Visualizing the interplay of dual electronic nematicities in kagome superconductors

By combining scanning tunneling microscopy with Ginzburg-Landau analysis, this study reveals that the kagome superconductor CsV$_3$Sb$_5$ hosts two distinct, coexisting electronic nematic orders—one tied to charge density waves and another to V-$d_{x^{2}-y^{2}}$ orbital distortions—that exhibit complex interplay, temperature-dependent alignment, and persistence even when long-range CDW order is suppressed.

The Gist

Imagine a crowded dance floor where the dancers are tiny electrons, and the floor itself is a special, honeycomb-like pattern called a kagome lattice. In a special material called CsV₃Sb₅, these electrons don't just dance randomly; they organize themselves into complex, synchronized patterns.

For a long time, scientists were puzzled by two specific "dance moves" (electronic orders) happening in this material:

1. The Big Wave (CDW): A massive, organized ripple that moves across the whole floor, breaking the symmetry of the room.
2. The Twisted Spin (Nematicity): A subtle twist where the dancers prefer to face one specific direction, like everyone suddenly deciding to look North instead of East.

The big mystery was: Are these two moves part of the same dance routine, or are they two different dancers trying to lead the floor at the same time?

The Experiment: Changing the Music

To solve this, the researchers acted like DJs. They took the material and swapped some of the original dancers (Vanadium atoms) with new ones (Titanium atoms). Think of this as changing the tempo or the genre of the music.

• Heavy Doping (x=0.18): They added so many new dancers that the "Big Wave" (the CDW) completely stopped. The floor was calm.

  • The Surprise: Even though the Big Wave was gone, the dancers were still twisted! They were still facing a specific direction. This revealed a second, hidden dance move that exists independently of the Big Wave. It's like finding out the dancers have a natural tendency to turn left, even when there's no music forcing them to do so.

• Medium Doping (x=0.12): They added a moderate amount of new dancers. Now, both the Big Wave and the Twisted Spin were present.

  • The Conflict: Here, the researchers noticed something weird. The Big Wave wanted the dancers to face North, but the Twisted Spin wanted them to face Northeast. They were fighting for control! The two "leaders" were misaligned, pulling the electrons in slightly different directions.

• No Doping (x=0.0): This is the original, pristine material.

  • The Harmony: In the original state, the Big Wave is very strong. It's so powerful that it forces the Twisted Spin to fall in line. Both moves now face the exact same direction. The electrons are perfectly synchronized, but only because the Big Wave is the boss.

The "Two-Headed" Monster

The paper's main discovery is that this material actually hosts two distinct types of electronic nematicity (twisted states):

1. The CDW-induced twist: This is the one caused by the Big Wave. It's like a crowd following a marching band.
2. The Intrinsic twist: This is a "native" preference of the electrons themselves, coming from a different part of their internal structure (specifically, their orbital shapes). It's like the dancers having a personal habit of leaning left, regardless of the band.

Why Does This Matter?

Think of it like a relationship between two people:

• Sometimes they are independent (when the Big Wave is gone, the intrinsic twist still exists).
• Sometimes they are at odds (at medium doping, they want to face different directions).
• Sometimes they are locked in a tight embrace (in the pristine material, the strong Big Wave forces them to align).

The researchers used a mathematical framework (Ginzburg-Landau theory) to map out this relationship, showing how the "coupling" between these two forces changes as you add more Titanium.

The Takeaway

This paper solves a long-standing puzzle by showing that the strange behavior of these superconductors isn't just one thing. It's a complex interplay between two different electronic personalities.

• The "Hidden" State: The fact that the intrinsic twist survives even when the Big Wave is destroyed suggests that the electrons have a deep, intrinsic desire to break symmetry.
• The "High-Temperature" Clue: This hidden twist is visible at much higher temperatures than the Big Wave. This suggests that before the material settles into its final, ordered state, there is a chaotic "fluctuation" phase where these two forces are fighting it out.

In short, the kagome superconductor is a stage where two different directors are trying to choreograph the same dance. Sometimes they agree, sometimes they fight, and sometimes one leaves the stage entirely, revealing that the other was there all along. Understanding this "dual nematicity" helps scientists unlock the secrets of how these materials become superconductors and might lead to new technologies in the future.

Technical Summary

1. Problem Statement

The layered kagome superconductors $AV_3Sb_5$ ($A$ = K, Rb, Cs) exhibit a complex landscape of intertwined electronic orders driven by geometric frustration and electron correlations. A central puzzle in the prototypical compound $CsV_3Sb_5$ is the nature of the electronic state below a characteristic temperature $T^* \approx 35$ K. While the material undergoes a $2 \times 2 \times 2$ charge density wave (CDW) transition at $T_{CDW} \approx 80-100$ K, a secondary symmetry-breaking state emerges at lower temperatures.

• The Core Question: Spectroscopic probes reveal two distinct features below $T^*$: a pronounced nematic CDW modulation and a twofold ($C_2$) symmetric distortion of the Fermi surface (electronic nematicity). It remains unclear whether these are consequences of a single ordering mechanism or if they originate from distinct, coexisting order parameters.
• The Challenge: Disentangling the relationship between the CDW-induced nematicity and the intrinsic electronic nematicity, particularly how they interact, align, or decouple under external perturbations like chemical doping.

2. Methodology

The authors employed Spectroscopic-Imaging Scanning Tunneling Microscopy (SI-STM) to investigate the local electronic structure of Ti-doped $CsV_{3-x}Ti_xSb_5$ samples with varying doping levels ($x = 0.0, 0.12, 0.18$) across a range of temperatures.

• Quasiparticle Interference (QPI) Mapping: High-precision Fourier transforms of differential conductance ($dI/dV$) maps were used to reconstruct the constant-energy contours and scattering vectors near the Fermi level.
• Orbital Resolution: The study distinguished between scattering channels associated with different orbitals:
  • $q_0$: Scattering involving out-of-plane $Sb$-$p_z$ orbitals.
  • $q_1, q_2$: Scattering involving in-plane $V$-$d_{x^2-y^2}$ orbitals (triangular pockets).
  • $q_3$: Scattering involving $V$-$d_{xy}$ orbitals (hexagonal pockets).
• Doping Strategy: Titanium substitution on Vanadium sites was used to systematically suppress the long-range CDW order while tracking the evolution of electronic nematicity.
• Theoretical Framework: A Ginzburg-Landau (GL) theory was developed to model the free energy of two coupled nematic order parameters: one associated with the CDW ($\phi$) and one intrinsic ($\eta$).

3. Key Contributions & Results

A. Discovery of a Distinct $q=0$ Nematic State

In highly doped samples ($x=0.18$), where the long-range CDW order is completely suppressed (no $2 \times 2$ modulation detected in STM or transport), the authors identified a robust intra-unit-cell ($q=0$) electronic nematic state.

• Evidence: QPI maps revealed a pronounced $C_2$ symmetry breaking in the $V$-$d_{x^2-y^2}$ Fermi pockets (scattering vectors $q_1$ and $q_2$), while the $Sb$-$p_z$ related vectors ($q_0$) remained isotropic.
• Temperature Stability: This nematic state persists up to high temperatures (up to 80 K), far exceeding the $T^* \approx 35$ K where quasiparticle coherence vanishes in the pristine sample.
• Intrinsic Nature: Statistical analysis confirmed that the orientation of this nematicity is uncorrelated with the local orientation of Ti impurities, proving it is an intrinsic electronic instability rather than a defect-induced effect.

B. Decoupling and Misalignment at Intermediate Doping

At intermediate doping ($x=0.12$), where the CDW order is partially suppressed but still present, the study revealed the coexistence of two distinct nematic orders:

1. CDW-induced Nematicity: Associated with the $2 \times 2$ charge modulation.
2. Intrinsic $q=0$ Nematicity: Associated with the $V$-$d_{x^2-y^2}$ pocket distortion.

• Critical Finding: The principal axes ($C_2$ symmetry axes) of these two orders are misaligned (rotated by $\pi/3$ relative to each other). This demonstrates that the two orders are independent entities that can exist simultaneously but are not locked together at this doping level.

C. Alignment in the Pristine Limit

In the pristine sample ($x=0$), the strong CDW order dominates. The study found that the intrinsic $q=0$ nematic distortion is pinned to the CDW axis.

• The $C_2$ axis of the QPI pattern aligns perfectly with the principal axis of the nematic CDW modulation.
• This suggests a strong coupling mechanism that locks the two orders together in the undoped limit.

D. Theoretical Modeling (Ginzburg-Landau)

The authors proposed a GL free energy model with two order parameters ($\phi$ for CDW-nematicity and $\eta$ for intrinsic nematicity) coupled via a bilinear term ($\beta_{\phi\eta}\phi \cdot \eta$).

• Doping Evolution:
  • High Doping ($x=0.18$): CDW is suppressed ($\phi \to 0$); only intrinsic $\eta$ remains.
  • Intermediate Doping ($x=0.12$): Both orders coexist, but a weak positive coupling ($\beta_{\phi\eta} > 0$) leads to a misaligned state (energy minimum at a relative rotation).
  • Pristine ($x=0$): Strong CDW predominates; the coupling term changes sign ($\beta_{\phi\eta} < 0$), generating an energetically favorable locked (aligned) state.
• Microscopic Origin: The evolving coupling is attributed to doping-induced changes in the hybridization between $V$-$d_{xy}$ and $V$-$d_{x^2-y^2}$ orbitals.

4. Significance

• Resolution of the "Intertwined Order" Puzzle: The work definitively proves that the electronic nematicity in $CsV_3Sb_5$ is not merely a byproduct of the CDW stacking. Instead, there are two distinct nematic order parameters originating from different kagome-lattice orbitals ($d_{xy}$ vs. $d_{x^2-y^2}$).
• New Material Platform: The study establishes $CsV_3Sb_5$ as a rare platform where dual nematic orders can coexist, decouple, and re-couple, offering a controlled environment to study the interplay of symmetry breaking.
• Implications for Superconductivity: The discovery of a high-temperature $q=0$ nematic state suggests that electronic nematicity is a robust parent state in kagome metals, potentially playing a crucial role in the mechanism of superconductivity and the formation of exotic phases like chiral charge orders.
• Theoretical Advancement: The successful application of a two-component GL theory to explain the doping-dependent rotation of nematic directors provides a new framework for understanding complex intertwined orders in frustrated quantum materials.

In summary, this paper disentangles the complex electronic landscape of kagome superconductors, revealing a rich phase diagram where intrinsic orbital-selective nematicity and CDW-driven nematicity compete, coexist, and interact, fundamentally altering the understanding of symmetry breaking in these materials.