Magnon thermal Hall effect in collinear antiferromagnets

This paper theoretically demonstrates that a non-zero magnon thermal Hall effect can occur in collinear antiferromagnets at zero external magnetic field, driven by symmetry-breaking mechanisms such as sublattice asymmetry in compensated Néel order or Dzyaloshinskii-Moriya interactions in weak ferromagnets, and proposes a model where an external electric field can modulate this effect by altering the system's symmetry.

The Gist

Here is an explanation of the paper "Magnon Thermal Hall Effect in Collinear Antiferromagnets" using simple language and creative analogies.

The Big Picture: Heat Traffic in a Magnetic City

Imagine a city built on a grid. In this city, the "traffic" isn't cars, but tiny waves of energy called magnons. These waves carry heat.

Usually, if you heat one side of a material, the heat flows straight to the cold side, like water flowing down a hill. But sometimes, due to the specific rules of the city's layout (symmetry) and some hidden "wind" (magnetic interactions), the heat gets pushed sideways. This sideways flow of heat is called the Thermal Hall Effect.

This paper asks a simple question: Can this sideways heat flow happen in a special type of magnetic material called an "antiferromagnet" when there is no external magnet pushing on it?

The answer is yes, but only if the city's layout is slightly "broken" or "tilted" in a specific way.

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The Cast of Characters

To understand the paper, we need to meet the main characters:

1. The Neighbors (The Sublattices): Imagine the city is divided into two neighborhoods, Red and Blue. In a perfect antiferromagnet, every Red house has a Blue house right next to it. They are perfect opposites (like a dance partner pair).
2. The Green Ghost (The Non-Magnetic Atom): There is a special, invisible "Green Ghost" atom sitting in the middle of the neighborhood. It doesn't have a magnetic personality itself, but it changes how the Red and Blue neighbors talk to each other.
3. The Magnons: These are the heat-carrying waves. Think of them as messengers running between the Red and Blue houses.
4. The Dzyaloshinskii-Moriya Interaction (DMI): This is a fancy name for a "twist" or a "whirlwind" that the Green Ghost creates. It makes the messengers spin as they run.

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The Three Scenarios

The author explores three different ways to arrange this city to see if the heat will flow sideways.

1. The Perfectly Balanced City (Genuine Antiferromagnet)

• The Setup: The Red and Blue neighborhoods are perfectly symmetrical. If you flip the city over or rotate it, it looks exactly the same. The Green Ghost sits perfectly in the center.
• The Result: The messengers (magnons) run in circles. For every messenger going left, another goes right. The sideways forces cancel each other out perfectly.
• The Verdict: No Thermal Hall Effect. The heat goes straight down the hill. Nothing interesting happens.

2. The Broken Symmetry City (Ferrimagnet)

• The Setup: The Green Ghost moves slightly off-center. Now, the Red neighborhood looks different from the Blue neighborhood. They are no longer mirror images. The "rules" of the road are different for Red vs. Blue.
• The Result: Because the neighborhoods are different, the messengers get confused. The "twist" (DMI) from the Green Ghost pushes the Red messengers one way and the Blue messengers another way, but because the neighborhoods aren't identical, these pushes don't cancel out.
• The Verdict: Yes! Thermal Hall Effect. The heat gets pushed sideways. The paper calls this a "Ferrimagnet" (a fancy word for a magnet that is almost balanced but has a tiny imbalance).

3. The Tilted City (Weak Ferromagnet)

• The Setup: Imagine the Green Ghost isn't just off-center; it's actually floating above the ground (lifted out of the plane). This changes the geometry of the whole city. Even if the Red and Blue neighborhoods look symmetric from above, the 3D structure breaks the rules.
• The Result: This tilt creates a specific "twist" that allows the messengers to flow sideways, even though the neighborhoods are technically connected by symmetry.
• The Verdict: Yes! Thermal Hall Effect. This is called a "Weak Ferromagnet." It's a special case where the 3D shape of the atoms forces the heat to turn a corner.

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The "Magic Switch": Using Electricity to Control Heat

The most exciting part of the paper is the suggestion that we can control this effect with an electric field.

• The Analogy: Imagine the Green Ghost is a heavy ball sitting on a trampoline (the crystal lattice).
• The Trick: If you apply an electric field, it's like blowing a strong wind on the trampoline. You can push the Green Ghost from the center to the side, or even lift it up.
• The Result: By moving the Green Ghost, you can switch the material from "No Heat Turn" to "Heat Turn Left" or "Heat Turn Right."
• Why it matters: This means we could build devices that use electricity to steer heat without using magnets. It's like having a traffic cop for heat that you can control with a light switch.

Summary in One Sentence

This paper proves that heat can flow sideways in magnetic insulators if the atomic "neighborhoods" are slightly unbalanced or tilted, and we can control this effect by using electricity to shuffle the atoms around.

Why Should You Care?

Currently, we use magnets to control electricity (hard drives, motors). This research suggests we might soon be able to use electricity to control heat in magnetic materials. This could lead to new, super-efficient computer chips that don't overheat, or new ways to manage energy in tiny electronic devices.

Technical Summary

Here is a detailed technical summary of the paper "Magnon thermal Hall effect in collinear antiferromagnets" by Vladimir A. Zyuzin.

1. Problem Statement

The paper addresses the theoretical conditions under which a magnon thermal Hall effect (THE) can occur in insulating collinear antiferromagnets (AFMs) in the absence of an external magnetic field.

• Context: While the THE is well-studied in ferromagnets and non-collinear AFMs, its existence in collinear AFMs is often debated. Previous studies suggested that if the two magnetic sublattices are connected by specific symmetries (e.g., time-reversal combined with translation or inversion), the THE vanishes.
• Gap: The paper seeks to systematically determine the symmetry requirements for a non-zero magnon THE in collinear AFMs, distinguishing between "genuine" antiferromagnets, "weak" ferromagnets, and "ferrimagnetic" AFMs where sublattice symmetries are broken.

2. Methodology

The author employs a theoretical approach combining spin-wave theory and symmetry analysis:

• Model System: Two 2D insulating AFM models on honeycomb-like lattices are constructed (Fig. 1).
  • Lattice Structure: Red and blue sites represent spin-up ($+S$) and spin-down ($-S$) magnetic sublattices. A green non-magnetic atom sits in the plane (or is lifted out of the plane) to modulate interactions.
  • Hamiltonian: The system includes nearest-neighbor Heisenberg exchange (HEI) and Dzyaloshinskii-Moriya interaction (DMI), as well as anisotropic second-nearest neighbor interactions. The Hamiltonian is derived considering spin-orbit coupling effects on exchange constants.
• Quantization: The Holstein-Primakoff transformation is used to map spins to bosonic operators ($a, b$) representing magnons on the two sublattices.
• Calculation:
  • The magnon Hamiltonian is diagonalized to obtain the energy spectrum $\hat{E}_k$.
  • The Berry curvature ($\Omega_{xy}$) of the magnon bands is calculated.
  • The thermal Hall conductivity ($\sigma_{THE}$) is computed using the standard formula involving the integral of the Berry curvature weighted by the Bose-Einstein distribution function.
• Symmetry Analysis: The results are cross-referenced with Dzyaloshinskii's classification of AFMs. The author checks for the existence of Dzyaloshinskii invariants (terms of the form $M_\alpha L_\beta$, where $M$ is magnetization and $L$ is the Néel vector) allowed by the crystal symmetry.

3. Key Contributions

The paper makes three primary theoretical contributions:

1. Symmetry Breaking Requirement for Ferrimagnets: It establishes that a non-zero magnon THE in a collinear AFM with no symmetry between the two magnetic sublattices (a ferrimagnet) requires the simultaneous presence of:

   • Spin-momentum splitting of the magnon spectrum due to DMI (acting as a vector potential).
   • Anisotropic second-nearest neighbor exchange interactions ($J'$) that differ between the two sublattices.
   • Crucially, if only DMI exists without the anisotropic exchange (or vice versa), the effect can be gauged away or vanishes upon integration.

2. THE in Weak Ferromagnets: It demonstrates that collinear weak ferromagnets (where sublattices are connected by symmetry but allow a finite magnetic moment) also exhibit a non-zero magnon THE. This is driven by the same mechanisms (DMI and anisotropic exchange) consistent with the specific Dzyaloshinskii invariants of the weak ferromagnetic phase.

3. Electric Field Control: The paper proposes a mechanism to tune the magnon THE using an external electric field. By displacing the non-magnetic "green" ion within the unit cell (changing the lattice symmetry from a high-symmetry phase to a low-symmetry phase), one can switch the system between a state with zero THE and a state with finite, sign-reversible THE.

4. Results

• Model I (Genuine Antiferromagnet):

  • Symmetry: The two sublattices are connected by mirror symmetry and time-reversal.
  • Result: The Berry curvature integrates to zero. No magnon THE is observed.
  • Invariant: No Dzyaloshinskii invariant of the form $M_z L_z$ or $M_z L_x$ is allowed.

• Model II (Collinear Ferrimagnet - Type I):

  • Symmetry: The green atom is in the plane but displaced, breaking all symmetries connecting the sublattices.
  • Result: A non-zero magnon THE appears.
  • Mechanism: The effect arises from the interplay of DMI (shifting momentum) and anisotropic second-nearest neighbor exchange ($T_k$). The Berry curvature has an even-in-momentum component that does not vanish upon integration.
  • Invariant: Allows for an $M_z L_z$ invariant. The magnetization $M_z$ is generated purely by magnon fluctuations (no canting of the Néel vector).

• Model III (Collinear Ferrimagnet - Type II & Weak Ferromagnet):

  • Symmetry: The green atom is lifted out of the plane.
  • Result: A non-zero magnon THE is observed.
  • Invariant: Allows for an $M_z L_x$ invariant. This distinguishes it from Type I; here, the Néel order can cant to produce magnetization, and the system belongs to the weak ferromagnet class.
  • Electric Field Tuning: Simulations show that rotating an in-plane electric field moves the green ion, causing the magnon THE to oscillate and cross zero six times per $2\pi$ rotation (Fig. 4).

• Spectral Features:

  • DMI Effect: Breaks the degeneracy of spin-up and spin-down magnon branches at the $\Gamma$ point (momentum shift).
  • Anisotropic Exchange ($T_k$): Breaks degeneracy away from the $\Gamma$ point. Both are necessary for a robust THE.

5. Significance

• Fundamental Physics: The work clarifies the microscopic origin of the thermal Hall effect in insulating AFMs, resolving the ambiguity regarding the role of sublattice symmetry. It proves that collinear AFMs can host THE without canting, provided specific symmetry-breaking conditions (ferrimagnetism or weak ferromagnetism) are met.
• Experimental Guidance: The paper identifies specific material classes (e.g., $RuO_2$, $CrSb$, $\alpha$-Fe$_2$O$_3$) that likely exhibit this effect.
• Device Application: The proposal of using electric fields to control the magnon thermal Hall effect offers a promising route for magnon-based thermoelectric devices. It suggests a way to switch thermal currents on and off or reverse their direction without magnetic fields, which is crucial for low-power spintronic applications.
• Classification: It reinforces the utility of Dzyaloshinskii's invariant theory in predicting transport properties in complex magnetic systems.