Einstein-de Haas effect and induced rotation in an evolving magnetized QCD matter

This paper investigates the Einstein-de Haas effect in an expanding quark-gluon plasma using a quasiparticle model, revealing that the induced angular velocity grows with proper time and reaches significant magnitudes near the crossover temperature, thereby establishing a distinct transition between spin-dominated and inertia-dominated regimes driven by magnetic fields.

The Gist

Imagine a giant, super-hot fireball created when two heavy atomic nuclei smash into each other at nearly the speed of light. Inside this fireball, called a Quark-Gluon Plasma (QGP), the usual rules of matter break down. Protons and neutrons melt into a soup of their smaller parts: quarks and gluons.

This paper explores a fascinating phenomenon happening inside that soup, driven by two things: magnetic fields and spin.

The Setup: A Magnetic Storm and Spinning Tops

When these nuclei collide, they don't just hit head-on; they often graze each other. This creates two things:

1. A Massive Magnetic Field: The electrically charged protons flying past each other generate a magnetic field stronger than anything found in the universe (except maybe a neutron star).
2. Spinning Particles: Inside the plasma, quarks act like tiny spinning tops. Each quark has a "spin," which is an intrinsic form of angular momentum.

The Core Idea: The Einstein-de Haas Effect

The paper focuses on a classic physics principle called the Einstein-de Haas (EdH) effect.

Think of it like this: Imagine you are standing on a perfectly smooth, frictionless turntable holding a spinning bicycle wheel.

• If you flip the wheel over so it spins the opposite way, you (and the turntable) will start spinning in the opposite direction to keep the total "spin" of the system balanced.
• The Rule: Nature demands that the total amount of spin (angular momentum) stays the same. If the internal spin of the particles changes direction or alignment, the whole object must physically rotate to compensate.

In this study, the "turntable" is the expanding fireball of the QGP, and the "bicycle wheels" are the quarks.

What Happens in the Fireball?

1. Alignment: When the intense magnetic field is turned on, it acts like a giant magnet. It tries to line up all the tiny quark "spinning tops" in the same direction, just like iron filings aligning near a magnet.
2. The Reaction: As the quarks align their spins, the total internal spin of the system changes. To obey the law of conservation (the rule that total spin can't just disappear), the entire fireball must start physically rotating in the opposite direction.
3. The Result: The magnetic field doesn't just align the particles; it actually makes the whole fireball spin.

The Surprising Findings

The authors used a computer model to track how this happens as the fireball expands and cools down. They found some interesting patterns:

• Timing is Everything: The effect is strongest when the fireball is cooling down to a specific "critical" temperature (where the plasma turns back into normal matter). At this moment, the magnetic field is still strong enough to align the spins, but the fireball has cooled enough that the particles aren't jiggling around too wildly to break the alignment.
• The "Crossing" Point: They discovered a strange "tipping point."
  • At lower temperatures: Stronger magnetic fields make the fireball spin faster. This makes sense; more magnetism means more alignment.
  • At higher temperatures: Surprisingly, making the magnetic field stronger actually makes the fireball spin slower. Why? Because at high temperatures, the energy required to keep the particles in their magnetic "tracks" (a quantum effect called Landau quantization) becomes so huge that it acts like a heavy weight, making the fireball harder to spin. It's like trying to spin a heavy, frozen wheel versus a light, warm one.
• Size Matters: The bigger the fireball, the slower it spins. This is because the "spin" from the particles has to be shared across a much larger mass.

Why Does This Matter?

The paper concludes that this effect is significant. The rotation caused by the Einstein-de Haas effect is strong enough to be noticed in experiments at places like the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC).

It suggests that when scientists measure how fast the fireball is spinning (by looking at how particles align), they aren't just seeing the spin from the initial collision. They are also seeing a "bonus" spin generated by the magnetic field itself. It's a direct demonstration that in the extreme world of the early universe, magnetism can literally create motion.

Summary Analogy

Imagine a crowd of people (quarks) in a giant, expanding room (the fireball).

• A giant magnet (the magnetic field) suddenly turns on, forcing everyone to face North.
• Because everyone turned their bodies to face North, the entire room has to twist slightly to the South to keep the balance of the building.
• The paper calculates exactly how much the room twists, finding that it twists the most when the room is cooling down, and that the size of the room and the strength of the magnet change the rules of how much it twists.

Technical Summary

Technical Summary: Einstein–de Haas Effect in Evolving Magnetized QCD Matter

Problem Statement
The paper addresses the dynamics of angular momentum conservation in magnetized Quantum Chromodynamics (QCD) matter, specifically within the Quark-Gluon Plasma (QGP) formed in non-central heavy-ion collisions. While the generation of intense transient magnetic fields ($eB \sim m_\pi^2$) and the observation of global hyperon polarization (indicating high fluid vorticity) are established phenomena, the specific mechanism by which magnetic fields induce collective rotation via the Einstein–de Haas (EdH) effect in a dynamically expanding QGP has not been quantitatively investigated. Previous studies have examined the EdH effect in static hadron gases or the reciprocal Barnett effect in rotating matter, but a systematic analysis of spin-to-orbital angular momentum conversion in an evolving, magnetized QGP phase is lacking. The authors aim to determine if the EdH effect generates a non-negligible angular velocity ($\omega_{EdH}$) that could modify the vorticity inferred from final-state polarization measurements.

Methodology
The authors employ a Quasiparticle Model (QPM) to describe the thermodynamic properties of the magnetized QGP. The theoretical framework incorporates the following key elements:

• Dynamical Evolution: The system undergoes a boost-invariant longitudinal Bjorken expansion, with temperature cooling described by $T(\tau) = T_0(\tau_0/\tau)^{1/3}$. The external magnetic field is modeled with a phenomenological exponential decay, $eB(\tau) = eB_0 \exp(-\tau/\tau_B)$, accounting for the finite electrical conductivity of the QGP.
• Quasiparticle Properties: Interacting quarks and gluons are treated as massive quasiparticles with thermal masses derived from the QCD running coupling. The model interpolates between bare and thermal masses to capture the transition near the deconfinement temperature ($T_c \approx 0.155$ GeV).
• Magnetic Effects: The formalism explicitly includes Landau quantization for charged quarks (discrete transverse energy levels) and Zeeman splitting for intrinsic spin alignment. The authors isolate the "spin-only" contribution to magnetization by differentiating the pressure with respect to the magnetic field, retaining only terms dependent on Zeeman splitting, to distinguish intrinsic spin effects from orbital cyclotron currents.
• Angular Momentum Conservation: The induced angular velocity is calculated by enforcing the conservation of total angular momentum ($J_{tot} = J_{spin} + J_{orb}$). The spin angular momentum density ($S_z$) generated by field alignment is equated to a compensating orbital angular momentum ($I\omega_{EdH}$), where the moment of inertia $I$ is determined by the energy density and fireball geometry.

Key Contributions and Results
The study provides the first quantitative estimation of the EdH-induced angular velocity in a dynamically expanding, magnetized QGP. The primary findings include:

1. Magnitude and Evolution: The induced angular velocity $\omega_{EdH}$ grows with proper time as the system cools, reaching substantial magnitudes near the QCD crossover temperature ($T_c$). The calculated values range from $0.05$ to $2$ MeV, which is comparable to typical fluid vorticity estimates derived from hyperon polarization at RHIC and LHC energies.
2. Temperature Dependence: $\omega_{EdH}$ exhibits a non-monotonic dependence on temperature, peaking near $T_c$. At higher temperatures, thermal fluctuations suppress net spin density, while at lower temperatures (near $T_c$), the large effective quark masses enhance Zeeman splitting, facilitating efficient spin alignment.
3. The Crossing Phenomenon: A critical finding is the existence of a "crossing temperature" ($T_{cross}$) that separates two dynamical regimes. Below $T_{cross}$, stronger magnetic fields produce faster rotation (spin-dominated regime). Above $T_{cross}$, the trend reverses ($\omega_{strong} < \omega_{weak}$) because the increasing orbital energy density and effective rotational inertia (due to Landau quantization) suppress the rotational response. This indicates that stronger magnetic fields do not necessarily yield stronger vorticity in the hottest plasma.
4. Parameter Sensitivity:
   • Baryon Chemical Potential ($\mu_B$): The effect is more pronounced at higher $\mu_B$ (lower collision energies), where Pauli blocking and dense matter enhance the spin density.
   • Fireball Radius ($R$): The induced rotation scales as $\omega_{EdH} \propto R^{-2}$, implying that smaller systems or more peripheral collisions generate larger instantaneous rotation.
   • Magnetic Field Decay: The effect is maximized when the magnetic field decay timescale coincides with the cooling timescale near $T_c$, allowing significant spin alignment to persist into the hadronic phase.

Significance and Claims
The paper claims to establish the Einstein–de Haas effect as a manifestation of angular momentum conservation in magnetized QCD matter. The authors argue that this mechanism provides a distinct channel for converting spin angular momentum into global mechanical rotation, independent of initial hydrodynamic vorticity.

Key implications highlighted by the authors include:

• Unified Picture: The EdH effect operates across both the partonic (QGP) and hadronic phases, suggesting a unified mechanism for spin-to-orbital conversion throughout the fireball's evolution.
• Modification of Observables: The backreaction of the EdH effect implies that the total vorticity measured via polarization ($\omega_{total} = \omega_{hydro} + \omega_{EdH}$) may be quantitatively modified relative to the initial hydrodynamic vorticity. Depending on the relative orientation of the magnetic-field-induced spin alignment and the hydrodynamic flow, the EdH contribution could either enhance or suppress the net observed rotation.
• Experimental Signatures: The distinct scaling behaviors—specifically the $\omega \propto R^{-2}$ dependence and the non-monotonic temperature profile—offer potential experimental signatures to discriminate EdH-induced rotation from conventional vorticity in future measurements at RHIC, LHC, FAIR, and NICA.

The authors maintain a modest tone regarding their approximations, noting that the study assumes instantaneous spin relaxation and rigid-body rotation as an effective description. They acknowledge that a full spin-hydrodynamic treatment and the quantitative determination of spin relaxation timescales ($\tau_s$) in magnetized matter remain open directions for future work.