Revealing Hund superdispersion with tunneling spectroscopy

This paper combines tunneling spectroscopy with advanced theoretical calculations to provide direct experimental evidence of "superdispersion" in Sr$_2$RuO$_4$, revealing a distinct spectroscopic signature of Hund coupling that defies the conventional Mott-Hubbard paradigm.

The Gist

The Big Picture: A Traffic Jam in the Electron World

Imagine a crowded dance floor where everyone is trying to move to the music. In most materials, electrons (the dancers) move smoothly, following a predictable path. But in special "quantum materials," the dancers are so crowded and reactive that they bump into each other, creating a chaotic traffic jam.

Scientists have known about two main types of traffic jams:

1. The "Waterfall": In some materials (like cuprate superconductors), the electrons move fast, then suddenly hit a wall and crash into a chaotic mess. It looks like a waterfall in a graph.
2. The "Hund Metal": In materials like Sr₂RuO₄ (the star of this study), the electrons are governed by a rule called Hund's coupling. This is like a strict dance instructor who tells the dancers to spin in specific ways. This rule creates a unique, weird kind of traffic jam that doesn't fit the old "waterfall" model.

The authors of this paper wanted to prove that this weird "Hund" traffic jam actually exists and has a specific signature they call "Superdispersion."

The Mystery: A Reversal of Direction

Usually, when you push a car (an electron), it speeds up as you give it more energy. In a normal material, the "speed" of the electron (its dispersion) goes up steadily.

However, the theory predicted that in a Hund metal, something bizarre happens:

• The electrons slow down (get "renormalized").
• Then, suddenly, they speed up faster than they should have.
• Even stranger, in a tiny range of energy, they seem to reverse direction.

The authors call this "Superdispersion." Think of it like driving a car that, instead of just slowing down in traffic, suddenly hits a patch of road where the physics of the car flips, and you start moving backward before shooting forward again.

The Challenge: Seeing the Invisible

The problem is that this "reverse gear" happens in the unoccupied states.

• Occupied states: Electrons are already there (like cars parked in a lot). We can see them easily with cameras (like Angle-Resolved Photoemission Spectroscopy, or ARPES).
• Unoccupied states: These are empty spots where electrons could go. Traditional cameras can't see empty spots.

It's like trying to map a city by only looking at the buildings that are currently lit up, but the "Superdispersion" feature is in the dark, empty lots.

The Solution: The "Tunneling" Flashlight

To see these empty spots, the team used Tunneling Spectroscopy (STM). Imagine a very sensitive needle that hovers just above the material. It can "tunnel" electrons into the empty spots and measure how hard it is to push them in. This acts like a flashlight that can illuminate the empty lots.

However, interpreting this data is tricky. The surface of the material (Sr₂RuO₄) is slightly different from the inside (the bulk). It's like the top layer of a cake has been rotated slightly compared to the layers underneath. This rotation changes the "map" of the dance floor.

The Method: A Three-Part Detective Story

The team combined three tools to solve the mystery:

1. DFT (Density Functional Theory): They built a digital 3D model of the material's surface, accounting for that rotated top layer.
2. DMFT (Dynamical Mean-Field Theory): They used a super-computer simulation to calculate how the electrons interact with each other (the "Hund's coupling" rules). This gave them the "traffic rules" for the electrons.
3. cLDOS (Continuum Local Density of States): They combined the model and the rules to predict exactly what the tunneling needle should see.

The Discovery: Matching the Prediction

When they compared their complex computer prediction with the actual data from their tunneling microscope, the match was perfect.

• The "Kink": In the experimental data, they saw a distinct "kink" or dip in the signal at exactly 160 millielectron-volts (a specific energy level).
• The Proof: This kink appeared only when they included the "Hund's coupling" rules in their computer model. When they turned off the Hund rules (simulating a normal material), the kink disappeared.

This kink is the fingerprint of the Superdispersion. It proves that the electrons are indeed doing that weird "reverse direction" dance predicted by theory.

Why It Matters (According to the Paper)

This paper doesn't claim to build a new battery or a faster computer. Instead, it claims to have:

1. Proven a Theory: It provided the first direct experimental evidence that "Hund superdispersion" is real.
2. Validated a Method: It showed that you can combine surface models with bulk physics simulations to understand complex materials.
3. Opened a New Window: It demonstrated that tunneling spectroscopy can now be used to study "unoccupied" electron states with high precision, allowing scientists to test theories about how electrons behave in other complex materials (like iron-based superconductors) in the future.

In short, the team used a high-tech needle and a super-computer to catch a glimpse of electrons doing a backflip in a crowded quantum dance floor, confirming a decades-old prediction about how they move.

Technical Summary

Technical Summary: Revealing Hund Superdispersion with Tunneling Spectroscopy

Problem Statement
The physics of Hund coupling in multi-orbital quantum materials presents a challenge distinct from the conventional Mott–Hubbard paradigm. While single-orbital systems exhibit characteristic "waterfall" spectral features where quasiparticle dispersion transitions into broad Hubbard bands, Hund metals (systems with partially filled, degenerate orbitals) are predicted to display qualitatively different behavior. Specifically, dynamical mean-field theory (DMFT) predicts the emergence of "superdispersive" features in the spectral function of Hund metals. In these systems, the interaction-induced renormalization of band structure becomes non-monotonous with energy. This leads to regimes where band velocities are not only reduced ("renormalized") but can exceed the non-interacting band velocity ("unrenormalized") or even reverse direction locally.

Despite theoretical predictions, experimental verification of Hund superdispersion has been elusive. In the prototypical Hund metal $\text{Sr}_2\text{RuO}_4$, these features are predicted to occur in unoccupied electronic states (above the Fermi level), which are inaccessible to standard angle-resolved photoemission spectroscopy (ARPES). Furthermore, interpreting tunneling spectroscopy (STM/STS) data for such correlated systems is complicated by surface reconstruction effects and the need to account for energy-dependent self-energies rather than simple phenomenological band renormalization.

Methodology
The authors employ an integrated theoretical and experimental framework combining density functional theory (DFT), DMFT, and continuum local density of states (cLDOS) calculations to analyze tunneling spectroscopy data of $\text{Sr}_2\text{RuO}_4$.

1. Theoretical Modeling:

   • Bulk Self-Energy: The self-energy ($\hat{\Sigma}$) is calculated using DFT+DMFT with the numerical renormalization group (NRG) as the impurity solver. This captures non-perturbative correlation effects, including the specific role of Hund coupling ($J$).
   • Surface Electronic Structure: Recognizing that the $\text{Sr}_2\text{RuO}_4$ surface undergoes a $9^\circ$ rotation of the $\text{RuO}_6$ octahedra (unlike the bulk), the authors construct a surface-specific tight-binding model based on DFT.
   • Self-Energy Extraction: To validate the theoretical approach, the authors extract the real part of the self-energy ($\Sigma'$) from high-resolution ARPES data of the reconstructed surface. They compare this with bulk DMFT calculations, finding excellent agreement without additional fitting parameters, thereby justifying the use of the bulk self-energy to describe surface correlations.
   • Tunneling Simulation: The calculated self-energy is embedded into a continuum Green's function formalism (cLDOS) to simulate scanning tunneling microscopy (STM) spectra. This approach accounts for the energy dependence of the self-energy and tunneling matrix elements, utilizing the Feenstra function ($L(V)$) to normalize experimental differential conductance ($dI/dV$) and current ($I$) data.

2. Experimental Measurement:

   • Tunneling spectroscopy measurements were performed on $\text{Sr}_2\text{RuO}_4$ using a home-built STM in a dilution refrigerator at temperatures below 100 mK. Differential conductance spectra were acquired to probe both occupied and unoccupied states with high energy resolution.

Key Contributions and Results

• Identification of Superdispersion Mechanism: The study elucidates the mechanism behind Hund superdispersion. In the presence of finite Hund coupling ($J$), the real part of the self-energy ($\Sigma'$) exhibits a non-monotonous energy dependence, characterized by a dip slightly above the Fermi level followed by a region where $\partial_\omega \Sigma' > 0$. According to the pole equation for quasiparticle energies, this positive slope leads to "unrenormalization" (velocities exceeding the non-interacting limit) and, when $\partial_\omega \Sigma' > 1$, to a reversal of dispersion direction. This is distinct from the single-orbital Hubbard model where similar effects occur only at much higher energies (Hubbard band formation).
• Quantitative Agreement in Tunneling Spectra: By combining the surface DFT band structure with the bulk DMFT self-energy, the authors simulate the continuum local density of states. The simulated spectra for $J > 0$ reproduce the experimental tunneling data with high fidelity.
  • Low Energy: The model correctly reproduces the spin-orbit coupling induced gap at the Fermi level, requiring a correlation-enhanced spin-orbit coupling strength ($\lambda \approx 200$ meV, roughly double the DFT value).
  • High Energy (The Signature): Crucially, the simulation reveals a pronounced dip (minimum) in the normalized tunneling spectrum $L(V)$ at approximately $+160$ meV. This feature corresponds directly to the onset of the superdispersive regime predicted by DMFT.
• Validation of the Signature: The authors demonstrate that this $160$ meV feature is absent in simulations where Hund coupling is set to zero ($J=0$), which instead show a monotonic self-energy and a rapid suppression of spectral weight above the quasiparticle band maximum. The robustness of this feature against shifts in the chemical potential confirms it arises from the self-energy (correlation effects) rather than the bare band structure.

Significance
The paper claims to provide the first direct experimental evidence of Hund-induced superdispersion in a quantum material. By successfully bridging the gap between DMFT predictions for unoccupied states and experimental tunneling spectroscopy, the work validates the DMFT-derived self-energy for $\text{Sr}_2\text{RuO}_4$.

The significance of this work lies in establishing a new methodological framework that enables quantitative tests of DMFT calculations across the entire energy spectrum, from coherent quasiparticles to incoherent states. This approach allows tunneling spectroscopy and quasiparticle interference (QPI) measurements to probe correlation effects in unoccupied states with sub-meV resolution. The authors suggest this framework can be extended to other systems, such as $\text{Sr}_2\text{MoO}_4$ (where superdispersion is predicted in occupied states) and iron-based superconductors, offering a pathway to distinguish between different scenarios of quantum criticality and understand the role of Hund coupling in emergent phenomena.