configurations with antiparallel nuclear spins are expected to be lower in energy. When coupling makes antiparallel states more favorable, J is considered to have a positive value. The one-bond coupling constant, 1J,...
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In bromoethane, the three methyl protons are coupled to the two methylene protons that are three bonds away. In accordance with the n+1 rule, the signal from the methyl protons is split into three peaks with 1:2:1 relative intensities. The methylene protons appear as a quartet, with the relative intensities of 1:3:3:1.
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Vicinal or three-bond coupling is commonly observed between protons attached to adjacent carbons. Here, nuclear spin information is primarily transferred via electron spin interactions between adjacent C‑H bond orbitals. This generally favors the antiparallel arrangement of spins, so 3J values are usually positive.
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The spin state of an NMR-active nucleus can have a slight effect on its immediate electronic environment. This effect propagates through the intervening bonds and affects the electronic environments of NMR-active nuclei up to three bonds away; occasionally, even farther. This phenomenon is called spin–spin coupling or J-coupling. Coupling interactions are mutual and result in small changes in the absorption frequencies of both nuclei involved. While nuclei of the same element are involved...
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Extracting Topological Spins from Bulk Multipartite Entanglement.

Yarden Sheffer1, Ady Stern1, Erez Berg1

  • 1Weizmann Institute of Science, Department of Condensed Matter Physics, Rehovot 7610001, Israel.

Physical Review Letters
|September 10, 2025
PubMed
Summary
This summary is machine-generated.

We developed new entanglement measures to identify topological phases of matter. These methods use multiple copies of a quantum state to extract topological invariants, offering a more refined way to classify phases.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Information Theory

Background:

  • Topologically ordered phases are exotic states of matter characterized by long-range entanglement.
  • Identifying these phases is crucial for understanding quantum materials and developing quantum technologies.

Purpose of the Study:

  • To develop novel methods for identifying 2+1 dimensional topologically ordered phases.
  • To introduce new entanglement measures capable of distinguishing between different topological phases.

Main Methods:

  • Utilizing measurements on the ground-state wave function.
  • Defining bulk multipartite entanglement measures based on permutation operators acting on multiple replicas of the wave function.
  • Calculating topological invariants (quantum dimension and topological spin) of anyons.

Main Results:

  • Introduced a series of entanglement measures that successfully extract topological invariants ∑_{a}d_{a}^{2}θ_{a}^{r} for nonchiral topological order.
  • Demonstrated that these measures provide information beyond conventional methods like topological entanglement entropy.
  • Showed that the proposed procedure is optimal in terms of the number of wave function replicas required.

Conclusions:

  • The developed entanglement measures offer a powerful and refined tool for classifying topological phases of matter.
  • The findings are generalizable to chiral topological states.
  • This work advances the understanding of topological order and its detection.