Nonadiabatic Theory of Phonon Magnetic Moments in Insulators and Metals

This paper develops a unified nonadiabatic theory for phonon magnetic moments in both insulators and metals using a gauge-covariant Wigner expansion, which successfully explains the experimentally observed large magnetic moments in Pb$_{1-x}$Sn$_x$Te by revealing significant contributions from Fermi-surface processes and resonant interband transitions beyond the adiabatic limit.

The Gist

Imagine a crystal lattice as a giant, three-dimensional trampoline made of atoms. Usually, when these atoms vibrate (creating what physicists call "phonons"), they bounce up and down or side-to-side in perfect, symmetrical patterns. In a world without magnetic fields, these vibrations are neutral; they don't have a magnetic personality.

However, this paper introduces a new way to understand what happens when you put a magnet near this vibrating trampoline. The authors, Haoran Chen and colleagues, have developed a new set of rules—a "nonadiabatic theory"—to explain how these vibrations can suddenly start acting like tiny magnets.

Here is the breakdown of their discovery using everyday analogies:

1. The Old Rules vs. The New Rules

For a long time, scientists used "adiabatic" rules to predict how these vibrations behave. Think of the adiabatic approach like watching a slow-motion movie. It assumes the electrons (the tiny particles orbiting the atoms) are so fast and lazy that they just instantly adjust to the atoms' movements, like a shadow that perfectly follows a dancer's slow steps.

This worked fine for insulators (materials that don't conduct electricity) when the vibrations were slow. But recent experiments in metals and doped semiconductors showed something strange: the vibrations were acting much more magnetic than the old "slow-motion" rules predicted. It was as if the dancers were suddenly spinning wildly, and the shadow was reacting with a force the old rules couldn't explain.

The authors say the old rules failed because they ignored two things:

1. Speed: Sometimes the vibrations are fast enough that the electrons can't just "keep up" instantly.
2. The Crowd: In metals, there are free-moving electrons (like a crowd of people at a concert) that can interact with the vibrations in a way insulators (where everyone is stuck in their seats) cannot.

2. The Two Sources of the "Magnetic Spin"

The paper explains that the magnetic moment (the "magnetic personality") of a vibrating atom comes from two main sources, which they call the Fermi-sea and the Fermi-surface.

• The Fermi-sea (The Deep Ocean): Imagine the electrons in a material as a deep ocean. Even in a calm state, the water is moving. When the atoms vibrate, they create ripples in this deep ocean. The old theories mostly looked at these deep, underlying ripples.
• The Fermi-surface (The Surface Waves): In metals, there is a distinct "surface" where the electrons are free to move around. The authors discovered that when atoms vibrate, they create waves right on this surface.

The Big Discovery: In metals, the "surface waves" (Fermi-surface contribution) are not just a small ripple; they are a massive tsunami compared to the deep ocean ripples. The authors found that this surface effect is what was missing from previous theories. It is so powerful that it can make the magnetic effect of the vibration 100 times stronger than previously thought.

3. The "Resonance" Effect

The paper also highlights a phenomenon called resonance. Imagine pushing a child on a swing. If you push at just the right rhythm, the swing goes higher and higher.

The authors found that if the frequency of the atomic vibration matches the energy gap between electron states (like pushing the swing at the perfect moment), the magnetic effect explodes. This "resonant" boost happens even in insulators if the energy gap is narrow, but it becomes the dominant force in metals.

4. Testing the Theory: The Pb1-xSnxTe Experiment

To prove their new rules work, the authors applied them to a specific material called Pb1-xSnxTe (a mix of Lead, Tin, and Tellurium).

• The Experiment: Scientists had measured how magnetic the vibrations were in this material as they changed the amount of Tin (Sn) in the mix.
• The Problem: The old "slow-motion" theories predicted very small magnetic effects, but the experiments showed huge effects (reaching the scale of a Bohr magneton, $\mu_B$).
• The Solution: When the authors applied their new "nonadiabatic" theory, which included the powerful "Fermi-surface" contribution, their calculations matched the experimental data almost perfectly. They showed that the extra magnetic strength came entirely from the free-moving electrons on the surface of the electron sea.

Summary

In simple terms, this paper fixes a broken calculator. For years, scientists used a calculator that assumed atoms vibrate slowly and electrons just sit still. This calculator worked for some materials but failed miserably for metals.

The authors built a new calculator that accounts for:

1. Fast vibrations (where electrons can't keep up instantly).
2. Free-moving electrons (the "surface waves" in metals).

By adding these factors, they finally explained why vibrations in metals are so much more magnetic than anyone expected, bridging the gap between theory and real-world experiments.

Technical Summary

Technical Summary: Nonadiabatic Theory of Phonon Magnetic Moments in Insulators and Metals

Problem Statement
While phonons carrying finite angular momentum (chiral phonons) are known to influence thermodynamic, spectral, and transport properties, a quantitative understanding of their magnetic moments remains incomplete. Existing theoretical frameworks, including those based on adiabatic current pumping, density functional perturbation theory, and density-matrix perturbation approaches, are primarily formulated for insulating systems. These adiabatic theories fail to account for experimental observations in doped semiconductors and metals, where phonon magnetic moments reach the scale of the Bohr magneton ($\mu_B$), far exceeding theoretical predictions. Furthermore, phenomenological extensions for Fermi-surface electrons in Dirac semimetals suffer from model dependence. The core issue is the lack of a unified theory that treats insulators and metals on equal footing, particularly when phonon frequencies are comparable to electronic energy scales or when finite carrier densities are present.

Methodology
The authors develop a nonadiabatic theory that expresses the phonon magnetic moment as a linear response of ionic forces to ion velocities in a magnetic field. The methodology proceeds through the following steps:

1. Linear Response Formulation: The time-reversal symmetry (TRS) breaking effects in lattice dynamics are identified as arising from two sources: direct Lorentz coupling of ionic charges to the magnetic field and electron-phonon coupling (EPC). The EPC contribution is formulated as a Kubo-type response function, $\hat{G}$, relating the force on an ion to the velocity of another ion.
2. Gauge-Invariant Expansion: To handle magnetic fields in a gauge-invariant manner, the response function is transformed into the Wigner representation. Using a gauge-covariant Wigner expansion, the authors derive corrections to the Green's functions and the product of Green's functions with force operators up to linear order in the magnetic field $B$.
3. Derivation of the Magnetic Moment: The phonon magnetic moment is defined via the derivative of the response function with respect to the magnetic field. The resulting expression naturally separates contributions from the Fermi sea (interband transitions) and the Fermi surface (intraband transitions).
4. Application to a Model System: The theory is applied to a massive anisotropic Dirac $k \cdot p$ model describing the topological crystalline insulator $\text{Pb}_{1-x}\text{Sn}_x\text{Te}$. This model includes spin and orbital degrees of freedom and allows for the tuning of the band gap and Fermi level to simulate both insulating and metallic regimes.

Key Contributions and Results

• Unified Framework: The derived formula is applicable to both insulators and metals. In the low-frequency limit for gapped systems ($\hbar\omega \ll \Delta_{\text{gap}}$), the theory rigorously reduces to previous adiabatic expressions.
• Nonadiabatic Corrections: Beyond the adiabatic limit, the theory captures additional contributions:
  • Resonant Interband Processes: In insulators with narrow gaps, near-resonant transitions between occupied and unoccupied states (where the energy difference is close to the phonon energy $\hbar\omega$) substantially enhance the magnetic moment.
  • Fermi-Surface Contributions: In metals, an intraband term arising from the Fermi surface emerges. This term scales approximately as $(\hbar\omega + i\Gamma)^{-2}$ and can dominate the magnetic moment, especially in the low-frequency limit.
• Application to $\text{Pb}_{1-x}\text{Sn}_x\text{Te}$:
  • In the insulating regime with a narrow gap ($\Delta_{\text{gap}} = 10$ meV), the phonon magnetic moment is strongly enhanced as the phonon frequency approaches the gap size due to resonant processes.
  • Upon doping into the metallic regime, the Fermi-surface contribution rapidly becomes dominant.
  • Using parameters consistent with experimental measurements (including EPC strength and scattering rates), the theory predicts phonon magnetic moments on the order of $\mu_B$ (e.g., $\approx 1.47 \mu_B$ at $x=0.42$). These values reproduce the order of magnitude and trends observed in recent experiments, which previous adiabatic theories could not explain.
• Limitations: The current harmonic approximation supports only a single transverse optical doublet. Consequently, the theory does not capture the branch-dependent behavior (sign changes in specific modes) observed experimentally across the topological crystalline insulator transition, which arises from anharmonic splitting.

Significance
The paper claims to provide a systematic, nonadiabatic framework for describing phonon magnetic moments that resolves the discrepancy between theory and experiment in metallic and narrow-gap systems. By identifying the Fermi-surface contribution and resonant interband processes as essential physical ingredients, the theory explains why phonon magnetic moments in metals are significantly larger than adiabatic estimates. This framework offers a unified route for studying phonon magnetic moments in general systems, bridging the gap between insulating and metallic regimes. The authors note that extending this framework to incorporate anharmonic phonon effects remains an important direction for future work to fully capture branch-dependent phenomena.