Discovery of intertwined pair density and charge density wave orders in UTe2

Using vector magnetic field scanning tunneling microscopy, researchers discovered that UTe2 hosts intertwined pair density wave and charge density wave orders, where distinct modulations emerge above and below the critical temperature, establishing the material as a unique platform for directly resolving the fundamental and emergent manifestations of triplet PDW physics.

The Gist

Imagine a crowded dance floor where electrons are the dancers. In most materials, these dancers move in a predictable, uniform way. But in a special material called UTe2, the electrons are "strongly correlated," meaning they are incredibly sensitive to each other and form complex, synchronized patterns.

This paper is like a detective story where scientists used a super-powerful microscope (a Scanning Tunneling Microscope) to figure out exactly what these electrons are doing, especially when they are dancing in the dark (superconductivity) or when a magnetic "wind" blows through the room.

Here is the breakdown of their discovery using simple analogies:

1. The Two Types of Dance Moves

The scientists found that the electrons in UTe2 aren't just doing one thing; they are doing two different, intertwined dances at the same time:

• The Uniform Dance (Superconductivity): This is the main event where electrons pair up and flow without resistance (zero friction). Think of this as a smooth, continuous wave of dancers moving across the floor.
• The Patterned Dance (Charge Density Waves - CDW): This is where the electrons form a static, repeating pattern, like a checkerboard or a ripple in a pond.

2. The Mystery: The "Ghost" Patterns

Previously, scientists saw some of these patterns (called $q$-waves) but were confused. They noticed these patterns survived even when the main "Uniform Dance" (superconductivity) stopped. It was like seeing a ripple in a pond even after the water had frozen solid. This didn't make sense with the old theories.

They also saw new patterns (called $p$-waves and $h$-waves) that appeared only when the material was cold enough to superconduct, or when a magnetic field was applied.

3. The Big Discovery: The "Parent" and the "Child"

The team realized that these patterns aren't random. They are related like a parent and a child.

• The Parent (The Pair Density Wave or PDW): The scientists discovered a hidden "parent" dance that starts before the material becomes a superconductor. Imagine a group of dancers starting to form a specific, wavy pattern (the PDW) while the room is still warm. This pattern exists even before the "Uniform Dance" begins.
• The Child (The CDW):
  • The $q$-waves (The Older Child): These patterns are created purely by the "Parent" PDW. Because the Parent exists even when the room is warm, these patterns survive even after the superconductivity stops. This explains why they were seen at higher temperatures!
  • The $p$-waves (The Younger Child): These patterns are created when the "Parent" PDW meets the "Uniform Dance" (superconductivity). They need both parents to exist. That's why they disappear as soon as the material warms up and loses its superconductivity.

4. The Magnetic Wind Test

To prove this theory, the scientists used a "vector magnet," which is like a fan that can blow wind from any angle (up, down, left, right).

• The Result: They found that the "Older Child" patterns ($q$-waves) are very sensitive to the wind. If they blow the wind in a specific direction, the patterns vanish.
• The Clue: The direction where the patterns vanish matches the direction where the superconductivity is strongest. This confirmed that the "Parent" PDW is deeply linked to the superconducting state, even though it starts earlier.

5. The "Surface Secret"

One tricky part of the story is that the "Parent" dance (PDW) seems to start at a higher temperature than the bulk material should allow. The scientists suggest this might be a surface phenomenon.

Analogy: Imagine a block of ice. The inside might be frozen at 0°C, but the very thin outer skin might freeze at a slightly different temperature due to air exposure. The scientists think the "Parent" dance starts on the very surface of the crystal first, creating the patterns we see, before the whole block fully joins in.

Why Does This Matter?

This discovery is a "Rosetta Stone" for understanding exotic materials.

• Before: Scientists saw a jumble of patterns and didn't know which was the cause and which was the effect.
• Now: They have a clear map. They know there is a "Parent" wave (PDW) that creates "Child" waves (CDWs).

This helps us understand how superconductors work, which is the holy grail of physics. If we can understand how to control these "Parent" waves, we might one day build superconductors that work at room temperature, revolutionizing everything from power grids to quantum computers.

In short: The scientists found that in UTe2, a hidden "parent" wave of electrons forms first. This parent creates two types of "child" patterns: one that survives even when the material warms up, and one that only exists when the material is super-cold and super-conducting. They solved the puzzle by watching how these patterns react to magnetic winds.

Technical Summary

1. Problem Statement

The heavy-fermion compound UTe$_2$ is a rare spin-triplet superconductor with an unusually large and anisotropic upper critical field ($H_{c2}$). Previous studies identified an incommensurate Charge Density Wave (CDW) order (wavevectors $q_i$, where $i=1,2,3$) in early-stage UTe$_2$ crystals. It was hypothesized that these CDWs arise from a Pair Density Wave (PDW) state intertwined with uniform superconductivity.

However, several critical puzzles remained unresolved:

• Temperature Anomaly: If the $q_i$ CDWs were generated solely by the coupling of PDWs and uniform superconductivity, they should vanish when uniform superconductivity is lost (above $T_c \approx 2.1$ K). Yet, $q_i$ peaks persist well above $T_c$ (up to $\sim 5$ K).
• New Signatures: Recent studies reported additional spectral features ($h_1, h_2, p_1, p_2, p_3$) with reduced wavevectors. Their origin was debated: were they intrinsic CDWs, quasiparticle interference (QPI) from in-gap states, or harmonics of the $q_i$ orders?
• Mechanism: The microscopic relationship between the high-temperature $q_i$ orders, the low-temperature $p_i$ orders, and the bulk superconducting state was unclear. Specifically, it was unknown whether the PDW forms above or below $T_c$.

2. Methodology

The authors utilized next-generation, ultra-clean UTe$_2$ single crystals (grown via molten flux, $T_c \approx 2.1$ K) to minimize extrinsic disorder effects. The core experimental tool was a Scanning Tunneling Microscope (STM) equipped with a vector magnetic field capability.

• Vector Magnet: Enabled independent tuning of magnetic field components ($B_x, B_y, B_z$) relative to the crystallographic axes ($a, b, c$), allowing for precise mapping of anisotropic responses.
• Spectroscopic Imaging: The team acquired $dI/dV$ maps (local density of states, LDOS) at various energies, both within and far above the superconducting gap.
• Fourier Analysis: Fast Fourier Transforms (FFTs) of real-space topography and LDOS maps were used to identify momentum-space signatures ($q_i, p_i, h_i$).
• Systematic Variable Control: Measurements were performed across a wide range of:
  • Temperatures: From 300 mK up to 4.8 K (spanning $T_c$ and the normal state).
  • Magnetic Fields: Up to 8 T along different crystallographic directions.
• Theoretical Framework: The experimental data were interpreted using a Landau free-energy model coupling triplet PDW order parameters ($\mathbf{d}_{p_i}$) and uniform superconductivity ($\mathbf{d}_0$).

3. Key Contributions and Results

A. Identification of New CDW Orders ($p_i$ and $h_i$)

• Discovery: The study resolved a new set of nondispersive modulations: $p_1, p_2, p_3$ and $h_1, h_2$.
• CDW vs. QPI: Energy-dependent measurements confirmed that $h_{1,2}$ and $p_3$ peaks remain nondispersive at energies far above the superconducting gap. This definitively classifies them as intrinsic CDW orders, ruling out previous interpretations of them as quasiparticle interference (QPI) features.
• Wavevector Relations: The new peaks satisfy specific relationships:
  • $p_i = \frac{1}{2} q_i$ (e.g., $p_3$ is half of $q_3$).
  • $h_1 = q_3 - q_2$ and $h_2 = q_3 - q_1$.
• Defect Sensitivity: The authors demonstrated that surface defects (common in older samples) generate spectral weight near zero momentum that obscures the $p_i$ and $h_i$ signals, explaining why these orders were missed in previous studies.

B. Distinct Temperature and Field Dependencies

The study revealed a hierarchical behavior between the two sets of orders:

1. $p_i$ Orders (Low Temperature):
   • $p_3$ appears in zero field but vanishes rapidly as temperature approaches $T_c$ (disappearing above 2.5 K).
   • $p_1$ and $p_2$ are induced only by magnetic fields and also vanish above $T_c$.
   • Conclusion: These orders are tightly coupled to the uniform superconducting state.
2. $q_i$ Orders (High Temperature):
   • The $q_i$ peaks persist well above $T_c$ (up to $\sim 5$ K).
   • They are suppressed by magnetic fields, but the critical field for suppression ($H_{sup}$) follows the anisotropy of the bulk upper critical field ($H_{c2}^b > H_{c2}^c > H_{c2}^a$).
   • Conclusion: The $q_i$ orders exist in a regime where uniform superconductivity is absent, implying they are driven by a PDW that forms above the bulk $T_c$.

C. Theoretical Synthesis: A Refined PDW Model

The authors propose a unified Landau free-energy model that resolves the contradictions:

• Surface PDW: A spin-triplet PDW with wavevectors $p_i$ forms at a transition temperature $T_{PDW} \approx 5$ K, which is higher than the bulk superconducting $T_c$ ($\approx 2.1$ K). This state is likely localized to the surface.
• Regime $T > T_c$ (but $T < T_{PDW}$):
  • Only the PDW order ($\mathbf{d}_{p_i}$) is non-zero; uniform superconductivity ($\mathbf{d}_0$) is zero.
  • The PDW alone generates the $q_i$ CDW orders via the coupling term $\rho_{q_i} \propto \mathbf{d}_{p_i} \cdot \mathbf{d}_{p_i}^*$.
  • This explains why $q_i$ persists above $T_c$ and why its suppression field mirrors $H_{c2}$ (as the PDW is sensitive to the same magnetic anisotropy).
• Regime $T < T_c$:
  • Both PDW ($\mathbf{d}_{p_i}$) and uniform superconductivity ($\mathbf{d}_0$) coexist.
  • The coupling between them generates the $p_i$ CDW orders via $\rho_{p_i} \propto \mathbf{d}_{p_i} \cdot \mathbf{d}_0^* + \text{c.c.}$
  • This explains why $p_i$ orders vanish above $T_c$.
  • The magnetic field dependence of $p_1$ and $p_2$ (which are zero-field forbidden) is explained by the field rotating the $\mathbf{d}$-vectors to align them, enabling the coupling.
• Harmonic Orders: The $h_1$ and $h_2$ peaks are identified as higher-order composite harmonics derived from the interaction of $q_i$ orders (e.g., $h_1 = q_3 - q_2$).

4. Significance

• Resolution of Controversy: The paper definitively distinguishes between CDW orders and QPI features in UTe$_2$, clarifying the nature of the "reduced wavevector" peaks observed in recent literature.
• Evidence for Surface PDW: It provides the first direct experimental evidence of a PDW state that exists above the bulk superconducting transition temperature, suggesting that surface energetics can stabilize exotic orders not present in the bulk.
• Intertwined Order Hierarchy: The work establishes a clear hierarchy: PDW ($p_i$) $\to$ Composite CDW ($q_i$) $\to$ Coupled CDW ($p_i$ below $T_c$). This demonstrates how a single microscopic origin (triplet PDW) can generate a complex landscape of intertwined electronic orders.
• Platform for Future Study: By resolving both the "parent" PDW and its "descendant" CDW orders, UTe$_2$ is established as a premier platform for studying the fundamental physics of pair density waves and their role in high-temperature superconductivity and other correlated systems.

In summary, this work utilizes advanced vector-field STM to map the phase diagram of UTe$_2$, revealing a surface-localized triplet PDW that drives a cascade of charge orders, fundamentally reshaping the understanding of intertwined states in heavy-fermion superconductors.