Knight shift measurements probing Fermi surface changes under pressure in CeRhIn$_5$

NMR Knight shift measurements on CeRhIn$_5$ under pressure reveal that the suppression of the dipolar component at the In(1) site and the resulting pressure dependence of the shift are driven by an increase in 4f electron content at the Fermi surface, reflecting electronic structure changes near a Kondo breakdown quantum critical point.

The Gist

Imagine a crowded dance floor where two types of dancers are trying to move together.

• The "Heavy" Dancers (Cerium atoms): These guys are carrying heavy backpacks (their electrons). They like to stay in one spot, swaying slowly.
• The "Light" Dancers (Indium atoms): These are the free-spirited ones, zipping around the room, carrying the music (electricity).

In the material CeRhIn5, these two groups are trying to dance together. Sometimes, the heavy dancers get dragged along by the light ones, making the whole group move slowly and heavily. This is called a "heavy fermion" state. Scientists are fascinated by what happens when you squeeze this dance floor (apply pressure). Does the heavy group get lighter? Do they break up? Or do they get even heavier?

The Experiment: Listening to the Music

To figure out what's happening, the researchers used a technique called NMR (Nuclear Magnetic Resonance). Think of this as a super-sensitive microphone listening to the "heartbeat" (magnetic signal) of the Indium dancers.

There are two types of Indium dancers on this floor:

1. In(1): They stand right next to the heavy Cerium dancers.
2. In(2): They stand a bit further away, in a different spot.

The researchers measured how the "heartbeat" of these dancers changed as they squeezed the dance floor with increasing pressure.

The Big Surprise

Here is what they found, and it was quite unexpected:

• The "Away" Dancers (In(2)): When the pressure increased, their heartbeat stayed exactly the same. They didn't notice much change.
• The "Neighbor" Dancers (In(1)): This is where the magic happened. As the pressure increased, their heartbeat slowed down dramatically.

In physics terms, the "Knight Shift" (the heartbeat signal) for the In(1) site dropped significantly when squeezed, while the In(2) site stayed flat.

Why Did This Happen? (The Detective Work)

The scientists had to play detective to figure out why the In(1) dancers changed their rhythm. They considered a few suspects:

1. Suspect A: The Dance Moves Changed (Crystal Field). Maybe the pressure forced the heavy dancers to change their posture or the shape of their orbit.

   • Verdict: Not guilty. The researchers ran computer simulations and found that changing the "posture" didn't explain the massive drop in the signal.

2. Suspect B: The Connection Got Stronger (Hybridization). Maybe the heavy and light dancers started holding hands tighter.

   • Verdict: Not guilty. If they held hands tighter, the signal should have gone up, but it went down.

3. The Real Culprit: The Heavy Backpacks (4f Electrons).
   The researchers realized that the pressure caused the "heavy" electrons to spill out of their backpacks and join the "light" electrons on the dance floor.

   • The Analogy: Imagine the heavy dancers were wearing heavy, invisible backpacks. Under pressure, they started taking the backpacks off and wearing them as part of their regular clothes. Suddenly, the "heavy" electrons became part of the "light" crowd.
   • The Result: Because the heavy electrons are now part of the main crowd (the Fermi surface), the "neighbor" Indium dancers (In(1)) feel a different magnetic pull. Their signal drops because the nature of the crowd has fundamentally changed.

The "Kondo Breakdown" Moment

The paper suggests this happens near a Quantum Critical Point. Think of this as a tipping point.

• Before the tipping point: The heavy electrons are stuck in their backpacks (localized).
• After the tipping point: The backpacks break open, and the heavy electrons join the light ones (itinerant).

This is called a Kondo Breakdown. It's like a sudden realization where the heavy dancers decide, "We aren't heavy anymore; we are part of the flow!"

Why Does This Matter?

This discovery is a big deal for two reasons:

1. It solves a mystery: Scientists have been arguing for years about whether the "heavy" electrons stay stuck or join the flow at these critical points. This experiment provides strong evidence that they do join the flow, changing the shape of the entire "dance floor" (the Fermi surface).
2. A New Tool: The researchers showed that by listening to the "heartbeat" of the neighbors (Indium), you can detect these subtle changes in the heavy electrons without needing to see them directly. It's like knowing a storm is coming by feeling the wind on your face, even if you can't see the clouds yet.

In a Nutshell

By squeezing a special crystal, the scientists found that the "heavy" electrons inside it suddenly decided to join the "light" electron crowd. This change was detected by listening to the magnetic signals of nearby atoms. It proves that under pressure, the fundamental nature of the material's electrons can change drastically, offering a new way to understand how superconductors and other exotic materials work.

Technical Summary

Here is a detailed technical summary of the paper "Knight shift measurements probing Fermi surface changes under pressure in CeRhIn5" by Nian et al.

1. Problem Statement

Heavy fermion materials, such as the CeMIn5 family ($M$ = Co, Rh, Ir), exhibit complex quantum phase transitions where localized $f$-electrons hybridize with itinerant conduction electrons. A central debate in the field concerns the nature of the Quantum Critical Point (QCP) in these systems:

• The Question: Does the non-Fermi liquid behavior and unconventional superconductivity arise from antiferromagnetic spin fluctuations, or from a Kondo breakdown where the $f$-moments decouple from the conduction electrons?
• The Specific Issue: Theoretical models suggest that a Kondo breakdown QCP involves an abrupt reconstruction of the Fermi surface (from a "small" surface excluding $f$-electrons to a "large" surface including them). However, experimental evidence for this reconstruction is conflicting. While de Haas-van Alphen (dHvA) measurements suggest a Fermi surface jump, Angle-Resolved Photoemission Spectroscopy (ARPES) has challenged this interpretation.
• The Gap: Previous Nuclear Magnetic Resonance (NMR) Knight shift studies in CeRhIn5 were limited to magnetic fields aligned along the crystallographic $c$-axis. This prevented a full characterization of the hyperfine coupling tensor and its pressure dependence, leaving the mechanism behind observed changes in hyperfine couplings ambiguous (e.g., were they due to Crystal Field (CEF) changes or electronic structure changes?).

2. Methodology

The authors employed Nuclear Magnetic Resonance (NMR) to probe the local electronic environment of Indium (In) nuclei in single-crystal CeRhIn5 under hydrostatic pressure.

• Sample & Setup: High-quality single crystals were grown using In-flux methods. Experiments were conducted in a piston-cylinder pressure cell (up to ~2.4 GPa) using Daphne oil as a pressure medium.
• Measurement Technique:
  • Knight Shift ($K$): Measured for both In(1) (in the Ce-In plane) and In(2) (between Ce layers) sites.
  • Field Orientation: Unlike previous studies, the magnetic field ($H_0$) was applied perpendicular to the $c$-axis (in the $ab$-plane). This allowed for the separation of the Knight shift tensor components ($K_{aa}, K_{bb}, K_{cc}$).
  • Spectral Analysis: The authors analyzed the complex NMR spectra (involving quadrupolar splitting) to distinguish between the In(1) and In(2) sites and their specific sub-sites (In(2A) vs. In(2B)) based on the Electric Field Gradient (EFG) tensor orientation.
• Theoretical Modeling:
  • A tight-binding model was used to simulate the electronic structure, incorporating Ce 4$f$ and In 5$p$ orbitals, hybridization ($V_{pf}$), and Crystal Field (CEF) terms ($B_2^0, B_4^0$).
  • The model calculated the hyperfine coupling constants as a function of the fraction of $4f$character ($f$) at the Fermi surface, allowing the authors to distinguish between CEF-driven effects and Fermi surface reconstruction.

3. Key Results

A. Anisotropic Pressure Dependence of Knight Shifts

• In(1) Site (Ce Plane):
  • The Knight shift component along the $c$-axis ($K_{cc}$) decreases significantly (by a factor of two) as pressure increases from 0 to 2 GPa.
  • Crucially, the in-plane components ($K_{aa}$) exhibit little to no pressure dependence.
  • Decomposition of the shift tensor reveals that the dipolar component ($K_{dip}$) of the In(1) shift is strongly suppressed by pressure, while the isotropic component ($K_{iso}$) also shows a reduction.
• In(2) Site (Between Ce layers):
  • Both in-plane ($K_{aa}, K_{bb}$) and out-of-plane ($K_{cc}$) components show negligible pressure dependence.
  • The dipolar term dominates the In(2) shift but remains constant under pressure.

B. Electric Field Gradient (EFG)

• The EFG parameters ($\nu_{cc}$) for both sites show a gradual increase with pressure, consistent with previous data.
• The authors did not observe a discontinuity in the In(2) EFG near 1.5 GPa, which some previous studies interpreted as a signature of a QCP. This suggests the QCP, if present, is continuous or hidden by the superconducting dome.

C. Theoretical Interpretation

• Ruling out CEF: The tight-binding model demonstrated that varying the Crystal Field parameters ($B_2^0$ and $B_4^0$) to account for orbital anisotropy changes could only explain a ~2-11% change in hyperfine coupling. This is insufficient to explain the factor-of-two drop observed in $K_{cc}$ for In(1).
• Fermi Surface Reconstruction: The data strongly supports a model where the fraction of $4f$character ($f$) at the Fermi surface increases with pressure.
  • The hyperfine coupling is inversely related to the $4f$content in the hybridized state at the Fermi level. As pressure increases, the system moves toward a state where$f$-electrons become more itinerant (part of the Fermi surface), reducing the transferred hyperfine coupling to the In(1) nucleus.
  • The lack of change in In(2) suggests these sites couple to Fermi surface sheets that are already strongly hybridized at ambient pressure.

4. Key Contributions

1. Tensor Resolution: The study provides the first complete tensor analysis of the Knight shift in CeRhIn5 under pressure, revealing a stark anisotropy where only the $c$-axis component of the In(1) shift is sensitive to pressure.
2. Mechanism Identification: The authors definitively rule out Crystal Field parameter changes as the primary driver for the observed Knight shift evolution. Instead, they attribute the changes to a modification of the electronic structure at the Fermi surface.
3. Evidence for Kondo Breakdown: The pressure-induced suppression of the dipolar hyperfine coupling at the In(1) site is interpreted as evidence for an increase in the $4f$ electron content at the Fermi surface. This supports the Kondo breakdown scenario, where the Fermi surface reconstructs from "small" to "large" as the Kondo screening collapses.
4. Methodological Insight: The paper establishes transferred hyperfine coupling as a highly sensitive probe for detecting subtle changes in $f$-electron localization, potentially more sensitive than traditional X-ray or ARPES techniques in certain regimes.

5. Significance

This work provides critical experimental evidence supporting the Kondo breakdown quantum critical point scenario in heavy fermion systems. By demonstrating that the hyperfine coupling evolves in a manner consistent with a change in the Fermi surface topology (specifically, the inclusion of $f$-electrons), the study helps resolve the long-standing debate regarding the nature of the QCP in CeRhIn5.

The findings suggest that the "small" to "large" Fermi surface transition is a continuous evolution driven by pressure, rather than a sharp jump, and that the In(1) site acts as a specific probe for the $f$-electron delocalization process. This methodology offers a new pathway for investigating quantum criticality in other heavy fermion compounds (e.g., YbRh$_2$Si$_2$) where Fermi surface reconstruction is hypothesized but difficult to observe directly.