Relativistic theory for coupled orbital and spin angular momentum dynamics in magnetic systems

This paper develops a complete relativistic theory based on the Dirac-Kohn-Sham framework to derive the coupled dynamics of spin and orbital angular momenta in magnetic systems, demonstrating that while the total angular momentum is not conserved under external electromagnetic fields in the general case, it remains conserved under the atomistic Heisenberg approximation despite the non-conservation of individual spin and orbital components.

The Gist

Imagine a magnetic material, like a tiny piece of iron, as a bustling city filled with billions of tiny, spinning tops. These tops are electrons. In this city, every electron has two ways of moving: it spins on its own axis (like a spinning top), and it also orbits around the center of the city (like a planet around the sun).

In physics, we call the spinning "spin" and the orbiting "orbital" motion. Together, they make up the electron's total "angular momentum." Think of angular momentum as the total "oomph" or rotational energy the electron has.

For a long time, scientists studying how magnets react to super-fast laser pulses (which happen in trillionths of a second) mostly focused on the spinning tops. They often ignored the orbiting planets, thinking they were too small or too "stuck" to matter. However, this new paper argues that to truly understand what happens when you zap a magnet with a laser, you have to watch both the spin and the orbit, and how they talk to each other.

Here is the story the paper tells, broken down into simple parts:

1. The Rules of the Game (The Theory)

The authors built a new set of mathematical rules based on Einstein's theory of relativity (specifically the Dirac equation). Think of this as upgrading the rulebook for our city of spinning tops.

They started with the most accurate, high-speed description of electrons and then simplified it just enough to be useful, creating what they call an "Extended Pauli Hamiltonian." You can think of this as a new, more detailed instruction manual that accounts for how the spinning and orbiting parts of the electron interact with each other and with outside forces, like a laser pulse or a magnetic field.

2. The Dance Without Outside Help

First, they looked at what happens when the city is left alone, with no lasers or outside magnets interfering.

• The Spin and Orbit Swap: They found that the spinning tops and the orbiting planets are constantly trading energy. One spins faster while the other slows down, and vice versa. It's like two dancers holding hands; if one spins faster, the other has to adjust.
• The Total is Safe: Even though they are swapping energy back and forth, the total amount of "oomph" (total angular momentum) in the system stays exactly the same. Nothing is lost; it just moves from the spin to the orbit or back again.

3. The Laser Pulse (The Outside Intruder)

Next, they turned on the "laser" (an electromagnetic field). This is like someone walking into the city and starting to push the dancers around.

• The Total "Oomph" Changes: When the laser hits, the total angular momentum is no longer safe. The laser adds or removes energy from the system. It's like the dancers are now being pushed by an external wind; the total energy of the dance floor changes because of the wind.
• The Paper's Big Discovery: The authors showed that under these laser conditions, the total angular momentum is not conserved. This answers a big debate in the scientific community about whether angular momentum is strictly conserved during ultrafast demagnetization (when a magnet loses its magnetism very quickly). The paper says: "No, not if a laser is involved."

4. The Neighborhood Effect (Exchange Interaction)

Finally, the authors looked at how the electrons talk to their immediate neighbors. In magnets, electrons don't just act alone; they are influenced by the electrons right next to them. This is called "exchange interaction."

They tested two different ways of modeling this neighborhood:

• The General Neighborhood: If you assume the electrons interact in a complex, messy way (a general "Kohn-Sham" field), the total angular momentum is not conserved, even without a laser. The rules get too messy to keep the total count steady.
• The Atomic Neighborhood (Heisenberg Model): If you assume the electrons interact like a neat, organized neighborhood where each atom has a specific, localized spin (the "Heisenberg" approximation), something interesting happens.
  • The individual spins and orbits still swap energy and change.
  • But, when you add up everyone in the whole city, the total angular momentum is conserved again, even if a laser is hitting them!

The Bottom Line

This paper is like a detective story about the conservation of energy in a magnetic city.

1. Spin and Orbit are linked: You can't understand one without the other; they are constantly trading places.
2. Lasers break the rules: If you hit a magnet with a laser, the total angular momentum of the electrons changes. It is not a closed system anymore.
3. Neighborhood matters: How you model the interaction between atoms changes the outcome. If you treat the atoms as a specific, localized team (Heisenberg style), the total angular momentum of the whole group stays conserved, even under a laser. If you treat it as a messy, general cloud, it doesn't.

The authors conclude that to truly understand how magnets behave in ultrafast experiments, we must use this new, complete relativistic theory that tracks both the spin and the orbit, and we must be very careful about how we model the interactions between atoms.

Technical Summary

Technical Summary: Relativistic theory for coupled orbital and spin angular momentum dynamics in magnetic systems

Problem Statement
The ultrafast manipulation of spins in magnetic systems, particularly following femtosecond laser excitation, has revealed complex dynamics where magnetic moments quench on sub-picosecond timescales. A central unresolved issue in this field is the conservation of angular momentum. While the total angular momentum ($J = L + S$) is expected to be conserved in a closed system, experimental and theoretical studies have yielded conflicting results. Some theories suggest that the total angular momentum is not conserved during ultrafast spin dynamics, attributing this to simplified treatments of spin-orbit coupling (SOC) or the emission of electromagnetic radiation. Furthermore, the role of orbital angular momentum ($L$), often considered "quenched" in 3d-transition metals but significant in rare-earth systems, remains underexplored in the context of relativistic dynamics. Existing theories, such as the Landau-Lifshitz-Gilbert (LLG) equation and its inertial extensions, primarily focus on spin dynamics and often neglect the coupled evolution of orbital moments or treat them within non-relativistic approximations that fail to capture the full conservation laws.

Methodology
The authors develop a complete relativistic theory starting from the Dirac-Kohn-Sham (DKS) Hamiltonian for electrons in a magnetic system under an electromagnetic field.

1. Hamiltonian Formulation: They apply a Foldy-Wouthuysen (FW) unitary transformation to the DKS Hamiltonian to derive an extended Pauli Hamiltonian ($H_{EPH}$) valid up to order $1/c^2$. This effective Hamiltonian includes non-relativistic terms, relativistic corrections (such as the Darwin term and higher-order SOC), and the coupling to external electromagnetic fields (vector potential $A$ and scalar potential $\Phi$). Crucially, this formulation accounts for the spin-polarized Kohn-Sham exchange field ($B_{xc}$) and its relativistic corrections ($B_{xc}^{corr}$).
2. Equations of Motion: Using the Heisenberg equation of motion, the authors derive the time evolution of the orbital angular momentum ($L$), spin angular momentum ($S$), and total angular momentum ($J$). The derivation is performed in two stages:
   • First, for nonmagnetic materials (neglecting $B_{xc}$) to isolate the effects of external electromagnetic fields and intrinsic SOC.
   • Second, for magnetic systems, incorporating the exchange interaction. They specifically analyze a general Kohn-Sham exchange field and then adopt an atomistic Heisenberg approximation, where the exchange interaction is modeled as a bilinear coupling between localized atomic spins ($H_{Hei} = -\sum J_{ij} S_i \cdot S_j$).
3. Conservation Analysis: The commutation relations of $L^2$, $S^2$, and $J^2$ with the derived Hamiltonians are evaluated to determine which quantities are conserved under various conditions (absence/presence of external fields, general vs. Heisenberg exchange).

Key Contributions and Results

• Coupled Dynamics without Exchange: In the absence of a spin-polarized exchange field, the authors show that while the individual spin ($S$) and orbital ($L$) angular momenta are not conserved due to intrinsic spin-orbit coupling, the total angular momentum $J = L + S$ is conserved in the absence of external perturbations. However, the application of an external electromagnetic field (e.g., a laser pulse) breaks the conservation of $J$. The derived equations of motion reveal that $J$ precesses around the external field and undergoes transverse relaxation, with the non-conservation arising from terms involving the field and its time derivative.
• Conservation of Magnitudes: A notable finding is that within this relativistic framework, the magnitude of the spin angular momentum ($S^2$) remains conserved even in the presence of the interaction Hamiltonian, whereas the magnitude of the orbital angular momentum ($L^2$) is not conserved when external fields are present.
• Role of Exchange Interaction:
  • For a general Kohn-Sham exchange field, the total angular momentum is not conserved, even without external fields. The relativistic corrections to the exchange interaction and the spatial dependence of the field lead to a complex redistribution of angular momentum among spin, orbital, and lattice degrees of freedom.
  • For an atomic Heisenberg exchange interaction (isotropic, localized spins), the situation changes significantly. While the individual atomic spin and orbital angular momenta are not conserved (they exchange angular momentum via relativistic corrections and external fields), the total atomic angular momentum ($J_{tot} = \sum (L_i + S_i)$) remains conserved, even in the presence of an external electromagnetic field.
• Mechanism of Non-Conservation: The paper identifies that the non-conservation of total angular momentum in the general case (and in the presence of external fields for the non-Heisenberg case) stems from the non-commutation of the angular momentum operators with the exchange field and the external field terms in the Hamiltonian. Specifically, the interaction terms involving $E \times A$ and time derivatives of the magnetic field contribute to the breakdown of conservation.

Significance and Claims
The paper claims to provide a unified relativistic framework that clarifies the conditions under which angular momentum is conserved in ultrafast magnetic processes.

• It resolves the apparent contradiction regarding angular momentum conservation by demonstrating that the form of the exchange interaction is critical. The conservation of total angular momentum is not a universal feature of all magnetic models but depends on the specific approximation used for the exchange field.
• The authors argue that the widely debated non-conservation of angular momentum in ultrafast demagnetization (observed in some TD-DFT studies) may be attributed to the use of general, non-conserving forms of the exchange interaction or simplified SOC terms.
• By showing that the total angular momentum is conserved under the atomic Heisenberg approximation even with external fields, the work suggests that the "loss" of angular momentum in experiments might be a result of the specific theoretical model (general exchange field) rather than a fundamental physical violation, or that the transfer to the lattice (via the exchange field's spatial dependence) is the mechanism for apparent non-conservation in broader models.
• The derived coupled equations of motion offer a more rigorous basis for understanding ultrafast magnetization dynamics, including the origins of Gilbert damping, field-derivative torques, and optical spin-orbit torques, by explicitly including the orbital degree of freedom and relativistic corrections.

The work does not propose new experimental setups but rather provides a theoretical foundation to interpret existing ultrafast magnetization phenomena and the conservation laws governing them.