Unified pressure and field response across distinct charge-order regimes in Ti-doped CsV$_3$Sb$_5$

This muon spin rotation study of Ti-doped CsV$_3$Sb$_5$ reveals that despite distinct charge-order regimes (long-range vs. short-range) at different doping levels, the material exhibits remarkably similar superconducting responses under pressure, suggesting that the competition between superconductivity and charge order occurs on a local scale independent of long-range charge coherence.

The Gist

Imagine a bustling city built on a unique, triangular grid pattern called a Kagome lattice. In this city, electrons (the citizens) are constantly moving, sometimes forming orderly lines (charge order) and sometimes dancing together in pairs to flow without resistance (superconductivity).

For a long time, scientists thought these two behaviors were bitter rivals: if the citizens formed orderly lines, they couldn't dance in pairs, and vice versa. But a new study on a material called Ti-doped CsV3Sb5 (a fancy name for a specific type of superconductor) reveals a much more complex and surprising story.

Here is the breakdown of what the researchers found, using simple analogies:

1. The Two Neighborhoods: Long-Range vs. Short-Range

The researchers studied two different "neighborhoods" in this material by changing how much Titanium (Ti) they added:

• Neighborhood A (Underdoped): Here, the citizens form a massive, city-wide parade. Everyone is marching in perfect lockstep. This is called Long-Range Charge Order.
• Neighborhood B (Optimally Doped): Here, the big parade has broken up. The citizens are still moving in small, local groups, but there is no city-wide coordination. This is Short-Range Charge Order.

Usually, you would expect Neighborhood A and Neighborhood B to behave very differently. But the researchers discovered something strange: They are actually twins in disguise.

2. The "Ghost" in the Machine (Time-Reversal Symmetry Breaking)

Before the material even becomes a superconductor (when it's still just a normal metal), the researchers found a "ghost" haunting both neighborhoods.

• The Analogy: Imagine a room where everyone is supposed to be facing North. Suddenly, without anyone telling them to, a few people start spinning in a way that breaks the rules of physics (specifically, "time-reversal symmetry").
• The Discovery: Using a special tool called Muon Spin Rotation (think of it as a microscopic, invisible compass that detects tiny magnetic whispers), they found this "spinning" happening in both neighborhoods.
• The Twist: It didn't matter if the citizens were in a massive parade (Neighborhood A) or just small groups (Neighborhood B). The "ghost" was there in both. This suggests that the cause of this weird behavior is local—it happens between neighbors, regardless of whether the whole city is marching in sync.

3. The Pressure Cooker Experiment

Next, the scientists put both neighborhoods under hydrostatic pressure (squeezing them like a stress ball).

• The Result: Squeezing the material made the superconducting "dance" much stronger. The temperature at which the material becomes a superconductor jumped up significantly.
• The Connection: They found a perfect, straight-line relationship: The more the citizens danced (superfluid density), the higher the temperature they could dance at. This is a classic signature of "unconventional" superconductivity—the kind that isn't explained by standard textbook rules.

4. Changing the Dance Style

Perhaps the most fascinating part was how the "dance style" changed under pressure.

• At Normal Pressure: The electrons danced in a lopsided, anisotropic way. Imagine a dance where you spin fast in one direction but slow in another. It's uneven.
• Under High Pressure: As they squeezed harder, the dance became perfectly round and symmetrical (isotropic). The lopsidedness disappeared.
• Why it matters: This suggests that the "lopsided" dance was caused by the tension between the superconducting dance and the charge-order parade. Once the pressure suppressed the parade, the electrons were free to dance in a perfect, symmetrical circle.

The Big Takeaway

The main conclusion of this paper is a revelation about how these materials work:

The competition between the "parade" (charge order) and the "dance" (superconductivity) happens on a tiny, local scale.

It doesn't matter if the parade stretches across the whole city or is just a small group down the street. The superconductivity reacts to the local presence of the parade, not the global organization of it.

In simple terms:
Think of the material as a dance floor.

• In one version, everyone is lined up in a rigid formation.
• In the other, people are just milling about in small clusters.
• The researchers found that the "magic" of superconductivity (the frictionless flow) is triggered by the same local interactions in both cases.
• When you squeeze the floor (apply pressure), the rigid lines break down, and the dancers are free to perform a perfect, symmetrical routine that works at higher temperatures.

This discovery helps scientists understand that in these complex quantum materials, local interactions are the true boss, not the big-picture organization. It opens the door to designing better superconductors by focusing on these tiny, local relationships rather than trying to control the entire material at once.

Technical Summary

1. Problem Statement and Motivation

The interplay between charge order (CO) and superconductivity (SC) in kagome lattice materials, specifically the $AV_3Sb_5$ family ($A$ = K, Rb, Cs), remains a central puzzle in condensed matter physics. While these materials exhibit spontaneous time-reversal symmetry (TRS) breaking in their charge-ordered states, the microscopic nature of this symmetry breaking and its relationship to the superconducting state is debated.

Specifically, Ti-doped $CsV_3Sb_5$ presents a unique, non-monotonic phase diagram:

• At low doping ($x \approx 0.05$), the system retains long-range charge order (LRCO), and $T_c$ decreases.
• At higher doping ($x \approx 0.22$), LRCO collapses into short-range charge order (SRCO), yet $T_c$ increases significantly.
• Key Question: Do the superconducting states in these two distinct regimes (LRCO vs. SRCO) share the same microscopic pairing mechanism? Is the competition between SC and CO driven by long-range coherence or local electronic correlations?

2. Methodology

The authors employed Muon Spin Rotation/Relaxation ($\mu$SR) as a local magnetic probe to investigate two representative compositions:

• Underdoped: $Ti_{0.05}CsV_3Sb_5$ (retains LRCO).
• Optimally Doped: $Ti_{0.22}CsV_3Sb_5$ (LRCO suppressed, SRCO persists).

The study utilized three experimental configurations:

1. Zero-Field (ZF) and Longitudinal-Field (LF) $\mu$SR: To detect spontaneous internal magnetic fields and TRS breaking in the normal state without external field interference.
2. High Transverse-Field (TF) $\mu$SR: To measure the superfluid density (via magnetic penetration depth, $\lambda$) and probe the field-induced enhancement of relaxation rates.
3. High-Pressure $\mu$SR: Hydrostatic pressure was applied (up to $\sim$1.7 GPa) to tune the electronic structure, suppress charge order, and observe the evolution of superconducting properties.

3. Key Results

A. Normal State: Spontaneous TRS Breaking

• Observation: Both samples exhibit spontaneous TRS breaking in the normal state, evidenced by an enhancement of the muon spin relaxation rate ($\Gamma$) below the charge-order transition temperatures ($T_{LRCO} \approx 70$ K for $x=0.05$; $T_{SRCO} \approx 55$ K for $x=0.22$).
• Field Dependence: The relaxation rate enhancement scales linearly with applied magnetic field in both regimes. This indicates that the TRS breaking is driven by local charge-order correlations rather than long-range coherence.
• Significance: TRS breaking persists even when long-range order collapses into short-range correlations, suggesting the phenomenon is an intrinsic local feature of the kagome lattice.

B. Superconducting State at Ambient Pressure

• Bulk Superconductivity: Both samples show bulk superconductivity with $T_c \approx 2.52$ K ($x=0.05$) and $2.62$K ($x=0.22$).
• Low Superfluid Density: Both systems exhibit unusually low superfluid densities ($T_c/\lambda_{eff}^{-2} \approx 0.73 - 0.92$ K$\mu$m$^2$), characteristic of unconventional superconductors.
• Gap Symmetry: Analysis of the temperature dependence of the superfluid density reveals an anisotropic nodeless gap structure. The data fits best to an anisotropic s-wave model ($\Delta(\phi) = \Delta_0(1 + a \cos(4\phi))$) with large anisotropy parameters ($a \approx 0.56-0.59$), rather than a nodal d-wave or isotropic s-wave gap.

C. Response to Hydrostatic Pressure

• Enhancement of $T_c$ and Superfluid Density: Pressure significantly enhances both $T_c$ (up to $\sim$7 K) and the superfluid density (by a factor of $\sim$2.5) in both doping regimes.
• Linear Scaling: A robust linear correlation is observed between $T_c$ and the superfluid density ($\lambda^{-2}$). This scaling is a hallmark of unconventional pairing mechanisms.
• Gap Symmetry Crossover:
  • Below $\sim$1 GPa: The gap remains anisotropic nodeless.
  • Above $\sim$1 GPa: The anisotropy parameter $a$ approaches zero, indicating a crossover to an isotropic nodeless gap state.
• Implication: The suppression of charge order (both long-range and short-range) by pressure removes the competition that induces gap anisotropy, revealing a stable isotropic ground state.

4. Key Contributions

1. Unified Microscopic Framework: The study demonstrates that despite fundamentally different charge-order landscapes (LRCO vs. SRCO), the superconducting responses (TRS breaking, low superfluid density, gap anisotropy) are remarkably similar.
2. Local vs. Global Competition: The results suggest that the competition between superconductivity and charge order is governed by local electronic correlations and is largely independent of the long-range coherence of the charge-ordered state.
3. Pressure-Driven Phase Transition: The identification of a pressure-induced crossover from anisotropic to isotropic nodeless pairing provides direct evidence that charge order is the primary driver of gap anisotropy in these kagome superconductors.
4. TRS Breaking Mechanism: Confirmation that TRS breaking is a generic feature of the kagome family, persisting even in the absence of long-range charge order, and is strongly enhanced by external magnetic fields.

5. Significance

This work resolves a critical ambiguity in the phase diagram of Ti-doped $CsV_3Sb_5$. It establishes that the "double-dome" behavior of $T_c$ is not due to a change in the fundamental pairing mechanism, but rather a modulation of the competition between superconductivity and charge order.

The findings position $CsV_{3-x}Ti_xSb_5$ as a model system for studying intertwined orders in geometrically frustrated lattices. The observation that local charge correlations dictate the superconducting properties, regardless of whether the charge order is long- or short-range, offers a new perspective on the mechanisms of unconventional superconductivity in kagome metals. Furthermore, the pressure-induced transition to an isotropic state suggests that the "ideal" superconducting ground state in these materials is isotropic nodeless, with anisotropy arising solely from the competition with charge order.