Quantifying the quantum nature of high spin YSR excitations in transverse magnetic field

Using ultra-low temperature scanning tunneling microscopy and zero-bandwidth modeling, this study characterizes the quantum nature of Yu-Shiba-Rusinov excitations in high-spin manganese phthalocyanine molecules on a lead film under transverse magnetic fields, distinguishing between isolated and coupled spin systems to elucidate the roles of magnetic anisotropy and exchange interactions in their quantum phases.

The Gist

The Big Picture: Dancing Electrons in a Superconductor

Imagine a superconductor (like the thin lead film in this experiment) as a perfectly synchronized dance floor. In this dance, electrons pair up and move together in perfect harmony, creating a "gap" where no single dancer can move alone. This is the superconducting state.

Now, imagine dropping a single, slightly clumsy dancer (a Manganese molecule) onto this floor. This clumsy dancer disrupts the perfect rhythm. Because of this disruption, a few specific "moves" (energy states) become possible right in the middle of the dance floor where they usually aren't allowed. These special moves are called Yu-Shiba-Rusinov (YSR) states.

The scientists in this paper wanted to understand exactly how these special moves behave when they push the system with a magnetic field (like a strong wind blowing across the dance floor).

The Experiment: Two Types of Dancers

The researchers placed individual Manganese Phthalocyanine (MnPc) molecules on a super-thin sheet of lead. They found that the molecules landed in two different poses, which changed how they danced:

1. Type 1 (The Soloist): The molecule landed in a way that its "legs" (ligands) were aligned perfectly with the floor's grid.

   • What happened: It acted like a single, isolated dancer. When the magnetic wind blew, the dancer's moves changed in a predictable, non-linear way, but it stayed mostly the same.
   • The Analogy: Think of a spinning top. If you blow on it, it wobbles in a specific, understandable pattern. The scientists could easily predict this wobble using a simple math model.

2. Type 2 (The Duo): The molecule landed in a different orientation, where its "legs" were twisted relative to the grid.

   • What happened: This molecule acted like a coupled pair. It wasn't just the central metal atom dancing; the surrounding chemical "legs" were also spinning and interacting with the center.
   • The Analogy: Imagine two dancers holding hands. If you blow wind on them, they don't just wobble individually; they spin around each other, merge, split, and change their formation in complex, surprising ways.

The Mystery: When the Model Fails

The scientists tried to predict what would happen using a computer model (a "zero-bandwidth model"). Think of this model as a simplified video game physics engine.

• For the Soloist (Type 1): The video game physics worked perfectly. The simulation matched the real-life dance floor.
• For the Duo (Type 2): The video game physics broke down.
  • The Glitch: In the real experiment, when the magnetic wind blew, the two dancers would merge into one big blob and stay merged for a long time. In the video game model, they should have just crossed paths and kept dancing separately.
  • The Out-of-Bounds Move: In the real experiment, one of the dancers sometimes moved off the dance floor (outside the superconducting gap) but still influenced the dancers who stayed inside. The model couldn't explain this.

Why Does This Matter?

This paper is important because it shows us the limits of our current understanding.

1. We know the basics: We can explain simple magnetic atoms on superconductors.
2. We hit a wall: When things get complicated (like high-spin molecules with multiple interacting parts), our current "video game physics" isn't good enough.
3. The Future: The scientists are saying, "We need a better physics engine." We need to account for things like electrons tunneling through the molecule in weird ways or the molecule "pumping" energy into the system.

The Takeaway

Think of this research as learning the rules of a new sport.

• The scientists found that for simple players (Type 1), the rules are clear.
• But for complex teams (Type 2), the players are doing things the rulebook doesn't cover yet. They are merging, splitting, and moving in ways that suggest there are hidden rules we haven't discovered.

By understanding these "glitches" in the complex molecules, scientists hope to build better tools for quantum computing. If we can control these complex magnetic dances, we might be able to use them to store and process information in the future.

Technical Summary

1. Problem Statement

Yu-Shiba-Rusinov (YSR) states are local in-gap excitations formed when a magnetic impurity is exchange-coupled to a superconductor. While extensively studied for spin-$1/2$ impurities, high-spin impurities ($S > 1/2$) present a more complex quantum landscape involving competing energy scales: Kondo exchange ($J_K$), intra-atomic exchange (Hund's rules), on-site Coulomb interactions ($U$), and single-ion magnetic anisotropy.

A critical gap in current research is the understanding of how these high-spin systems behave under transverse magnetic fields. Most existing studies are performed at zero field or rely on models that fail to capture the non-trivial evolution of excitation pathways when rotational symmetry is broken by a magnetic field. Furthermore, standard theoretical models (often reduced to a "giant spin" approximation or zero-bandwidth limit) struggle to predict the complex, non-monotonic evolution of YSR spectra and the potential for quantum phase transitions (QPTs) in high-spin systems under magnetic perturbation.

2. Methodology

The researchers employed ultra-low temperature Scanning Tunneling Microscopy/Spectroscopy (STM/STS) to investigate individual Manganese Phthalocyanine (MnPc) molecules adsorbed on ultra-thin Lead (Pb) films.

• Sample Preparation:
  • Substrate: Si(111)-Ag($\sqrt{3} \times \sqrt{3}$) reconstruction.
  • Superconductor: 11-monolayer (ML) thick Pb films. These were chosen for their quantum confinement effects and, crucially, their exceptionally high in-plane upper critical field ($H_{c2} \approx 4$ T), allowing the superconducting gap to remain robust against transverse magnetic fields up to 4 T.
  • Impurity: Individual MnPc molecules deposited and laterally manipulated via STM to specific sites on the Pb terraces.
• Experimental Conditions:
  • Temperature: $T \approx 30$ mK.
  • Magnetic Field: Transverse (parallel to the sample surface) fields up to $B_{\parallel} = 4$ T.
  • Spectroscopy: STS measurements taken at the center of the molecules using a lock-in technique with a normal-metal tip to avoid deconvolution artifacts.
• Theoretical Modeling:
  • A zero-bandwidth spin model was utilized to simulate the system.
  • The Hamiltonian included terms for superconductivity ($H_{SC}$), Kondo exchange ($H_{J_K}$), magnetic anisotropy ($H_{MA}$), inter-spin exchange ($H_{J_{ex}}$), and Zeeman energy ($H_{Zeeman}$).
  • Two distinct models were tested:
    1. Single-spin model: One spin site ($S=1$) coupled to one superconducting site.
    2. Coupled-spin model: Two spin sites ($S_1=1$ on Mn, $S_2=1/2$ on ligand) coupled antiferromagnetically, representing the full molecular spin system.

3. Key Contributions

• Systematic Field-Dependent Study: The work provides a comprehensive experimental dataset tracking YSR excitations of high-spin impurities in a transverse magnetic field up to 4 T, a regime previously difficult to access due to the destruction of superconductivity in bulk materials.
• Geometry-Dependent Quantum States: The study identifies that the quantum nature of the YSR excitations is strictly dictated by the adsorption geometry (orientation of ligand axes relative to the Pb lattice), leading to two distinct classes of behavior (MnPc1 vs. MnPc2).
• Limitations of Current Models: The paper demonstrates that while zero-bandwidth models can qualitatively reproduce some features of single-spin systems, they fail to capture critical phenomena in coupled-spin systems, specifically the robust merging of states and non-linear splitting patterns, suggesting the need for theories beyond the zero-bandwidth approximation (e.g., including co-tunneling and renormalization).

4. Key Results

A. Two Distinct Molecular Types

The MnPc molecules exhibited two distinct YSR spectra based on their orientation relative to the Pb(111) lattice:

1. MnPc1 (Parallel Alignment): One ligand axis parallel to a high-symmetry direction.
   • Observation: At $B=0$, a single pair of YSR peaks. Under increasing $B_{\parallel}$, the peak splits asymmetrically around 0.5 T. One branch moves non-linearly toward the gap center (inflection point), while the other moves linearly toward the gap edge.
   • Modeling: Successfully reproduced by a single-spin model ($S=1$) with magnetic anisotropy. The ground state is a partially screened state with a specific fermion parity. The splitting is attributed to the Zeeman effect acting on a Kramer's degenerate excited state.
2. MnPc2 (Bisected Alignment): Ligand axes bisected by high-symmetry directions (tested at $\alpha = 15^\circ$ and $45^\circ$).
   • Observation: At $B=0$, three pairs of peaks. Under $B_{\parallel}$, the system exhibits complex behavior:
     • Non-linear splitting of peaks.
     • A change in the total number of observable excitations (e.g., 3 $\to$ 5 $\to$ 3 states).
     • Robust Merging: Distinct peaks merge at specific field strengths (e.g., $\approx 2.5$ T) and remain merged for a wide field range ($\Delta B \approx 1.5$ T) without crossing or splitting, contrary to standard Zeeman expectations.
     • Out-of-Gap Continuation: One branch of an excited state crosses the coherence peak and persists outside the superconducting gap, diminishing in intensity but remaining observable.
   • Modeling: Requires a coupled-spin model ($S_1=1$ on Mn, $S_2=1/2$ on ligand) to explain the multiplicity of states. However, the model failed to reproduce:
     • The non-linear splitting of specific peaks.
     • The robust, field-independent merging of states.
     • The persistence of excitations outside the gap.

B. Quantum Phase Transitions (QPTs)

The study analyzed potential QPTs (changes in ground state fermion parity or spin projection).

• No YSR excitation was observed crossing the gap center (zero energy), which would indicate a fermion parity change.
• The observed changes in the number of excitations suggest parity-preserving QPTs or transitions in the ground state spin projection, but the complex merging behavior suggests the system enters a regime where standard selection rules and independent state descriptions break down.

5. Significance and Conclusion

This research highlights the quantum nature of high-spin YSR excitations in magnetic fields, revealing that simple giant-spin models are insufficient for complex molecular systems.

• Insight into Spin-Ligand Coupling: The results confirm that for certain orientations (MnPc2), the spin polarization of the molecular ligands cannot be neglected, necessitating a multi-spin description.
• Theoretical Challenges: The inability of the zero-bandwidth model to explain the robust merging of states and out-of-gap excitations suggests that higher-order tunneling processes, co-tunneling, bath renormalization, and non-equilibrium effects (like spin pumping) play a critical role in these systems.
• Platform for Topological Quantum Computing: By characterizing the excitation pathways and quantum phases of high-spin impurities in magnetic fields, this work provides a vital platform for testing spin impurity models and advancing the understanding of topological quantum computing architectures based on magnetic atoms on superconductors.

In summary, the paper establishes that while single-spin models work for specific geometries, high-spin molecular systems in transverse fields exhibit rich, non-trivial quantum dynamics that challenge current theoretical frameworks and point toward the necessity of more sophisticated transport-based theories.