Collective excitations in magnetic topological insulators and axion dark matter search

This paper investigates collective excitations in magnetic topological insulators and finds that one identified "axionic" quasi-particle has an electromagnetic coupling that is either unchanged or suppressed by up to two orders of magnitude compared to previous estimates, significantly altering the sensitivity prospects for axion dark matter searches using these materials.

The Gist

The Big Picture: Hunting for Invisible Ghosts

Imagine the universe is filled with invisible, ghostly particles called Axions. Physicists think these ghosts make up "Dark Matter," the invisible glue holding galaxies together. But we've never seen them.

Recently, scientists had a brilliant idea: What if we could create a fake axion inside a special crystal? If we can make a crystal that acts like an axion, we might be able to "catch" the real cosmic axions by making them talk to our fake ones.

This paper is about two scientists, Koji and Kentaro, who decided to check the blueprints of these special crystals to see if the math actually works. They found that the previous plans were a bit off, and the "fake axions" might be much heavier and harder to catch than everyone thought.

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The Setting: A Crystal City with Two Neighborhoods

Think of the material they are studying (a Magnetic Topological Insulator) as a giant, 3D city made of atoms.

• The Residents: Electrons live here.
• The Neighborhoods: The city has two types of districts: Ferromagnetic (FM) and Antiferromagnetic (AFM).
  • In the FM district, all the residents (electrons) agree to face the same direction (like a crowd of people all looking North).
  • In the AFM district, the residents are neighbors who hate each other. One faces North, the next faces South, and they alternate perfectly (North, South, North, South).

The scientists wanted to know: Which neighborhood is the most stable? And, more importantly, what kind of "wiggles" or "waves" can travel through these neighborhoods?

The Discovery: The "Higgs" and the "Spin-Wave"

In this crystal city, when things get disturbed, they don't just sit still; they create waves. The scientists found two main types of waves:

1. The Magnon (The Spin-Wave): Imagine a line of people passing a ball. If one person changes their mind, the next one changes, creating a ripple. This is a Magnon. It's like a wave of "direction" moving through the crowd.
2. The Amplitude Mode (The "Higgs" or "Breathing" Mode): Imagine the whole crowd suddenly inhaling and exhaling together, changing the size of their steps in unison. This is the Amplitude Mode. It's a wave of "intensity" or "breathing."

The Big Surprise:
The scientists realized that one specific type of "breathing" wave (the AFM-type Amplitude Mode) acts exactly like the Axion we are looking for. It couples to light and magnetic fields just like the real cosmic axion should.

The Twist: The Price Tag Changed

Here is where the paper gets exciting (and a little disappointing for the search).

• The Old Plan: Previous researchers estimated that these "fake axions" would be very light (like a feather) and very easy to detect. They thought the "signal" would be loud and clear.
• The New Reality: Koji and Kentaro did a much more careful calculation. They found that while these fake axions do exist, they are likely much heavier (like a brick) and their connection to light is weaker than expected.

The Analogy:
Imagine you are trying to hear a whisper in a noisy room.

• Previous Estimate: You thought the whisper was loud and clear, so you bought a cheap earplug.
• This Paper: Koji and Kentaro say, "Actually, that whisper is very faint and muffled. You need a super-sensitive microphone, or you might not hear it at all."

They found that the "signal strength" (called the decay constant) could be 10 to 100 times weaker than previously thought. This means the experiments designed to catch these particles might need to be redesigned.

The "Unstable" Neighbors

The paper also looked at other types of waves (the other magnons and breathing modes).

• Some of these waves are unstable. Imagine trying to roll a ball up a hill; it just rolls right back down. In the crystal, these waves dissolve into "electron-hole pairs" (like a wave crashing into a puddle) before they can travel far.
• This means we can't use every wave in the crystal to hunt for axions. We have to be very picky and only look at the stable "breathing" waves in the "Antiferromagnetic" neighborhood.

The Good News: It Doesn't Matter What the Crystal Is Made Of

One of the most helpful findings is that the "fake axion" behavior doesn't depend on whether the crystal is a "Topological Insulator" or a normal insulator.

The Analogy:
Think of the crystal as a musical instrument. You might think you need a rare, magical violin (a Topological Insulator) to play the right note. But Koji and Kentaro found that even a standard, wooden flute (a normal insulator) can play the same note if you arrange the players (the magnetic order) correctly.

This is great news! It means scientists don't have to hunt for the rarest, most expensive materials. They can look at a much wider variety of magnetic materials to find the right "breathing" wave.

Conclusion: What Does This Mean?

1. The Hunt is Harder: The "fake axions" in these crystals are likely heavier and quieter than we hoped. The search needs to be more sensitive.
2. The Hunt is Broader: We don't need to limit ourselves to exotic materials. Any magnetic material with the right "Antiferromagnetic" arrangement might work.
3. The Math is Better: The scientists fixed the math used in previous years, giving us a much more accurate map of where to look and what to expect.

In short: They didn't find the axion, but they gave the search team a better map and a warning: "Don't bring a paper airplane to a hurricane; bring a rocket ship."

Technical Summary

1. Problem Statement

The search for axion dark matter has recently focused on "dynamical axions" (axion-like quasi-particles) existing within magnetic topological insulators (TIs). Previous proposals suggested using magnetic TIs (e.g., Bi$_2$Se$_3$ doped with magnetic impurities) to detect axions in the meV mass range. However, there was a significant discrepancy in the literature regarding the mass of these dynamical axions:

• Previous Estimate [9]: Estimated the axion mass to be $\sim 1$ meV.
• Contradictory Study [10]: Suggested the mass is typically of the order of eV, independent of the insulator's topology, though it could be suppressed near phase boundaries.

Furthermore, the effective coupling (decay constant) of these quasi-particles to electromagnetic fields, which determines the sensitivity of detection experiments, relied on approximations that had not been rigorously validated. The authors aim to resolve these discrepancies by revisiting the microscopic model of magnetic TIs, deriving the effective actions for collective excitations from first principles, and accurately calculating the axion mass and its coupling to electromagnetic fields.

2. Methodology

The authors employ a field-theoretic approach based on the Hubbard model augmented with the effective Hamiltonian for 3D TIs.

• Model Construction:
  • The total Hamiltonian is $H = H_{TI} + H_U$, where $H_{TI}$ describes the 3D TI (Bi$_2$Se$_3$ family) and $H_U$ is the Hubbard interaction term ($U \sum n_{i\uparrow}n_{i\downarrow}$).
  • They utilize the Stratonovich-Hubbard transformation to introduce auxiliary fields $\phi_{ai}$ (antiferromagnetic, AFM) and $\phi_{fi}$ (ferromagnetic, FM), representing spin moments.
• Effective Action Derivation:
  • Starting from the partition function, they integrate out the electron fields to derive the effective action ($S_{eff}$) for the bosonic fluctuations (amplitude modes and magnons).
  • The derivation relies on the dynamical susceptibility ($\chi$), calculated using the Matsubara formalism at finite temperature and taking the zero-temperature limit ($T \to 0$).
  • The inverse propagator for excitations is defined as $\tilde{\Gamma}(k) = 1 - \frac{U}{4}\tilde{\chi}(k)$. The poles of this propagator determine the mass (gap) and dispersion relations of the excitations.
• Classification of Excitations:
  • The study identifies eight collective modes under AFM and FM orders:
    • Amplitude Modes: AFM-type and FM-type.
    • Magnons: $\alpha$-type and $\beta$-type (distinguished by their symmetry properties under the magnetic order).
• Axion Identification:
  • The authors identify the AFM-type amplitude mode under the AFM order as the "axionic" quasi-particle.
  • They derive the effective action for this mode in Minkowski spacetime, explicitly calculating the mass ($m_a$) and the stiffness (related to the decay constant).

3. Key Contributions and Results

A. Magnetic Orders and Ground State

• By computing the effective potential, the authors confirm that both AFM and FM orders emerge as the Hubbard interaction $U$ increases.
• Ground State: The AFM order is the global minimum (ground state) for the system.
• Phase Transition: The transition to the AFM order is continuous (second-order), while the FM order emerges discontinuously (first-order) due to the negative quartic term in the effective potential.
• At large $U$ (saturated magnetization), the AFM and FM states become quasi-degenerate.

B. Stability and Dispersion of Excitations

• Amplitude Modes:
  • The AFM-type amplitude mode under the AFM order is stable. Its dispersion relation has a positive curvature around $k=0$, indicating a well-defined quasi-particle.
  • Other modes (FM-type amplitude mode under AFM, and all modes under FM order) are found to be unstable. They exhibit imaginary parts in their self-energy, indicating rapid dissipation into electron-hole pairs.
• Magnons:
  • The $\alpha$- and $\beta$-type magnons are generally stable with energy scales of $\mathcal{O}(\text{eV})$.
  • The $\alpha$-type magnon under AFM order possesses a gap generated by the breaking of $SO(3)$ symmetry to $SO(2)$ due to non-zero parameters $d_1, d_2$ (related to $A_2$). If $d_1=d_2=0$, the $\alpha$-mode becomes a massless Nambu-Goldstone boson.
  • The $\beta$-type magnon remains massive even if $d_1=d_2=0$.

C. Revised Axion Mass and Coupling

• Mass Calculation:
  • The mass of the axionic mode is derived as $m_a = U\phi_{a0}$.
  • Unlike previous works that expanded around zero frequency ($\omega=0$), the authors expand around the physical mass pole ($\omega=m_a$).
  • Result: The typical mass scale is $\mathcal{O}(\text{eV})$, consistent with Ref. [10], rather than the meV scale proposed in Ref. [9]. However, the mass can be suppressed to the meV range if the order parameter $\phi_{a0}$ is small (near the phase boundary or for specific parameter tuning).
• Effective Decay Constant ($f_Q$):
  • The effective coupling to electromagnetic fields is characterized by the decay constant $f_Q = m_a / c_a^{eff}$.
  • The authors calculate $f_Q$ to be in the range of $10^2 - 10^4$ eV.
  • Significance: This value is roughly the same as or up to two orders of magnitude larger than the previous estimate of $\sim 190$ eV. A larger decay constant implies a weaker coupling to photons, which drastically reduces the sensitivity of axion search experiments using these materials.
• Topology Independence:
  • The nature of the "axion" (the AFM-type amplitude mode) and its properties are found to be independent of the topological phase (Normal Insulator vs. Topological Insulator). The effective decay constant is qualitatively similar in both phases, suggesting that topological protection is not a prerequisite for this mechanism.

4. Significance and Implications

• Re-evaluation of Experimental Sensitivity: The finding that the effective decay constant is potentially two orders of magnitude larger than previously thought implies that the signal strength for axion detection in magnetic TIs is significantly weaker. This necessitates a re-evaluation of the feasibility of current and proposed experiments targeting the meV mass range in these materials.
• Material Selection: Since the "axion" nature is topology-independent, researchers can search for suitable materials based on magnetic properties and collective excitation stability rather than strictly requiring topological insulators. This broadens the candidate pool for axion searches.
• Theoretical Rigor: The paper corrects the perturbative expansion used in previous literature. By expanding around the physical mass pole rather than zero frequency, the authors provide a more accurate derivation of the stiffness and mass, resolving the discrepancy between the meV and eV mass estimates.
• New Detection Channels: While the primary axion mode is identified, the paper also discusses the potential (though challenging due to instability) interaction of magnons with electromagnetic fields, suggesting a secondary channel where two magnons could be excited by an axion field.

In conclusion, this work provides a rigorous microscopic derivation of collective excitations in magnetic TIs, identifying the AFM-type amplitude mode as the dynamical axion. It corrects previous mass estimates to the eV scale (with tunability to meV) and significantly revises the expected coupling strength, fundamentally impacting the strategy for axion dark matter searches in condensed matter systems.