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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.
Qualitatively, any spin plus-half nucleus polarizes the spins of its electrons to the minus-half state. Consequently, the paired electron in the hydrogen–carbon bond must...
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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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Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)01:20

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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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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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Wrapping Corrections for Long-Range Spin Chains.

Tamas Gombor1

  • 1MTA-ELTE "Momentum" Integrable Quantum Dynamics Research Group, Department of Theoretical Physics, Eötvös Loránd University, H-1117 Budapest, Pázmány Péter sétány 1/A, Hungary and Holographic QFT Group, Wigner Research Centre for Physics, H-1121 Budapest, Konkoly-Thege Miklós út 29-33, Hungary.

Physical Review Letters
|January 13, 2023
PubMed
Summary
This summary is machine-generated.

This study generalizes integrable spin chain transfer matrices to long-range models, defining conserved charges for finite chains. This framework captures wrapping corrections crucial for gauge-string duality research.

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Area of Science:

  • Theoretical physics
  • Quantum mechanics
  • String theory

Background:

  • Long-range spin chains are fundamental to understanding gauge-string duality.
  • Existing models primarily focus on infinite systems, limiting finite-size analyses.
  • Integrable medium-range spin chains have recently seen advancements in transfer matrix methods.

Purpose of the Study:

  • To generalize transfer matrices from integrable medium-range spin chains to long-range models.
  • To define conserved charges applicable to finite-size spin chains of any length.
  • To incorporate wrapping corrections into the study of integrable finite-size long-range spin chains.

Main Methods:

  • Generalization of transfer matrix techniques.
  • Construction of conserved charges for arbitrary spin chain lengths.
  • Analysis of the resulting spin chain spectra.

Main Results:

  • A novel definition for integrable finite-size long-range spin chains.
  • Identification of a large set of conserved charges valid for every chain length.
  • The developed spectrum inherently includes wrapping corrections.

Conclusions:

  • The introduced transfer matrices provide a powerful tool for studying finite-size effects in long-range spin chains.
  • This work bridges the gap between infinite and finite models in gauge-string duality.
  • The findings offer new avenues for exploring quantum integrable systems and their applications.