Long-range magnetic order and the darwin lagrangian

Area of Science:

• Computational Physics
• Condensed Matter Physics
• Quantum Mechanics

Background:

• Understanding electron behavior in confined systems is crucial for materials science.
• The Darwin magnetic interaction influences electron dynamics in confined systems.
• Electron crystallization into Wigner lattices is a key phenomenon at low densities.

Purpose of the Study:

• To simulate a finite system of N confined electrons, including Darwin magnetic interactions.
• To investigate the ground state properties of confined electrons in two and three dimensions.
• To determine the critical density for the transition from a static Wigner lattice to a dynamic state.

Main Methods:

• Simulations of finite electron systems (N electrons).
• Inclusion of Darwin magnetic interaction.
• Steepest descent quenching adapted for velocity-dependent potentials to locate lowest-energy states.
• Analysis in two and three dimensions.

Main Results:

• Below a critical density, the ground state is a static Wigner lattice.
• Above the critical density, the ground state exhibits non-zero kinetic energy.
• The critical density's dependence on N varies with confinement type (exponential vs. harmonic).
• An antiferromagnetic cluster forms for harmonic confinement in two dimensions.

Conclusions:

• The ground state of confined electrons is sensitive to density, confinement, and dimensionality.
• Electron systems can transition from ordered static states to dynamic states based on density.
• Confinement geometry and dimensionality dictate emergent magnetic properties like antiferromagnetism.