Related Experiment Video
Updated: Oct 20, 2025
09:06
Visualizing Uniaxial-strain Manipulation of Antiferromagnetic Domains in Fe1+YTe Using a Spin-polarized Scanning Tunneling Microscope
Published on: March 24, 2019
8.3K
Magnetization profiles at the upper critical dimension as solutions of the integer Yamabe problem
Alessandro Galvani1, Giacomo Gori2,3, Andrea Trombettoni1,3,4
1SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy.
Physical Review. E
|September 16, 2021
Summary
This study links magnetization profiles in scalar field models to the Yamabe problem, finding solutions in terms of Weierstrass elliptic functions. These findings confirm recent results and show magnetization profiles solve the Yamabe equation at critical dimensions.
Area of Science:
- Mathematical Physics
- Statistical Mechanics
- Differential Geometry
Background:
- Magnetization profiles in bounded domains are related to geometric problems.
- The Yamabe problem seeks metrics of constant curvature.
- Scalar field models with marginal interactions are relevant in physics.
Purpose of the Study:
- To investigate the connection between magnetization profiles and Yamabe problem solutions.
- To derive general formulas for magnetization profiles at upper critical dimensions.
- To confirm theoretical predictions with simulations.
Main Methods:
- Analysis of saddle-point equations for magnetization.
- Derivation of solutions using Weierstrass elliptic functions.
- Comparison with Monte Carlo simulations for the 4D Ising model.
Main Results:
- General formulas for magnetization profiles derived in terms of Weierstrass elliptic functions.
- Connection established between zeros/poles of solutions and boundary conditions.
- Magnetization profiles shown to be solutions of the integer Yamabe equation.
Conclusions:
- The study confirms theoretical predictions at upper critical dimensions.
- Explicit agreement found between theoretical results and Monte Carlo simulations.
- The findings offer a deeper understanding of the interplay between statistical mechanics and geometry.
Related Concept Videos
01:24Potential Due to a Magnetized Object
391
Magnetic dipoles in magnetic materials are aligned when placed under an external magnetic field. For paramagnets and ferromagnets, dipole alignment occurs in the direction of the magnetic field. However, the dipoles align opposite to the field in the case of diamagnets. This state of magnetic polarization due to the external field is called magnetization. Magnetization is defined as the dipole moment per unit volume. It plays a similar role to polarization in electrostatics.
The vector...
The vector...
391
01:14NMR Spectrometers: Resolution and Error Correction
820
When magnetic nuclei in a sample achieve resonance and undergo relaxation, the signal detected in NMR is an approximately exponential free induction decay. Fourier transform of an exponential decay yields a Lorentzian peak in the frequency domain. Lorentzian peaks in an NMR spectrum are defined by their amplitude, full width at half maximum, and position, where the peak width is governed by the spin-spin relaxation time alone. In real experiments, however, the applied magnetic field is rendered...
820
01:31Magnetic Susceptibility and Permeability
1.5K
In linear magnetic materials, like paramagnets and diamagnets, magnetization is proportional to the magnetic field intensity. The constant of proportionality, a dimensionless number, is called magnetic susceptibility. The value of the susceptibility depends on the type of material.
When diamagnetic materials are placed under an external magnetic field, the moments opposite to the field are induced. Hence, the susceptibility for diamagnets has a minimal negative value of 10-5–10-6. Since...
When diamagnetic materials are placed under an external magnetic field, the moments opposite to the field are induced. Hence, the susceptibility for diamagnets has a minimal negative value of 10-5–10-6. Since...
1.5K
01:23Atomic Nuclei: Nuclear Relaxation Processes
782
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.
782
01:28Magnetostatic Boundary Conditions
1.2K
An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...
1.2K
01:26Divergence and Curl of Magnetic Field
3.3K
The magnetic field due to a volume current distribution given by the Biot–Savart Law can be expressed as follows:
3.3K