Floquet engineering spin triplet states in unconventional magnets

This paper demonstrates that high- and low-frequency light drives can induce and control novel spin-triplet states and odd-frequency superconducting correlations in unconventional magnets and altermagnets, revealing new phases absent in static systems.

The Gist

Imagine a world where the tiny magnets inside a material usually cancel each other out, leaving the material with no overall magnetic pull. This is like a room full of people shouting in opposite directions; the noise cancels out, and the room seems quiet. In physics, we call these materials unconventional magnets (specifically, "altermagnets"). They are special because, even though they are quiet overall, the "shouting" (spin) inside is organized in a very specific, complex pattern based on the direction you look.

Now, imagine you want to wake up a specific type of "magnetic whisper" inside this quiet room—a spin triplet state. Normally, this is impossible because the material's internal rules forbid it.

This paper is about using light as a magical wand to break those rules and create these forbidden states. Here is how they did it, explained simply:

1. The Magic Wand: Floquet Engineering

Think of the material as a calm lake. If you just look at it, the water is still. But if you start tapping the surface rhythmically with a stick (shining a light on it), you create waves. In physics, this rhythmic shaking is called Floquet engineering.

The researchers found that if you tap the lake with a specific rhythm (using linearly polarized light, which is like light vibrating in a straight line), you can change the fundamental rules of the water. You aren't just making waves; you are temporarily changing the physics of the lake itself.

2. The Discovery: Waking Up the "Triplet"

Inside these special magnets, there are two types of "dancing pairs" of electrons:

• Singlets: Like a couple holding hands and spinning in opposite directions (canceling out).
• Triplets: Like a couple holding hands and spinning in the same direction.

In the natural, static state of these magnets, triplets are forbidden. They simply don't exist. It's like trying to get a group of people to dance in a circle when the music strictly forbids it.

The Breakthrough: The researchers discovered that by shining high-frequency, straight-line vibrating light on these magnets, they could force the triplets to appear.

• The Analogy: Imagine a strict bouncer at a club (the magnet's natural state) who only lets in couples dancing a specific way. The light acts like a "VIP pass" that temporarily changes the bouncer's rules, allowing the forbidden triplets to enter the dance floor.

3. The "Dip-Peak" Fingerprint

One of the coolest parts of the paper is how they can measure this.
When the light creates these new triplet states, it leaves a unique mark on the material's energy levels. The researchers found a "dip-peak" structure.

• The Analogy: Imagine you are trying to guess the weight of a hidden object by how much a trampoline bounces. If you drop a ball on the trampoline, it makes a specific dent (a dip) and then a high bounce (a peak). The distance between that dip and the peak tells you exactly how heavy the hidden object is.
• In the paper: The distance between the "dip" and the "peak" in the light's interaction tells scientists exactly how strong the magnet's internal field is and which way it is pointing. It's a new, super-precise ruler for measuring magnetic strength.

4. The Superconductor Twist

The paper also looked at what happens if you attach a superconductor (a material that conducts electricity with zero resistance) to these magnets.

• The Analogy: Think of the superconductor as a factory that only produces "Singlet" products (standard pairs). When you shine the light on the magnet next to it, the factory gets confused. Because the light changed the magnet's rules, the factory accidentally starts producing "Triplet" products too!
• The Result: They created a new type of superconducting pair that doesn't exist in nature under normal conditions. These are "odd-frequency" pairs, which is a fancy way of saying they dance to a beat that is out of sync with normal time.

5. Why This Matters

• New Tools: This gives scientists a way to turn "off" and "on" magnetic properties using light, like a switch.
• Better Computers: These new "triplet" states are the holy grail for spintronics (computing using electron spin instead of charge). They could lead to faster, more efficient, and more powerful computers.
• Universal Rule: The paper shows this isn't just a fluke for one specific magnet. It works for a whole family of these "unconventional" magnets, whether they are simple or very complex.

Summary

In short, the researchers used a rhythmic beam of light to shake up a special type of magnet. This shaking broke the material's natural rules, allowing a forbidden type of magnetic pairing (triplets) to emerge. They found a unique "fingerprint" (dip-peak) that lets them measure the magnet's strength with incredible precision. This opens the door to building new types of quantum devices that use light to control magnetism and superconductivity.

Technical Summary

1. Problem Statement

Unconventional magnets (UMs), particularly altermagnets (AMs) with $d$-wave parity and $p$-wave magnets, possess a unique electronic structure: they exhibit zero net magnetization (like antiferromagnets) but feature a non-relativistic, momentum-dependent spin splitting of energy bands (like ferromagnets). While the static properties of these materials have been extensively studied, the response of UMs to time-periodic light drives (Floquet engineering) remains an open question.

Specifically, the authors aim to determine:

1. Can light drives induce spin-triplet states (spin densities and Cooper pairs) in UMs that are absent in the static equilibrium regime?
2. How do the symmetry properties (parity) of the magnetism ($d$-wave vs. $p$-wave) influence the light-matter interaction?
3. Can these light-induced states serve as a spectroscopic tool to probe the strength and orientation of the underlying altermagnetic order?

2. Methodology

The authors employ a theoretical framework based on Floquet theory to analyze UMs under time-periodic electromagnetic fields.

• Models:
  • Normal State: Described by a Hamiltonian $H_j(\mathbf{k}) = \xi_{\mathbf{k}} + M^j_{\mathbf{k}}\sigma_z$, where $M^j_{\mathbf{k}}$ represents the momentum-dependent magnetic order for $j$-wave parities ($d$-wave and $p$-wave).
  • Superconducting State: Modeled using the Bogoliubov–de Gennes (BdG) formalism, coupling the UMs to a conventional spin-singlet $s$-wave superconductor.
• Light-Matter Coupling:
  • The light drive is introduced via minimal coupling substitution $\mathbf{k} \to \mathbf{k} + e\mathbf{A}(t)$.
  • Two polarization types are considered: Linearly Polarized Light (LPL) and Circularly Polarized Light (CPL).
  • The time-dependent Hamiltonian is expanded into a Fourier series (Floquet harmonics) to solve the eigenvalue problem in the frequency domain.
• Regimes Analyzed:
  • High-Frequency Regime: An effective Hamiltonian ($H_{\text{eff}}$) is derived using the high-frequency expansion (Magnus expansion), focusing on the leading order terms.
  • Low-Frequency Regime: Full Floquet band structures are analyzed to account for photon absorption/emission processes coupling different Floquet sidebands.
• Observables:
  • Spin Density ($\rho_z$): Calculated from the retarded Green's function.
  • Pair Amplitudes: Derived from the anomalous Green's function to identify spin-singlet ($F_s$) and spin-triplet ($F_t$) Cooper pairs, including their frequency (odd/even) and parity ($s, d, p$-wave) symmetries.

3. Key Contributions and Results

A. High-Frequency Regime: Induced Spin Densities and Triplet Pairs

• Linear Polarization is Key: The authors find that linearly polarized light induces a non-trivial light-matter coupling in even-parity UMs (specifically $d$-wave altermagnets), whereas circular polarization does not induce these effects in the high-frequency limit due to vanishing commutators in the effective Hamiltonian.
• Emergence of Spin-Triplet Density: In static $d$-wave altermagnets, the integrated spin density is zero due to symmetry. However, under high-frequency LPL, a finite spin-triplet density emerges.
  • This density arises from a light-induced Zeeman-like field ($M^d_\Omega$) that depends on the angle between the altermagnetic lobes and the light polarization.
  • Spectroscopic Signature: The spin density exhibits a distinct peak-dip structure in the frequency domain. The energy separation ($\delta\omega$) between the peak and dip is directly proportional to the altermagnetic strength ($M$). This provides a direct method to extract $M$ and the orientation angle $\theta_d$ via electron spin resonance or optical spectroscopy.
• Odd-Frequency Spin-Triplet Cooper Pairs: When coupled to an $s$-wave superconductor, the light drive induces odd-frequency spin-triplet pairs in $d$-wave altermagnets.
  • These pairs possess mixed parities: the static interaction induces $d$-wave parity, while the light drive induces an $s$-wave parity component.
  • The resulting pair amplitude shows a characteristic peak-dip structure similar to the normal state spin density, allowing the extraction of altermagnetic parameters via Andreev conductance measurements.

B. Low-Frequency Regime: Broadened Symmetries

• In the low-frequency regime, photon absorption/emission processes couple different Floquet bands.
• This coupling broadens the types of superconducting correlations available.
• Universal Triplet Generation: Unlike the high-frequency regime where effects are restricted to even-parity magnets, low-frequency driving enables the generation of spin-triplet pairs in both $d$-wave and $p$-wave UMs.
• New Pairing Classes: The system supports "Floquet" Cooper pairs with mixed symmetries (e.g., odd-Floquet, odd-frequency, odd-parity) that are strictly forbidden in the static regime.

C. Generalization to Higher-Order Parities

• The study extends to higher-order parities ($g$-wave, $i$-wave, etc.).
• Even-Parity Magnets ($d, g, i$): Exhibit the light-induced Zeeman field and the characteristic peak-dip structure in spin density/triplet amplitude. The magnitude of the splitting scales with the power of the driving amplitude corresponding to the parity order (e.g., $(eA_0)^4$ for $g$-wave).
• Odd-Parity Magnets ($p, f$): Do not exhibit the light-induced Zeeman term in the high-frequency limit; thus, no net spin density or integrated triplet amplitude is generated in the same manner as even-parity systems.

4. Significance and Implications

1. Probe for Altermagnetism: The paper proposes a robust experimental protocol to detect and quantify altermagnetism. The peak-dip structure in spin density or Andreev conductance serves as a "fingerprint" for the strength ($M$) and orientation ($\theta$) of the altermagnetic order, which is difficult to measure using conventional magnetic probes due to zero net magnetization.
2. Engineering Exotic Superconductivity: It demonstrates that light can be used to engineer odd-frequency spin-triplet superconductivity with tunable symmetries ($s$-wave and $d$-wave components) in materials that are not naturally triplet superconductors.
3. Universality of Light-Matter Coupling: The work highlights a fundamental distinction between even- and odd-parity unconventional magnets regarding their response to linearly polarized light, rooted in the non-relativistic spin-orbit coupling nature of altermagnets.
4. Experimental Feasibility: The authors estimate that the required driving fields ($E_0 \sim 0.4 - 6$ MV/cm) and frequencies (mid-infrared/terahertz) are within the range of current experimental capabilities (e.g., pump-probe spectroscopy, tr-ARPES) and have been successfully applied in similar systems like graphene and topological insulators. The effects are predicted to be robust against screening in metallic films.

In conclusion, this work establishes Floquet engineering as a powerful tool to manipulate and probe unconventional magnets, offering a pathway to create and detect non-equilibrium spin-triplet states that are invisible in static conditions.