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In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis.
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Nuclear relaxation restores the equilibrium population imbalance and can occur via spin–lattice or spin–spin mechanisms, which are first-order exponential decay processes.
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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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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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Atomic Nuclei: Nuclear Spin State Population Distribution01:14

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Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.
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Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses
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Non-perturbative effects in spin glasses.

Michele Castellana1, Giorgio Parisi2

  • 1Joseph Henry Laboratories of Physics and Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544, United States.

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|March 4, 2015
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Summary
This summary is machine-generated.

Numerical simulations of the hierarchical Edwards-Anderson model reveal discrepancies between Monte Carlo and renormalization-group methods in non-mean-field regions. Strong non-perturbative effects appear to govern the spin glass transition.

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Area of Science:

  • Statistical mechanics
  • Condensed matter physics
  • Computational physics

Background:

  • Spin glasses exhibit complex magnetic behavior.
  • Hierarchical interactions introduce complexity beyond standard models.
  • Understanding phase transitions in disordered systems is crucial.

Purpose of the Study:

  • To numerically investigate the hierarchical Edwards-Anderson (HEA) model with an external magnetic field.
  • To compare results from Monte Carlo (MC) simulations with renormalization-group (RG) analysis.
  • To explore the behavior of the HEA model in both mean-field (MF) and non-mean-field (NMF) regions.

Main Methods:

  • Monte Carlo (MC) simulations were performed for the d-dimensional HEA model.
  • Renormalization-group (RG) analysis was employed, treating the critical fixed point as a perturbation.
  • Simulations and RG were conducted for d ≥ 4 (MF) and d < 4 (NMF) dimensions.

Main Results:

  • MC and RG methods show agreement in the MF region, predicting a transition and compatible critical exponents.
  • Significant disagreement arises in the NMF region: MC data suggests a transition, while RG predicts no perturbative critical fixed point.
  • MC estimates for the critical exponent ν in the NMF region are approximately double the classical value, despite system dimension being close to the upper-critical dimension.

Conclusions:

  • The hierarchical Edwards-Anderson model exhibits distinct behaviors in MF and NMF regimes.
  • Strong non-perturbative effects are indicated to govern the phase transition in the NMF region.
  • Discrepancies highlight limitations of perturbative RG approaches for complex spin glass systems.