🔬 mesoscale physics

Electron state tomography from quasiparticle interference maps

This paper introduces a tomography method that reconstructs electron state density matrices and quantum geometric tensors from quasiparticle interference maps around on-site impurities by exploiting the distinct symmetry representations of orbital contributions, thereby enabling orbital-resolved band structure characterization in scanning tunneling microscopy using conventional unpolarized tips.

Original authors: A. Razanajatovo, J. Cayssol, C. Dutreix

Published 2026-06-18

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CC BY 4.0

Original authors: A. Razanajatovo, J. Cayssol, C. Dutreix

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Taking a "3D X-Ray" of Invisible Electrons

Imagine you are trying to understand the shape of a complex, invisible sculpture made of light. You can't touch it, and you can't see it directly. However, if you throw a pebble at it, the ripples (waves) that bounce off the sculpture carry information about its shape.

This paper introduces a new way to "see" the hidden internal structure of electrons in certain materials (like graphene) by analyzing the ripples they make when they hit a tiny obstacle.

The Problem: The "Blurry Camera"

In the world of quantum physics, electrons have a "shape" or "personality" called a wave function. This isn't just about where the electron is; it's about which "orbital" (a specific type of energy room) it lives in and how it spins.

Usually, scientists use a tool called a Scanning Tunneling Microscope (STM) to look at these electrons. Think of the STM tip as a camera.

The Limitation: Most STM cameras are "unpolarized." This is like taking a photo with a black-and-white lens that blurs together different colors. You can see the electron's location, but you can't tell if it's wearing a "Red Hat" (Orbital A) or a "Blue Hat" (Orbital B). You lose the crucial details about its internal structure.

The Solution: The "Echo Chamber" Trick

The authors propose a clever trick to recover those lost details using Quasiparticle Interference (QPI).

The Setup: Imagine a perfectly smooth dance floor (the material). Suddenly, you place a single, small chair (an impurity) in the middle.
The Dance: Electrons are like dancers moving across the floor. When they hit the chair, they bounce back.
The Ripples: As the electrons bounce, they create standing waves (ripples) on the floor. These ripples form a pattern of rings, much like the rings you see when you drop a stone in a pond.
The Secret: Even though the camera is blurry (unpolarized), the way the ripples form depends on whether the electron was wearing a "Red Hat" or a "Blue Hat" before it hit the chair.
- The "Red Hat" electrons create ripples that stay the same if you rotate the room.
- The "Blue Hat" electrons create ripples that twist or change color when you rotate the room.

The Magic: Sorting the Ripples

The paper's main breakthrough is a mathematical method to "unmix" these ripples.

The Analogy: Imagine you have a bowl of mixed red and blue jelly beans. You can't see the colors individually because they are all jumbled together. However, if you spin the bowl in a specific way (using the symmetry of the material), the red beans stay in one spot, while the blue beans move to a different spot.
The Method: By looking at the ripple patterns from three different angles (using the material's natural symmetry), the scientists can mathematically separate the "Red Hat" signal from the "Blue Hat" signal.

Once they separate these signals, they can reconstruct the Density Matrix.

What is this? Think of the Density Matrix as a complete ID card for the electron. It tells you exactly how much "Red Hat" and "Blue Hat" the electron has, and how they are mixed together.

What They Found

The team tested this idea using computer simulations of a honeycomb material (like graphene).

They created a fake "impurity" (the chair) and simulated the electron ripples.
They applied their "unmixing" math to the blurry data.
The Result: They successfully reconstructed the electron's ID card perfectly. They could see not just where the electron was, but its exact internal quantum state, including its "quantum geometry" (a fancy way of describing the shape of the space the electron lives in).

Why It Matters (According to the Paper)

This is a big deal because it means scientists don't need expensive, super-complex "polarized" cameras to see these details. They can use standard, conventional microscopes and just use this new math trick to see the hidden "colors" of the electron.

It turns a simple impurity (a defect in the material) into a powerful tomographic probe—essentially turning a tiny flaw into a 3D scanner that reveals the full quantum nature of the material.

Summary

The Goal: See the hidden internal structure of electrons.
The Obstacle: Standard tools blur the details.
The Trick: Use the ripples caused by a single impurity.
The Secret Sauce: Use the material's symmetry to mathematically separate the different types of electron "clothing" (orbitals) from the blurry signal.
The Outcome: You can now build a complete 3D map of an electron's quantum state using standard equipment.

Problem Statement
Characterizing electronic band structures requires precise knowledge of wave functions and their quantum geometry, encapsulated by the Quantum Geometric Tensor (QGT). While the imaginary part of the QGT (Berry curvature) and the real part (quantum metric) are fundamental to phenomena ranging from polarization and orbital magnetism to topological classification and supercurrent localization, their experimental measurement in solid-state systems remains exceedingly rare. Current methods often rely on engineered platforms or angle-resolved photoemission spectroscopy (ARPES). Although Quasiparticle Interference (QPI) maps obtained via Scanning Tunneling Microscopy (STM) have proven effective for reconstructing band features and global topological properties, conventional unpolarized STM tips lack the orbital resolution necessary to access the local quantum geometry. Resolving the orbital character of wave functions from standard LDOS measurements remains a significant experimental challenge.

Methodology
The authors introduce a tomography method to reconstruct the density matrix of electron states from QPI maps generated by single, on-site impurities in two-orbital models on a honeycomb lattice. The approach relies on the following theoretical framework:

Model System: The study focuses on low-energy band structures near the Brillouin zone (BZ) corners ( $K$ and $K'$ ) of honeycomb lattices, relevant to graphene heterostructures and direct-gap semiconductors. The system is described by a 2x2 Hamiltonian with spinless orbitals $A$ and $B$ .
Scattering Mechanism: Elastic scattering from a $\delta$ -localized on-site impurity is treated using a $T$ -matrix approach. The resulting Local Density of States (LDOS) modulations are analyzed using the stationary phase approximation, which identifies dominant backscattering contributions between time-reversed stationary points on Constant Energy Contours (CECs).
Symmetry-Based Disentanglement: While unpolarized STM tips measure the total LDOS ( $\Delta\rho = \Delta\rho_A + \Delta\rho_B$ $Δ ρ = Δ ρ_{A} + Δ ρ_{B}$ ), the authors exploit the distinct transformation properties of orbital contributions under the $C_3$ $C_{3}$ point group symmetry.
- The orbital contribution $\Delta\rho_A$ (associated with the impurity site) transforms under the trivial irreducible representation (irrep).
- The orbital contribution $\Delta\rho_B$ transforms under a non-trivial irrep (acquiring a phase factor $\omega = e^{\pm i 2\pi/3}$ ).
- By measuring the total signal at three intervalley vectors related by $C_3$ symmetry, the authors project the total signal onto orthogonal irrep vectors. This allows for the mathematical separation of $\Delta\rho_A$ and $\Delta\rho_B$ from a single unpolarized QPI map.
Reconstruction: Once separated, the orbital factors are shown to map directly to the populations ( $P_\alpha$ ) and coherences ( $C_{AB}$ ) of the scattering state density matrix. This enables the reconstruction of the full density matrix $\hat{\rho}(k)$ and, subsequently, the QGT (quantum metric and Berry curvature).

Key Contributions

Tomography from Unpolarized Probes: The paper demonstrates that full density matrix reconstruction is possible using conventional, unpolarized STM tips, removing the need for specialized orbital-polarized probes.
Symmetry-Driven Extraction: The method utilizes the orthogonality of point-group representations to disentangle orbital contributions that are otherwise averaged out in standard QPI measurements.
Direct Access to QGT: The technique provides a pathway to extract the Quantum Geometric Tensor (both the quantum metric and Berry curvature) directly from experimental QPI data.
Generality: The method is applicable to various two-band systems with time-reversal and hexagonal symmetries, including monolayer and multilayer rhombohedral graphene, gapped graphene, and transition metal dichalcogenides.

Results
The authors validated the method using numerical simulations based on tight-binding models:

Numerical Verification: They generated unpolarized QPI maps for massive Dirac quasiparticles ( $N=1$ ) and verified that the reconstructed density matrix matches the theoretical band structure.
Reconstruction Accuracy: The reconstructed population ( $P_A$ ) correctly approaches 1 near the conduction band minimum (where states are orbital-polarized), and the coherence modulus ( $|C_{BA}|$ ) vanishes as expected. The phase of the coherence exhibits the linear variation associated with pseudospin-momentum locking.
QGT Mapping: The method successfully reconstructed the quantum metric ( $g_{xy}$ ) and Berry curvature ( $\Omega_{xy}$ ) in the $K'$ valley.
Comparison: Quantitative comparisons between the QPI-reconstructed quantities and those derived directly from the bare band structure showed excellent agreement.
Limitations: The authors note that the method's resolution is limited by the inverse quasiparticle lifetime ( $\eta$ ). When the CEC width becomes comparable to the CEC radius, backscattering cannot be resolved, leading to "blank areas" near high-symmetry points at low energies. The method also remains robust even when CECs exhibit trigonal warping at higher energies.

Significance
The paper establishes impurities as effective tomographic probes for band structures in STM experiments. By enabling the reconstruction of the full electron density matrix and the QGT from unpolarized LDOS measurements, this method offers a practical route to accessing the local quantum geometry of band structures in two-dimensional materials. The authors suggest this approach could pave the way for investigating complex honeycomb systems (such as Kagome metals and high- $T_c$ superconductors) where orbital character is crucial for understanding topological states, pairing mechanisms, and pseudo-gaps. Furthermore, the authors note the potential relevance of this local probe tomography beyond electronic bands, extending to engineered lattices where LDOS measurements are accessible.