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Chirality is most prevalent in carbon-based tetrahedral compounds, but this important facet of molecular symmetry extends to sp3-hybridized nitrogen, phosphorus and sulfur centers, including trivalent molecules with lone pairs. Here, the lone pair behaves as a functional group in addition to the other three substituents to form an analogous tetrahedral center that can be chiral.
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Molecules that possess multiple chiral centers can afford a large number of stereoisomers. For instance, while some molecules like 2-butanol have one chiral center, defined as a tetrahedral carbon atom with four different substituents attached, several molecules like butane-2,3-diol have multiple chiral centers. A simple formula to predict the number of stereoisomers possible for a molecule with n chiral centers is 2n. However, there can be a lower number where some of the stereoisomers are...
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Cyclohexane does not exist in a planar form due to the high angle and torsional strain it would experience in the planar structure. Instead, it adopts non-planar chair and boat conformations.
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¹H NMR: Complex Splitting01:13

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A proton M that is coupled to a proton X results in doublet signals for M. However, NMR-active nuclei can be simultaneously coupled to more than one nonequivalent nucleus. When M is coupled to a second proton A, such as in styrene oxide, each peak in the doublet is split into another doublet.
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In the AX proton spin system, proton A can sense the two spin states of a coupled proton X, resulting in a doublet NMR signal with two peaks of equal (1:1) intensity. When proton A is coupled to two equivalent protons (AX2 spin system), the spin states of each X can be aligned with or against the external field, creating three possible scenarios. This results in a 1:2:1  triplet signal, where the central peak corresponds to the chemical shift of A and is twice as large or intense as the...
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In this lesson, we delve into the role of ring conformation and its stability, which determines the spatial arrangement and, consequently, the molecular symmetry and stereoisomerism of cyclic compounds. 1,2-Dimethylcyclohexane is used as a case study to evaluate the possible number of stereoisomers. Here, given the multiple (n = 2) chiral centers, there are 2n = 4 possible configurations that lack a plane of symmetry, as the ring skeleton exists in a non-planar chair conformation. In addition,...
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Double-Scaling Limit in the Principal Chiral Model: A New Noncritical String?

Vladimir Kazakov1, Evgeny Sobko2, Konstantin Zarembo3

  • 1Laboratoire de physique de l'École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 24 rue Lhomond, 75005 Paris, France and Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland.

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We present a non-perturbative study of the large-N expansion in the two-dimensional principal chiral model (PCM). Our findings reveal a double-scaling limit similar to noncritical string theory, suggesting a dual string with emergent dimensions.

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Area of Science:

  • High Energy Physics
  • Quantum Field Theory
  • String Theory

Background:

  • The principal chiral model (PCM) is a key theoretical framework in quantum field theory.
  • Understanding the large-N expansion is crucial for non-perturbative studies.
  • Previous solutions for PCM were limited, particularly at finite chemical potentials.

Purpose of the Study:

  • To systematically investigate the large-N expansion of the two-dimensional SU(N)×SU(N) PCM non-perturbatively.
  • To analyze the ground state properties at fixed chemical potential.
  • To explore the model's behavior at strong coupling and its potential dualities.

Main Methods:

  • Utilizing an iterative procedure to solve Bethe ansatz equations order-by-order in 1/N.
  • Computing the first few orders of the large-N expansion explicitly.
  • Applying near-threshold resummation techniques to the expansion.

Main Results:

  • A systematic enhancement pattern was observed at strong coupling.
  • The computed orders necessitate resummation of the large-N expansion.
  • The resulting double-scaling limit shows strong resemblance to c=1 noncritical string theory.

Conclusions:

  • The double-scaled PCM is potentially dual to a noncritical string theory.
  • This duality involves a (2+1)-dimensional target space.
  • An additional dimension dynamically emerges from the SU(N) Dynkin diagram in this duality.