Anomalous thermoelectric and thermal Hall effects in irradiated altermagnets

Imagine a world where materials can be "tuned" like a radio station, not just to change their sound, but to fundamentally change how they move heat and electricity. This is the story of a new paper exploring a special type of magnetic material called an altermagnet.

Here is the breakdown of what the researchers discovered, using simple analogies.

1. The Material: The "Perfectly Balanced" Magnet

First, let's meet the star of the show: the Altermagnet.

Ferromagnets (like your fridge magnet) are like a crowd of people all facing North. They have a strong magnetic pull.
Antiferromagnets are like a crowd where half face North and half face South, perfectly canceling each other out. They have no net magnetic pull.
Altermagnets are the "Goldilocks" of magnets. They are like a checkerboard where the spins alternate in a complex pattern. They have zero net magnetic pull (so they don't stick to your fridge), but they still have a hidden "spin" structure that makes electrons behave in unique ways.

In their natural state, these materials have "highways" for electrons (called Dirac cones) that are wide open and gapless. Because these highways are open and symmetrical, electrons flow straight, and no sideways "Hall" effects happen. It's like a perfectly flat, frictionless road where cars only go forward.

2. The Catalyst: The "Dance Floor" Light

The researchers asked: What if we shine a special light on this material?

They used elliptically polarized light (think of it as light that spins in a corkscrew motion rather than just vibrating up and down). They shined this high-frequency light onto the altermagnet.

The Analogy: Imagine the electrons are dancers on a floor.

Without light: The dancers are moving in a symmetrical pattern. If you push them from the left, they move left. No one moves sideways.
With light: The spinning light acts like a DJ changing the beat. It breaks the symmetry of the dance floor. Suddenly, the "floor" changes shape. The open highways (Dirac cones) get blocked off, creating gaps.

3. The Result: Creating a "Chern Insulator"

By breaking the symmetry with light, the material transforms into a Chern Insulator.

What is that? Think of it as a material that is an insulator (electricity can't flow through the middle) on the inside, but acts like a super-highway on the edges.
The Magic: Because the light broke the symmetry, the electrons now have a "preferred direction" to turn, even without a magnet. This creates a Berry Curvature.
- Analogy: Imagine a bowling ball rolling down a hill. Normally, it goes straight. But if the hill is shaped like a spiral slide (the Berry curvature), the ball is forced to curve sideways as it rolls down.

4. The Effects: Heat and Electricity Taking a Detour

The paper focuses on two specific effects that happen when you heat up one side of this light-irradiated material:

A. The Anomalous Thermoelectric Hall Effect (The "Heat-Induced Detour")

The Setup: You heat one side of the material. Heat naturally wants to flow to the cold side.
The Effect: Because of the "spiral slide" shape created by the light, the heat doesn't just go straight; it gets pushed sideways.
The Discovery: The researchers found that this sideways "heat current" is incredibly sensitive.
- If the energy gap (the blocked highway) is open, the sideways signal disappears.
- But right at the edges of that gap, the signal spikes up or dips down like a heartbeat.
- Analogy: It's like a metal detector. If you walk over a flat field, it beeps nothing. But if you step right on the edge of a buried treasure chest, it screams. This effect allows scientists to "map" exactly where the energy gaps are.

B. The Thermal Hall Effect (The "Quantized Heat Flow")

The Setup: Same as above, but looking at the total amount of heat moving sideways.
The Effect: In the gap regions, this sideways heat flow becomes quantized.
- Analogy: Imagine water flowing through a pipe. Usually, you can have any amount of water. But in this quantum world, the water can only flow in specific, discrete "buckets." You can have 1 bucket, 2 buckets, or 3 buckets, but never 1.5.
Why it matters: This "bucket" count is a direct signature of the material's topological nature. It proves the material has been successfully turned into a Chern insulator.

5. The Real-World Application: The "Bi2Se3–MnTe Sandwich"

The paper suggests a way to actually test this. Imagine a sandwich:

Top Bun: A material called Bi2Se3 (a topological insulator).
Meat: A layer of MnTe (the altermagnet).
The Experiment: Shine the spinning light on this sandwich.
The Measurement:
1. Heat one side of the sandwich.
2. Measure the voltage on the side (not the front or back). If you get a voltage, the "Anomalous Thermoelectric Hall" is working.
3. Measure the temperature difference on the side. If you get a temperature difference, the "Thermal Hall" is working.

The Big Takeaway

This paper shows that we don't need permanent magnets or complex wiring to create these exotic quantum effects. We can simply shine a light on a specific magnetic material to "switch on" a topological state.

Thermoelectric Hall acts like a sensitive ruler, measuring the exact edges of the energy gaps.
Thermal Hall acts like a topological badge, proving the material has entered a new, quantum state where heat flows in quantized steps.

It's a step toward "Floquet Engineering"—using light to build new materials on the fly, turning ordinary magnets into quantum super-highways for heat and electricity.

Here is a detailed technical summary of the paper "Anomalous thermoelectric and thermal Hall effects in irradiated altermagnets" by Fang Qin and Xiao-Bin Qiang.

1. Problem Statement

Altermagnets are a recently discovered class of collinear magnetic materials that combine the zero net magnetization of antiferromagnets with the spin-split electronic bands of ferromagnets. While they exhibit rich transport phenomena, their intrinsic topological responses (such as the anomalous Hall effect) are often symmetry-protected to be zero in the ground state due to specific crystal symmetries (e.g., $\hat{C}_4\hat{T}$ symmetry).

The central problem addressed in this work is:

Can a $d$ -wave altermagnet be driven into a non-trivial topological phase (specifically a Chern insulator) without breaking time-reversal symmetry via an external magnetic field?
How do the resulting topological band structures manifest in intrinsic anomalous thermoelectric (Nernst) and thermal Hall transport coefficients?
Can these transport signatures serve as sensitive probes for the light-induced band gaps and topological transitions?

2. Methodology

The authors employ a combination of Floquet theory, semiclassical transport theory, and topological invariants to analyze the system.

Model System: A two-dimensional effective low-energy Hamiltonian describing a $d$ -wave altermagnet:
$\hat{H} = v(k_y\sigma_x - k_x\sigma_y) + J_d(k_y^2 - k_x^2)\sigma_z$
where $v$ is the spin-orbit coupling strength and $J_d$ characterizes the altermagnetic order.
Floquet Engineering: The system is subjected to a high-frequency, elliptically polarized optical field ( $\mathbf{E}(t)$ ). Using the Magnus expansion (valid for off-resonant, high-frequency driving where $\hbar\omega \gg$ bandwidth), the time-dependent Hamiltonian is mapped to an effective static Floquet Hamiltonian ( $\hat{H}^{(F)}$ ).
Symmetry Breaking: The elliptical polarization breaks the $\hat{C}_4\hat{T}$ symmetry, inducing a mass term (Dirac mass) at the Dirac points located at the $\Gamma$ and $M$ points of the Brillouin zone.
Transport Formalism:
- The intrinsic anomalous thermoelectric ( $\alpha_{xy}^{in}$ ) and thermal Hall ( $\kappa_{xy}^{in}$ ) conductivities are derived using the semiclassical equations of motion for wavepackets, incorporating the Berry curvature ( $\Omega_{n,k}$ ).
- The Sommerfeld expansion is applied to derive low-temperature ( $T \to 0$ ) limits, establishing connections to the Mott relation and Wiedemann-Franz law.
Numerical Implementation: The continuum model is mapped to a square-lattice tight-binding model to calculate the Berry curvature and Chern numbers numerically.

3. Key Contributions

Floquet-Induced Chern Insulator: The paper demonstrates that elliptically polarized light can transform a $d$ -wave altermagnet into a Chern insulator with a total Chern number $|C|=1$ . This is achieved by opening energy gaps at both the $\Gamma$ and $M$ points, each contributing $|C|=1/2$ .
Critical Topological Phase Transition: The authors identify a critical light amplitude ( $A_0^c$ ) where a topological phase transition occurs. As the light amplitude increases, the gap at the $M$ point closes and reopens (band inversion), while the gap at the $\Gamma$ point remains open.
Distinct Transport Signatures:
- Thermoelectric Hall Effect: The coefficient $\alpha_{xy}$ vanishes within the bulk energy gap but exhibits pronounced peaks and dips at the gap edges (near $\Gamma$ and $M$ ). This makes it a sensitive probe for the electronic band structure and bandwidth.
- Thermal Hall Effect: The coefficient $\kappa_{xy}$ becomes quantized within the gapped regions, directly reflecting the underlying topological character (Chern number) of the phase.
Experimental Proposal: The authors propose a concrete experimental setup using a Bi $_2$ Se $_3$ –MnTe heterostructure, where the topological insulator (Bi $_2$ Se $_3$ ) provides strong Rashba spin-orbit coupling and the altermagnet (MnTe) provides the magnetic order, to measure these effects.

4. Results

Band Structure Evolution:
- No Light: The system possesses gapless Dirac cones at $\Gamma$ and $M$ . Intrinsic Hall responses are zero.
- With Light: Gaps open at both points. The size of the gap depends on the light amplitude $A_0$ and polarization.
- Phase Transition: At a critical amplitude $A_0^c \approx 1.4 \text{ nm}^{-1}$ (for specific parameters), the gap at the $M$ point closes, signaling a topological transition, before reopening with inverted bands.
Low-Temperature Transport:
- Thermoelectric Hall Coefficient ( $\alpha_{xy}$ ): Shows a linear dependence on temperature ( $T$ ). Crucially, it is zero inside the energy gap but spikes at the band edges. This behavior confirms that $\alpha_{xy}$ is sensitive to the density of states and Berry curvature distribution near the Fermi level.
- Thermal Hall Coefficient ( $\kappa_{xy}$ ): Also shows linear $T$ dependence but, unlike the thermoelectric case, it is quantized ( $\kappa_{xy} \propto C$ ) within the gapped regions. This quantization is a robust signature of the topological nature of the light-induced phase.
Universal Relations: The results satisfy the Mott relation ( $\alpha_{xy} \propto \partial \sigma_{xy}/\partial \mu$ ) and the Wiedemann-Franz law ( $\kappa_{xy}/\sigma_{xy} \propto T$ ) in the low-temperature limit, linking the thermal and electrical responses to the topological band structure.

5. Significance

New Platform for Topology: This work establishes altermagnets as a versatile platform for realizing light-induced topological phases (Chern insulators) without the need for external magnetic fields, which is advantageous for spintronic applications.
Probing Mechanisms: The study distinguishes between two powerful probes:
- Thermoelectric Hall transport is identified as a tool to map band edges and bandwidths (structural details).
- Thermal Hall transport is identified as a robust tool to detect topological phases (quantized invariants).
Experimental Feasibility: By proposing a specific material system (Bi $_2$ Se $_3$ –MnTe) and realistic parameters (high-frequency optical fields), the paper bridges the gap between theoretical Floquet engineering and experimental observation in condensed matter physics.
Fundamental Physics: It highlights the role of symmetry breaking via light in generating intrinsic transverse responses in magnetic materials that are otherwise forbidden by symmetry, expanding the understanding of non-equilibrium topological matter.