Signatures of three-state Potts nematicity in spin excitations of the van der Waals antiferromagnet FePSe$_3$

Neutron scattering experiments on the van der Waals antiferromagnet FePSe$_3$ under uniaxial strain reveal that tensile strain induces a transition to $C_2$ symmetry in both magnetic order and spin excitations, providing direct evidence that the observed three-state Potts nematicity in the paramagnetic phase arises from vestigial order associated with the low-temperature zigzag antiferromagnetic state.

The Gist

The Big Picture: A Magnetic Dance Floor

Imagine a crystal called FePSe₃ as a crowded dance floor made of tiny magnets (iron atoms). In this material, the magnets are arranged in a honeycomb pattern, like a beehive.

At high temperatures, these magnets are chaotic, spinning in random directions like people milling about at a loud party. But as the crystal cools down, they suddenly decide to organize. They form a specific pattern called a "zigzag" order, where they line up in rows, alternating directions.

The Problem: Three Equal Choices

The honeycomb dance floor has a special property: it looks the same if you rotate it by 120 degrees. Because of this, when the magnets decide to line up, they have three equally good options for how to arrange their zigzag rows. Let's call these options Direction A, Direction B, and Direction C.

In a normal, unstressed crystal, the magnets are fair. They pick all three directions equally. If you look at the whole crystal, the three directions cancel each other out, and the system looks perfectly symmetrical (like a triangle). This is called a three-state Potts state.

The Experiment: Pushing the Dance Floor

The scientists wanted to see what happens if they force the magnets to choose. They built a special device that gently stretches the crystal (like pulling on a rubber band) along one specific direction.

Think of this like a dance floor that is slightly tilted. If you tilt the floor, dancers who want to stand in one specific direction might feel unstable, while those standing in the other two directions feel more comfortable.

What happened when they stretched the crystal?

1. Breaking the Tie: The stretch (about 0.6% strain) was enough to make "Direction B" very uncomfortable. The magnets in that direction stopped forming.
2. The Winners: The magnets in "Direction A" and "Direction C" became the dominant groups.
3. The Result: The crystal lost its perfect triangular symmetry and became more like an oval (two-fold symmetry). The scientists could see this clearly using neutron beams, which act like a high-speed camera taking pictures of the magnetic patterns.

The Surprise: The Ghost of Order

Here is the most interesting part. The scientists heated the crystal back up, past the point where the magnets usually stop ordering (a temperature called $T_N$, roughly 108 K).

Usually, once you pass this temperature, the magnets go back to being chaotic and random, and the crystal should look perfectly symmetrical again (like a circle).

But it didn't.

Even though the long-range "zigzag" order was gone, the magnetic waves (the "spin excitations") still remembered the stretch. They still showed a preference for the two surviving directions and ignored the third one.

The Analogy:
Imagine a crowd of people at a party who were previously dancing in three distinct lines. The music stops (the temperature goes up), and everyone starts dancing randomly again. However, if you look closely at how they are moving, you can still see a slight "tilt" in their energy. They aren't dancing in a perfect circle; they are still subtly favoring the two directions that were comfortable before the music stopped.

This "ghost" of the previous order is what the paper calls vestigial nematicity. It suggests that even when the magnets aren't fully ordered, they are still "talking" to the crystal structure, creating a hidden preference that lasts for a tiny bit of time above the freezing point.

Why This Matters

The paper proves that in this material, the way the atoms move (the lattice) and the way the magnets spin are tightly coupled. You can't change one without affecting the other.

By using neutron scattering (which looks directly at the magnetic waves), the scientists provided the first direct proof that this "three-way choice" symmetry breaking exists in the magnetic waves themselves, not just in the static arrangement of the atoms. They showed that the "nematic" state (the directional preference) is a fundamental property of how these spins interact, persisting even when the main magnetic order disappears.

Summary

• The Material: A magnetic crystal with a honeycomb shape.
• The Setup: Scientists stretched the crystal to force the magnetic "dancers" to drop one of their three possible formation choices.
• The Discovery: The stretch worked, forcing the magnets into a two-direction pattern.
• The Twist: Even after heating the crystal until the main order vanished, the magnetic waves still remembered the stretch and kept the two-direction pattern for a short while.
• The Conclusion: This proves a strong link between the crystal's shape and its magnetic behavior, revealing a hidden "nematic" phase in the spin excitations.

Technical Summary

Problem and Context
In two-dimensional quantum materials, electron correlations can induce electronic nematic phases characterized by the spontaneous breaking of rotational symmetry. While nematicity breaking fourfold ($C_4$) symmetry in square-lattice systems (such as iron-based superconductors) is well-documented, the exploration of three-state Potts nematicity in systems with threefold ($C_3$) symmetry, such as honeycomb lattices, remains less understood. Previous studies on the van der Waals antiferromagnet (AFM) FePSe$_3$ using optical and thermodynamic methods under uniaxial strain suggested the existence of a vestigial three-state Potts nematic order associated with the underlying zigzag AFM order. However, these macroscopic probes could not directly determine the microscopic origin of the $C_3$ symmetry breaking, the wave vector of the magnetic order, or the momentum dependence of spin dynamics in the nematic state. A critical open question was whether the nematicity observed above the Néel temperature ($T_N$) in the paramagnetic state is a vestigial order of the low-temperature AFM phase or driven by other mechanisms.

Methodology
To address these questions, the authors employed inelastic neutron scattering (INS) to study the magnetic order and spin excitations of FePSe$_3$ single crystals under applied uniaxial tensile strain.

• Sample Preparation: High-quality FePSe$_3$ single crystals were synthesized via chemical vapor transport. Approximately 1 gram of co-aligned crystals was attached to thin aluminum plates mounted on an Invar alloy frame.
• Strain Application: Upon cooling, the mismatch in thermal expansion coefficients between the Invar frame and aluminum plates induced a uniaxial tensile strain of approximately 0.6% along the $[-0.5K, K, 0]$ direction (perpendicular to the nearest Fe-Fe bonds). This strain was calibrated using cryogenic digital image correlation (CDIC).
• Experimental Setup: Neutron scattering experiments were conducted on the time-of-flight spectrometer MERLIN at the ISIS Neutron and Muon Source. Data were collected at multiple temperatures (6.0 K to 113.0 K) using incident neutron energies of 61.9, 21.3, and 11.1 meV.
• Analysis: The study focused on the intensity of magnetic Bragg peaks to monitor domain populations and analyzed spin excitation spectra to determine symmetry properties. The intensity difference $\Delta I(Q)$ was defined as an effective nematic order parameter to quantify $C_3$ symmetry breaking.

Key Results

1. Domain Detwinning and Symmetry Reduction: In the AFM ordered state (below $T_N \approx 108.6$ K), the applied 0.6% tensile strain significantly suppressed one of the three zigzag magnetic domains (ZZ2) while promoting the other two (ZZ1 and ZZ3). This resulted in a detwinning ratio of $\eta \approx 75.7\%$. Consequently, the magnetic order and spin waves below $T_N$ exhibited $C_2$ symmetry rather than the intrinsic $C_3$ symmetry of the honeycomb lattice.
2. Critical Behavior: The magnetic phase transition under strain was found to be weakly first-order but approaching second-order behavior, with a critical exponent $\beta \approx 0.10$, closer to the 2D Ising limit ($1/8$) than the unstrained case.
3. Nematicity in the Paramagnetic State: Crucially, the study observed that the breaking of $C_3$ symmetry in spin excitations persisted slightly above $T_N$ (up to $\approx 109.5$ K). In this narrow temperature regime within the paramagnetic state, spin excitations remained anisotropic ($C_2$ symmetric) along specific momentum directions, despite the absence of long-range static magnetic order.
4. Energy Gap Evolution: At 108.1 K (just below $T_N$), a spin-wave energy gap of $\sim 13$ meV was observed. This gap closed completely at 108.8 K and above, indicating that the observed anisotropic excitations above $T_N$ are gapless paramagnetic fluctuations rather than gapped spin waves.
5. Symmetry Restoration: The $C_3$ symmetry of the spin excitation spectra was fully restored at 113.0 K, well above the transition temperature, confirming that the anisotropy observed just above $T_N$ is an intrinsic magnetic response rather than a mechanical artifact of the strain.

Significance and Claims
The authors claim that these results provide direct microscopic evidence of magnetoelastic coupling in FePSe$_3$. The persistence of $C_2$-symmetric spin excitations in the paramagnetic state immediately above $T_N$ suggests the existence of a vestigial three-state Potts nematic order. This nematicity arises from the fluctuations associated with the low-temperature zigzag AFM order.

The study distinguishes the behavior of FePSe$_3$ from iron-based superconductors (e.g., FeSe, BaFe$_2$As$_2$). In those systems, nematicity often spans a broad temperature range above $T_N$ and is linked to a tetragonal-to-orthorhombic lattice distortion. In contrast, the nematic phase in FePSe$_3$ exists in an extremely narrow temperature regime above $T_N$ and occurs without a known structural lattice distortion associated with the AFM transition. This implies that while a small but finite magnetoelastic coupling exists, the order-disorder phase transition in FePSe$_3$ may not be driven primarily by lattice distortion. The work establishes FePSe$_3$ as a platform for studying spin-driven three-state Potts nematicity and demonstrates that neutron scattering can resolve the microscopic origins of symmetry breaking in nematic phases where optical and thermodynamic methods fall short.