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Related Concept Videos

Potential Due to a Magnetized Object01:24

Potential Due to a Magnetized Object

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...
Ferromagnetism01:31

Ferromagnetism

Materials like iron, nickel, and cobalt consist of magnetic domains, within which the magnetic dipoles are arranged parallel to each other. The magnetic dipoles are rigidly aligned in the same direction within a domain by quantum mechanical coupling among the atoms. This coupling is so strong that even thermal agitation at room temperature cannot break it. The result is that each domain has a net dipole moment. However, some materials have weaker coupling, and are ferromagnetic at lower...
Paramagnetism01:30

Paramagnetism

Paramagnets are materials with unpaired electrons that possess a finite magnetic moment. In the absence of a magnetic field, these moments are randomly oriented, and thus the net moment is zero. Under an external field, a torque acting on the moments tends to align them along the field's direction. However, the random thermal motion of electrons produces a torque opposite to the external field and tries to disorient the moments. These two competing effects align only a few moments along the...
Magnetostatic Boundary Conditions01:28

Magnetostatic Boundary Conditions

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...
Magnetic Susceptibility and Permeability01:31

Magnetic Susceptibility and Permeability

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...
Atomic Nuclei: Nuclear Relaxation Processes01:23

Atomic Nuclei: Nuclear Relaxation Processes

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. This...

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Magnetometric Characterization of Intermediates in the Solid-State Electrochemistry of Redox-Active Metal-Organic Frameworks
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Constraint effective action and critical correlation functions at fixed magnetization.

Félix Rose1,2, Adam Rançon3,4, Ivan Balog5

  • 1CNRS, CY Cergy Paris Université, Laboratoire de Physique Théorique et Modélisation, F-95302 Cergy-Pontoise, France.

Physical Review. E
|June 19, 2026
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Summary

We extended the functional renormalization group (FRG) method to calculate critical properties of the Ising model. This approach accurately computes momentum-dependent observables, confirming its robustness for critical phenomena research.

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Area of Science:

  • Statistical Physics
  • Condensed Matter Physics
  • Computational Physics

Background:

  • The functional renormalization group (FRG) is a powerful tool for studying critical phenomena.
  • Calculating momentum-dependent observables at fixed magnetization presents a significant challenge.
  • Existing methods often struggle with accuracy in lower dimensions.

Purpose of the Study:

  • To extend the FRG framework for computing critical probability distributions and momentum-dependent observables.
  • To apply the extended FRG to the Ising universality class at fixed magnetization.
  • To benchmark the FRG results against Monte Carlo simulations.

Main Methods:

  • Derivation of exact flow equations for the constraint effective action at fixed magnetization.
  • Numerical solution of flow equations at the second order of the derivative expansion (DE2).
  • Application to two- and three-dimensional Ising systems.

Main Results:

  • Extraction of universal rate functions and momentum-dependent correlation functions.
  • Accurate reproduction of critical observables in 3D, demonstrating method convergence.
  • DE2 is shown to be necessary for accurate critical point description in 2D, unlike lower-order approximations.

Conclusions:

  • The extended FRG approach is robust for calculating both zero- and finite-momentum critical observables at fixed magnetization.
  • The DE2 provides accurate results in 3D and is essential for 2D critical phenomena.
  • The study confirms the utility of FRG for complex critical phenomena studies.