Nonlinear thermal gradient induced magnetization in $d^{\prime }$, $g^{\prime }$ and $i^{\prime }$ altermagnets

This paper demonstrates that a finite magnetization can be induced by a second-order nonlinear temperature gradient in $d^{\prime }$, $g^{\prime }$, and $i^{\prime }$ altermagnets due to their specific spin-split band structures, whereas such a response is absent in their $d$, $g$, and $i$ counterparts as well as in various odd-parity magnets.

The Gist

Imagine you have a very special kind of magnet. Unlike a regular fridge magnet that sticks to your door, this one is a "hidden" magnet. If you look at it from the outside, it appears to have no magnetic pull at all because its internal north and south poles cancel each other out perfectly. Scientists call these Altermagnets. They are exciting because they could be the future of super-fast, super-dense computer memory.

But here is the big question this paper asks: Can we wake up this "sleeping" magnet just by heating it unevenly?

The Setup: The Temperature Gradient

Usually, if you heat one side of an object and cool the other, you create a "temperature gradient" (a slope of heat). Think of it like a hill where the top is hot and the bottom is cold.

In most materials, if you just apply a gentle slope of heat, nothing magnetic happens. The laws of physics (specifically, symmetry) forbid it. It's like trying to roll a ball uphill without pushing it; it just won't move.

However, the authors of this paper asked: What if we don't just push gently, but we push hard and non-linearly?

Imagine the temperature hill isn't a smooth slope. Imagine it's a bumpy, wavy road where the heat changes in a complex, curvy way. This is a nonlinear temperature gradient. The paper investigates what happens when you apply this "wavy heat" to these special magnets.

The Cast of Characters: The "Wave" Magnets

The paper looks at a family of these special magnets, named after the shapes of their internal waves. Think of them like different musical instruments or wave patterns:

• The "d-wave" and "g-wave" magnets: These are like waves that look like a cross or a cloverleaf.
• The "d'-wave" and "g'-wave" magnets: These are the "twins" of the first group, but their waves are rotated or flipped, like a mirror image.

The paper also looks at "odd-parity" magnets (p-wave, f-wave), which are a different species entirely.

The Discovery: The "Mirror Image" Matters

The researchers ran the numbers (and simulations) to see what happens when you apply that "wavy heat" to these different magnets.

1. The "d-wave" and "g-wave" magnets: When you apply the wavy heat, nothing happens. They remain perfectly still. The symmetry of their internal structure is like a perfectly balanced seesaw; no matter how you wiggle the heat, the magnetization stays zero.
2. The "d'-wave" and "g'-wave" magnets: BAM! Suddenly, a magnetic field appears! Even though the material has no net magnetism at rest, the specific way the heat waves interact with the "flipped" internal structure of these magnets causes a tiny magnetic field to pop into existence.

The Analogy:
Think of the "d-wave" magnet as a perfectly symmetrical snowflake. If you blow hot air on it from the side, it just melts a bit, but it doesn't start spinning or moving in a specific direction.
Now, think of the "d'-wave" magnet as a snowflake that has been slightly twisted or has a secret pattern hidden inside. When you blow that same hot air on it, the heat interacts with the twist, causing the snowflake to suddenly start spinning (creating magnetization).

Why Does This Happen?

The paper explains that this is a second-order effect.

• Linear (First-order): A gentle slope of heat = No magnetism. (Forbidden by physics).
• Nonlinear (Second-order): A complex, curvy slope of heat = Magnetism!

It turns out that for the "prime" versions (d', g', i'), the math works out so that the heat energy gets converted into magnetic energy. For the non-prime versions, the math cancels itself out.

The "Néel Vector" Detective

One of the coolest parts of the discovery is that the direction of this newly created magnetism points exactly along the Néel vector.

• What is the Néel vector? It's the "secret compass" inside these hidden magnets that tells you which way the internal north and south poles are pointing.
• Why is this a big deal? Usually, you can't see the Néel vector because the magnet looks empty from the outside. But this paper shows that if you heat these specific magnets in a specific way, they will "speak" by generating a magnetic field that points exactly where their internal compass is. It's like a silent person suddenly shouting their location when you tap them on the shoulder in a specific rhythm.

The Bottom Line

This paper proves that heat can create magnetism in specific types of advanced materials, but only if:

1. The material is a specific "twisted" version of an altermagnet (d', g', i').
2. The heat isn't just a simple slope, but a complex, nonlinear wave.

Why should we care?

• New Tech: This could lead to new ways of reading and writing data in computers without using electricity, just heat.
• No "Rashba" Needed: Previous methods to create magnetism often required a specific, weak interaction called the "Rashba effect." This new method doesn't need that; it relies on the material's natural structure, which is much stronger and more efficient.
• Measurable: The authors calculated that the magnetic signal is strong enough to be detected by current lab equipment (SQUIDs), meaning this isn't just theory—it's something we can actually test in a lab.

In short: If you want to wake up a sleeping magnet, don't just push it gently. Give it a complex, wavy heat massage, but only if it's the "twisted" kind of magnet!

Technical Summary

1. Problem Statement

The central question addressed is whether a magnetization can be induced by a nonlinear temperature gradient ($\nabla T$) in the absence of any linear component.

• Context: While nonlinear electric conductivity and nonlinear spin conductivity are well-studied, the nonlinear thermal response of magnetization remains unexplored.
• Symmetry Constraints:
  • Linear Response: Forbidden in altermagnets because they preserve inversion symmetry ($\mathcal{I}$). Magnetization ($\mathbf{M}$) is an axial vector (odd under time-reversal $\mathcal{T}$, even under $\mathcal{I}$), while a temperature gradient ($\nabla T$) is a polar vector (odd under $\mathcal{I}$, even under $\mathcal{T}$). Thus, a linear term $\mathbf{M} \propto \nabla T$ violates inversion symmetry.
  • Nonlinear Response: A second-order response $\mathbf{M} \propto (\nabla T)^2$ is allowed by inversion symmetry (since $\mathcal{I}(\nabla T)^2 = \mathcal{I}(\nabla T) \cdot \mathcal{I}(\nabla T) = (-\nabla T)(-\nabla T) = (\nabla T)^2$). However, the system must break time-reversal symmetry ($\mathcal{T}$), which altermagnets do.
• Gap: It is unclear if this second-order response exists for all types of altermagnets or if it depends on the specific symmetry of the spin-split band structure (e.g., $d$-wave vs. $d'$-wave).

2. Methodology

The author employs a combination of symmetry analysis, analytical derivation, and numerical verification.

• Theoretical Framework:
  • The magnetization expectation value is derived using the non-equilibrium Fermi distribution function determined by the Boltzmann equation.
  • A general formula for the $\ell$-th order nonlinear thermal gradient induced magnetization is derived:
    $$\mathbf{M}^{(\ell)} = g\mu_B \left( -\frac{\tau}{\hbar} \partial_x T \right)^\ell \int d^3k \, \mathbf{S}(\mathbf{k}) \left( \frac{\partial \varepsilon}{\partial k_x} \right)^\ell \frac{\partial^\ell f^{(0)}}{\partial T^\ell}$$
    where $\tau$ is the relaxation time, $\varepsilon$ is energy, and $f^{(0)}$ is the equilibrium Fermi distribution.
  • High-Temperature Expansion: An analytic expression is derived for the high-temperature regime ($k_B T \gg |\varepsilon - \mu|$), simplifying the derivative of the distribution function.
• Model Systems:
  • Altermagnets: Investigated $d, g, i$-wave (described by $\sin N_X \phi$) and $d', g', i'$-wave (described by $\cos N_X \phi$) altermagnets.
  • Odd-Parity Magnets: Investigated $p, f, p', f'$-wave magnets for comparison.
  • Hamiltonians: Two-band models were constructed with a kinetic term ($H_{kin}$) and a spin-splitting term ($J f_X(\mathbf{k}) \sigma_z$). Tight-binding models were used for lattice simulations (square and triangular lattices).
• Numerical Validation: Numerical calculations were performed to verify the analytic high-temperature expansion results across various parameters (chemical potential $\mu$, coupling $J$, temperature $\beta$).

3. Key Contributions

1. Discovery of Symmetry-Dependent Nonlinear Response: The paper establishes a fundamental distinction between different classes of altermagnets based on their spin-splitting functions:
   • $d, g, i$-wave altermagnets (functions $\propto \sin N_X \phi$): These possess zero second-order nonlinear magnetization response. The integrand in the magnetization formula becomes odd with respect to $k_x$, causing the integral to vanish.
   • $d', g', i'$-wave altermagnets (functions $\propto \cos N_X \phi$): These exhibit a finite second-order nonlinear magnetization response. The symmetry of the $\cos$ term allows the integrand to be non-zero.
2. Analytic Expressions: The author derives closed-form analytic expressions for the induced magnetization in $d'$, $g'$, and $i'$-wave altermagnets in the high-temperature limit.
3. Exclusion of Odd-Parity Magnets: The study demonstrates that odd-parity magnets (which break inversion symmetry but preserve time-reversal symmetry) do not exhibit this even-order nonlinear response, as the time-reversal symmetry forbids a net magnetization from an even-order thermal perturbation.
4. Connection to Néel Vector: The resulting magnetization is shown to be proportional to the Néel vector (specifically the coupling constant $J$ and its sign), suggesting a new experimental method to detect the Néel vector via magnetization measurements.

4. Key Results

• Vanishing Response in Standard Altermagnets: For $d$-wave ($d_{xy}$), $g$-wave, and $i$-wave altermagnets, $M^{(2)}_z = 0$. This is due to the odd parity of the spin-splitting function $f_X(\mathbf{k})$ with respect to $k_x$ combined with the even parity of the kinetic energy.
• Finite Response in Prime Altermagnets:
  • $d'$-wave ($d_{x^2-y^2}$): The induced magnetization is given by:
    $$M_z \propto -\frac{J(m\mu - 2)}{m^2 k_B T^3}$$
  • $g'$-wave and $i'$-wave: Explicit analytic formulas were derived (Eqs. 27 and 32), showing a complex dependence on $J$, $\mu$, and $T$, but confirming a non-zero value.
• Numerical Agreement: The analytic results derived from the high-temperature expansion (cyan curves in Figs. 1-3) show excellent agreement with full numerical calculations (red curves) without expansion, validating the theoretical approach.
• Magnitude Estimation:
  • Using typical parameters ($\tau = 3\times 10^{-12}$ s, $v_F = 10^6$ m/s, $\nabla T = 1$ K/mm, $T=300$ K), the induced magnetization density is estimated at $\sim 0.3$ A/m.
  • For a $1$ mm$^3$ sample, the total magnetic moment is $\sim 3 \times 10^{-10}$ A$\cdot$m$^2$.
  • This magnitude is detectable by standard SQUID magnetometers (sensitivity $\sim 10^{-11}$ to $10^{-14}$ A$\cdot$m$^2$).

5. Significance

• New Detection Mechanism: This work proposes a novel, purely thermal method to detect the Néel vector in altermagnets. Unlike electrical methods (which often require Rashba spin-orbit coupling), this mechanism relies on the intrinsic spin-splitting of the altermagnet, which is typically much larger ($\sim 100$ meV) than Rashba interactions ($\sim$ meV).
• Symmetry Classification: It refines the classification of X-wave magnets, demonstrating that the specific angular dependence ($\sin$ vs. $\cos$) of the spin-splitting function dictates the existence of nonlinear thermal responses. This highlights the importance of $d'$, $g'$, and $i'$ wave altermagnets, which have been less studied than their $d, g, i$ counterparts.
• Thermoelectric Spintronics: The findings open a new avenue for thermoelectric spintronics, where temperature gradients can be used to manipulate and detect magnetic order in materials with zero net magnetization, potentially leading to ultrafast and ultradense magnetic memory applications.
• Experimental Feasibility: The estimated signal strength suggests that this effect is not just a theoretical curiosity but is experimentally observable with current technology.