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Related Concept Videos

¹H NMR: Long-Range Coupling01:27

¹H NMR: Long-Range Coupling

The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
In alkenes, spin information is communicated via σ–π overlap, as seen in allylic (four-bond) and homoallylic (five-bond) couplings. These coupling interactions are stronger when the σ bond is parallel to the alkene π orbitals.
Homogeneous Equilibria for Gaseous Reactions02:15

Homogeneous Equilibria for Gaseous Reactions

Homogeneous Equilibria for Gaseous Reactions
For gas-phase reactions, the equilibrium constant may be expressed in terms of either the molar concentrations (Kc) or partial pressures (Kp) of the reactants and products. A relation between these two K values may be simply derived from the ideal gas equation and the definition of molarity. According to the ideal gas equation:
Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)01:20

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

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.
The central atom need not be NMR-active because its electrons are affected by the electron polarization of the spin-active atoms. However, spin information is transmitted less effectively than in one-bond coupling, and 2J values are usually weaker than 1J values. The energy of...
Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion03:48

Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion

Although gaseous molecules travel at tremendous speeds (hundreds of meters per second), they collide with other gaseous molecules and travel in many different directions before reaching the desired target. At room temperature, a gaseous molecule will experience billions of collisions per second. The mean free path is the average distance a molecule travels between collisions. The mean free path increases with decreasing pressure; in general, the mean free path for a gaseous molecule will be...
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The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger...
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Van der Waals Interactions

Atoms and molecules interact with each other through intermolecular forces. These electrostatic forces arise from attractive or repulsive interactions between particles with permanent, partial, or temporary charges. The intermolecular forces between neutral atoms and molecules are ion–dipole, dipole–dipole, and dispersion forces, collectively known as van der Waals forces.Polar molecules have a partial positive charge on one end and a partial negative charge on the other end of the molecule,...

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Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform
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Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Published on: August 2, 2019

Local interactions and non-abelian quantum loop gases.

Matthias Troyer1, Simon Trebst, Kirill Shtengel

  • 1Theoretische Physik, Eidgenössische Technische Hochschule Zürich, 8093 Zürich, Switzerland.

Physical Review Letters
|December 31, 2008
PubMed
Summary
This summary is machine-generated.

Gapped non-Abelian quantum loop gases require nonlocal interactions, unlike their Abelian counterparts. Perturbing a non-Abelian state with local interactions drives it to the Abelian phase.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Information Theory
  • Topological Phases of Matter

Background:

  • Two-dimensional quantum loop gases are fundamental models for topological ground states.
  • These states host exotic excitations, classified as Abelian or non-Abelian anyons.

Purpose of the Study:

  • To investigate the conditions required for realizing gapped non-Abelian quantum loop gases.
  • To understand the stability of non-Abelian phases against local perturbations.

Main Methods:

  • Analysis of Hamiltonians and ground-state wave functions for quantum loop gases.
  • Perturbation theory to study phase transitions.
  • Measurement of loop Hausdorff dimensions to characterize topological order.
  • Investigation of local operator correlations.

Main Results:

  • Gapped non-Abelian loop gases necessitate nonlocal interactions or nontrivial inner products.
  • Local perturbations applied to anticipated non-Abelian ground states drive the system into the Abelian phase.
  • Local quantum critical behavior is observed when all equal-time correlations of local operators decay exponentially.

Conclusions:

  • Achieving stable, gapped non-Abelian topological phases in quantum loop gases is challenging due to the need for nonlocal interactions.
  • The transition from non-Abelian to Abelian phases can be detected through topological and correlation measurements.