Correlated charge order intertwined with time-reversal symmetry-breaking nodal superconductivity in the dual flat band kagome superconductor CeRu$_{3}$Si$_{2}$

This study identifies CeRu$_{3}$Si$_{2}$ as a unique kagome superconductor where intertwined Ru $d$- and Ce $f$-electron flat bands drive a complex hierarchy of charge orders and establish a direct correlation between normal-state time-reversal symmetry breaking and intrinsic nodal superconductivity.

The Gist

Imagine a bustling city built on a unique, honeycomb-like grid called a Kagome lattice. In this city, electrons (the tiny particles that carry electricity) usually zip around freely. But in certain materials, the city's layout is so special that it creates "traffic jams" where electrons get stuck in flat, motionless zones called flat bands.

Scientists have known about these traffic jams for a while, but they usually only looked at one type of driver: the d-electrons (think of them as standard city commuters). This new paper introduces a new character to the city: the f-electrons (think of them as heavy, slow-moving trucks carrying heavy cargo).

The researchers studied a specific material, CeRu₃Si₂, and discovered that when you mix these standard commuters with the heavy trucks in the same traffic jam, something magical and chaotic happens. Here is the story of their discovery, broken down simply:

1. The Traffic Jam Gets Organized (Charge Order)

In this material, the electrons don't just sit still; they start organizing themselves into a strict pattern, like cars parking in a perfect grid.

• The Discovery: The team found that the electrons form a dominant pattern (a "1/2 charge order") that is very strong, along with a weaker, secondary pattern ("1/3 charge order").
• The Analogy: Imagine a parking lot where cars suddenly decide to park in a specific, repeating zig-zag pattern. Even though the cars are parked, the lot is still usable (the material is still a metal), but the rules of the road have changed.
• The Twist: The heavy "trucks" (the f-electrons) are actually the ones holding the steering wheel. Without them, the pattern wouldn't form. This shows that the heavy electrons and the standard electrons are working together to create this order.

2. The "Ghost" Magnetism (Time-Reversal Symmetry Breaking)

Usually, if you look at a material before it becomes a superconductor (a material that conducts electricity with zero resistance), it behaves normally. But in similar materials, scientists found a "ghost" magnetism appearing before the superconductivity started.

• The Surprise: In this new material, CeRu₃Si₂, there is no ghost magnetism in the normal state. The city is calm.
• The Trigger: However, as soon as you apply a tiny external magnetic field (like a gentle breeze), the "ghost" appears. The material becomes slightly magnetic only when pushed.
• The Connection: The researchers found a universal rule across the whole family of these materials: The stronger the "ghost" magnetism (or the push needed to create it), the higher the temperature at which the material becomes a superconductor. It's like a seesaw: more magnetic tension in the normal state leads to a stronger superconducting state.

3. The Superconducting Switch (Nodal vs. Nodeless)

When the material finally cools down enough to become a superconductor, it usually has a "shield" (an energy gap) that protects the electrons from scattering.

• The Oddity: In most similar materials, this shield is solid all around (nodeless). But in CeRu₃Si₂, the shield has holes (nodes) in it when the magnetic field is low. It's like a shield with a few weak spots.
• The Magic Trick: When the researchers increased the magnetic field, those holes magically disappeared, and the shield became solid again.
• Why it matters: This is the first time scientists have seen a material switch from having "holes" in its superconducting shield to being "solid" just by changing the magnetic field. It suggests that the material is a complex mix of different electronic behaviors fighting for control.

4. The Grand Conclusion: A New Paradigm

The most exciting part of this paper is the realization that mixing different types of "flat bands" creates a rich, complex world of physics.

• The Metaphor: Think of it like mixing two different types of music. If you play just one genre (d-electrons), you get a predictable song. But if you mix in a heavy, complex jazz track (f-electrons), you get a new, unpredictable, and incredibly rich symphony of electronic states.
• The Impact: This proves that by engineering materials where these different electron types interact, we can create new quantum states that we've never seen before. It opens the door to designing future materials with custom-made properties for advanced electronics or quantum computers.

In a nutshell:
This paper tells the story of a material where heavy and light electrons team up to create a complex dance. They organize into patterns, react strangely to magnetic fields, and create a superconducting state that changes its shape depending on the environment. It's a breakthrough that shows us how to build better quantum materials by mixing different "flavors" of electron behavior.

Technical Summary

1. Problem Statement and Motivation

Kagome materials are renowned for hosting exotic quantum states driven by flat electronic bands, such as unconventional superconductivity and charge density waves. However, previous research has predominantly focused on d-electron flat bands (e.g., in $ARu_3Si_2$ where $A = \text{La, Y}$). A fundamental open question remains: Can qualitatively new physics emerge when flat bands of distinct origins—specifically kagome-derived d-electron bands and heavy-fermion f-electron bands—interact within a single system?

The authors investigate CeRu$_3$Si$_2$, a member of the $ARu_3Si_2$ family, to address this. While LaRu$_3$Si$_2$ and YRu$_3$Si$_2$ exhibit high-temperature charge order, field-induced magnetism, and time-reversal symmetry breaking (TRSB) in the normal state, CeRu$_3$Si$_2$ shows a drastically reduced superconducting transition temperature ($T_c \sim 1$ K). The study aims to elucidate how the incorporation of Ce 4f electrons alters the electronic landscape, charge ordering, and superconducting mechanisms compared to its La and Y counterparts.

2. Methodology

The study employs a multi-modal experimental and theoretical approach:

• Synchrotron X-ray Diffraction (XRD): High-resolution measurements (10–300 K) to identify structural transitions and charge order (CO) wave vectors.
• First-Principles Calculations (DFT+U): Density Functional Theory calculations incorporating Hubbard $U$ corrections to model Ce 4f electron correlations, phonon dispersion, and electronic band structures.
• Magnetotransport: Measurements of electrical resistivity, magnetoresistance (MR), and Hall effect to probe normal-state electronic properties.
• Muon Spin Rotation/Relaxation ($\mu$SR):
  • Zero-Field (ZF): To detect spontaneous internal magnetic fields and TRSB.
  • Transverse-Field (TF): To measure magnetic penetration depth ($\lambda$), superfluid density, and vortex lattice properties.
  • High-Field (HTF): To investigate field-induced magnetic responses.

3. Key Contributions and Results

A. Dual Flat Band Coexistence and Correlated Charge Order

• Electronic Structure: DFT calculations reveal the coexistence of Ru d-orbital kagome flat bands and heavy-fermion flat bands derived from Ce$^{4+}$ 4f states. The Fermi level lies between these two sets of bands.
• Charge Order (CO): XRD identifies a dominant 1/2 charge order ($q = 1/2$) and a weaker 1/3 charge order ($q = 1/3$) persisting up to room temperature.
  • The 1/2 order corresponds to a $Pmma$ space group (stripe-like Ru distortion).
  • The 1/3 order corresponds to an $Imma$ space group (Star-of-David pattern).
• Role of Correlations: DFT+U calculations demonstrate that treating Ce 4f electrons as valence states (rather than core) is essential to reproduce the experimental charge order. A large Hubbard interaction ($U > 6$ eV) is required to stabilize the 1/2 order as the ground state, highlighting the strongly correlated nature of the charge ordering driven by the interplay of d- and f-electrons.

B. Normal-State Anomalies and Magnetoresistance

• Transport Signatures: Resistivity measurements show metallic behavior with two characteristic temperatures: $T^*_1 \approx 90$ K (onset of MR and Hall sign reversal) and $T^*_2 \approx 35$ K (peak in $d\rho/dT$).
• Magnetoresistance: A finite magnetoresistance emerges below 90 K and strengthens significantly below 30 K, reaching ~18% at 14 T.
• TRS in Normal State: Unlike LaRu$_3$Si$_2$ and YRu$_3$Si$_2$, zero-field $\mu$SR shows no TRSB in the normal state of CeRu$_3$Si$_2$. However, an applied magnetic field induces a weak, bulk magnetic response that scales with field strength, indicating a field-induced magnetic state rather than intrinsic normal-state TRSB.

C. Nodal Superconductivity and Intrinsic TRSB

• Superconducting Gap: Analysis of the superfluid density ($1/\lambda^2$) via TF-$\mu$SR reveals a field-dependent crossover:
  • At low fields (5–10 mT), the data fits a nodal gap model.
  • At higher fields (50 mT), the behavior shifts to an anisotropic nodeless gap.
  • This makes CeRu$_3$Si$_2$ the first 132-type kagome compound to host nodal superconductivity at low fields.
• Intrinsic TRSB in SC State: Below $T_c$, ZF-$\mu$SR detects a continuous increase in the relaxation rate, corresponding to spontaneous internal magnetic fields of $\sim 0.3$ G. This provides definitive evidence for intrinsic TRSB in the superconducting state, distinct from the normal state.

D. Universal Scaling Across the $ARu_3Si_2$ Family

• The study establishes a striking linear scaling relationship across the series ($A = \text{Ce, Y, La}$):
  • $T_c$ scales linearly with the onset temperature of normal-state TRSB ($T_{TRSB}$).
  • $T_c$ scales linearly with the magnitude of the field-induced magnetic response ($\Delta\sigma_{HTF}$).
• This correlation suggests a direct positive link between normal-state symmetry breaking/magnetic fluctuations and the pairing mechanism, implying an electronically driven superconductivity.

4. Significance

• New Paradigm for Correlated States: The paper demonstrates that the coexistence of kagome d-electron and heavy-fermion f-electron flat bands creates a unique platform for intertwined quantum orders. This "dual flat band" system generates a rich hierarchy of phenomena not seen in single-band systems.
• Mechanism of Superconductivity: The identification of nodal superconductivity with intrinsic TRSB, coupled with the universal scaling of $T_c$ with magnetic response, strongly points to a pairing mechanism driven by magnetic fluctuations and electronic correlations rather than conventional phonon mediation.
• Material Design: The findings introduce a new strategy for engineering correlated quantum states by intentionally combining distinct flat-band systems (kagome and heavy-fermion) within a single material.
• Resolution of CeRu$_3$Si$_2$ Anomalies: The study resolves previous ambiguities regarding CeRu$_3$Si$_2$ by clarifying the role of Ce 4f electrons in stabilizing charge order and establishing the nature of its unconventional superconductivity.

In summary, CeRu$_3$Si$_2$ is established as a unique nodal, time-reversal symmetry-breaking superconductor where the interplay between kagome flat bands and heavy-fermion correlations drives a complex phase diagram involving robust charge order, field-induced magnetism, and unconventional superconductivity.