🔬 mesoscale physics

Imaging the topological sign of a Dirac mass with spin-polarized tunneling

This paper proposes a method to locally image the topological sign of a Dirac mass in magnetic topological surfaces by utilizing spin-polarized tunneling on a $C_3$ axis and zeroth Landau level measurements, thereby enabling the direct mapping of spatial variations in topological character beyond mere spectral gap detection.

Original authors: Payman Kazemikhah, Ruolun Zhang, Byunghoon Kim, Xiangyu Luo, Zachary Ducorsky, Yu-Peng Zhu, Md. Shafayat Hossain, M. Zahid Hasan

Published 2026-10-14

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CC BY 4.0

Original authors: Payman Kazemikhah, Ruolun Zhang, Byunghoon Kim, Xiangyu Luo, Zachary Ducorsky, Yu-Peng Zhu, Md. Shafayat Hossain, M. Zahid Hasan

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

In the quantum world, electrons moving on the surface of certain special materials behave as if they are massless, racing along at incredible speeds in a pattern known as a Dirac cone. When these materials are made magnetic, this massless state acquires a gap, a forbidden energy zone that stops the electrons from flowing freely. This gap is not just a simple barrier; it carries a hidden topological character, a kind of internal sign that dictates how the electrons respond to magnetic fields and how they flow along the edges of the material. This sign determines the direction of a special, one-way current that can flow without resistance, a phenomenon crucial for future electronic devices that do not waste energy as heat. For years, scientists could measure the size of this energy gap with extreme precision, but they could not tell which way the internal sign pointed. It was like knowing the size of a lock but not knowing which way the key turns, leaving the true nature of the material's surface hidden from direct view.

A team of researchers at Princeton University and the University of California, Los Angeles has now developed a way to see this hidden sign directly. They used a technique called spin-polarized tunneling, which involves bringing an incredibly sharp tip, tipped with a single atom, close to the surface of the material. This tip acts like a microscopic probe that can detect the spin, or intrinsic angular momentum, of the electrons. The challenge was that the relationship between the electron's actual spin and the theoretical "pseudospin" that defines the topological sign is usually muddled by the complex way the electron's motion is entangled with its orbit around the atom. This entanglement often scrambles the signal, making it impossible to tell if the topological sign is positive or negative just by looking at the spin.

The researchers discovered that this confusion disappears at specific points on the crystal surface where the material has a three-fold rotational symmetry. If the probe is positioned exactly over these high-symmetry points, the laws of physics force the electron state to become purely spin-polarized. In this specific location, the tip sees a clean, unambiguous signal that directly reveals the sign of the topological mass. By measuring the difference in how the tip interacts with electrons of different spins, the team can determine whether the material's surface has a positive or negative topological character. This method works independently of the complex orbital details that usually obscure the view, providing a direct readout of the material's fundamental nature.

To ensure their findings were robust, the team also explored a second, independent method involving magnetic fields. When a magnetic field is applied to the surface, the energy levels of the electrons rearrange into a ladder-like structure. The very first rung of this ladder, known as the zeroth Landau level, sits at a specific energy that depends on the sign of the topological mass. By observing where this level appears relative to the energy gap, the researchers can confirm the sign without needing a spin-polarized tip at all. They further validated their approach by looking at the boundaries between regions of opposite topological signs. At these boundaries, a special channel forms where electrons can flow in a single direction. The researchers showed that the properties of this channel, including its spin direction and energy, provide a self-checking mechanism that confirms the signs of the neighboring regions are indeed opposite.

The study, which combines theoretical derivations with extensive computer simulations, demonstrates that it is possible to map the topological sign of a Dirac mass across a material with atomic-scale resolution. This capability moves beyond simply identifying where a gap exists; it allows scientists to see exactly where the topological character of the material changes. The researchers tested their ideas on known magnetic topological materials, such as manganese bismuth telluride, and found that their method works even in the presence of disorder and imperfections. By establishing a clear protocol for calibrating the measurement and ruling out potential errors caused by the probe itself, they have provided a reliable route for imaging these quantum structures. This breakthrough opens the door to visualizing the real-space arrangement of topological domains, offering a new way to understand and potentially engineer the quantum geometry of materials at the most fundamental level.

Technical Summary: Imaging the Topological Sign of a Dirac Mass with Spin-Polarized Tunneling

Problem Statement
Magnetic order gaps the surface Dirac cone of three-dimensional topological insulators, creating a massive Dirac state that supports a half-quantized surface Hall response and serves as the building block for the quantum anomalous Hall effect. Crucially, the topological character of this state is defined not only by the magnitude of the exchange mass $|M|$ but also by its sign, $\text{sgn}(M)$ . The sign of $M$ determines the sign of the occupied-band Berry curvature, the direction of the surface Hall conductivity, and the chirality of modes bound to magnetic domain walls.

Existing local spectroscopy techniques, such as Scanning Tunneling Microscopy (STM), can resolve the gap magnitude $|M(r)|$ with atomic resolution but cannot distinguish between $+M$ and $-M$ , as the energy spectrum is identical for both. While spin-polarized STM adds sensitivity to local spin textures, the physical electron spin in topological surface states is generally entangled with orbital motion. This spin-orbital entanglement obscures the relationship between the measured spin polarization and the Dirac pseudospin, often reducing the measured polarization or even reversing its sign relative to the pseudospin texture. Consequently, measured spin contrast alone cannot identify the sign of the Dirac mass. Conversely, transport and optical measurements can determine Hall signs but typically average over many microscopic domains, lacking real-space resolution. There is a missing capability for a local measurement that distinguishes spectroscopically identical gaps carrying opposite Berry curvature.

Methodology and Theoretical Framework
The authors establish a framework for determining the local topological sign of $M$ using spin-resolved tunneling spectroscopy, grounded in symmetry arguments and numerical validation.

Symmetry and Sign Groups: The authors analyze the tunneling signal under three symmetry operations: time reversal ( $\Theta_T$ ), a perpendicular spin rotation ( $\Theta_\perp$ ), and a $C_z$ rotation ( $R_z$ ). They demonstrate that tunneling data can determine at most three independent sign combinations: $\text{sgn}(MP_z)$ , $\text{sgn}(vMP_\parallel)$ , and $\text{sgn}(MB)$ . To assign an absolute Hall sign, an external reference (such as a known magnetic field $B$ or a defect moment) is required.
The On-Axis Selection Rule: A central theoretical result is that crystalline rotation symmetry ( $C_n$ with $n \ge 3$ ) resolves the ambiguity caused by spin-orbital entanglement. On a $C_n$ axis (e.g., atop or hollow sites on a $C_{3v}$ surface), the band-edge state is an eigenstate of the rotation operator. This forces the state to be fully spin-polarized along the axis, regardless of the orbital composition. Specifically, for a tip aligned with the $C_n$ axis, the effective polarization $c_{\text{eff}}^z$ equals exactly 1. Consequently, the sign of the tunneling asymmetry at the band edge directly reveals the sign of the occupied-band Berry curvature, independent of orbital weights.
Orbital Filtering as Lorentz Boosts: Away from the band edge or off-axis, the tip acts as an orbital filter, weighting the pseudospin components unequally. This effect is mathematically equivalent to a Lorentz boost of the Bloch vector along the $z$ -axis. While this can shift the measured signal away from the true pseudospin value, the band-edge poles remain fixed points, preserving the sign information at the gap edges.
Independent Routes:
- Zeroth Landau Level: In a perpendicular magnetic field, the position of the zeroth Landau level relative to the gap is determined by $\text{sgn}(MB)$ . This provides a mass-sign measurement using an unpolarized tip, referenced to the laboratory field.
- Domain Walls: Opposite magnetic domains and their connecting domain walls provide internal consistency checks. The Jackiw-Rebbi mode at the wall has a spin perpendicular to the wall and exhibits exact antisymmetry in spin signals between opposite domains.

Key Results

Spin-Polarized Tunneling: The authors demonstrate that on a $C_3$ axis, an axial tip couples to a fully spin-polarized state. The tunneling asymmetry $A = \rho_z/\rho_0$ at the band edge is given by $A = \pm P_z \text{sgn}(\Omega_{\text{occ}})$ , where $\Omega_{\text{occ}}$ is the Berry curvature. This holds for any linearly dispersing surface and any strength of spin-orbit coupling.
Numerical Validation: Tight-binding models of $p$ -orbitals with spin-orbit and orbital Rashba coupling confirm that the on-axis spin of the upper band edge always equals $\text{sgn}(\Omega_{\text{occ}})$ , whereas the orbital-traced spin (measured away from the axis or without symmetry constraints) has the opposite sign in approximately 57% of random parameter sets.
Robustness: The selection rule is robust against lateral offsets, tip tilt, and height variations, with deviations decaying exponentially with tip height.
Noise and Tip Perturbations: The authors derive statistical bounds for sign assignment confidence, showing that with realistic noise levels, the sign can be determined with high probability. They also quantify tip-induced mass shifts (exchange and Zeeman effects), showing that two-sided fits (comparing upper and lower band edges) or $\pm B$ averaging effectively cancel these artifacts. Antiferromagnetic tips are proposed to eliminate stray field effects entirely.
Disorder: Large-scale simulations on lattices with mass disorder show that reading the sign at each site's local band edge (rather than at a fixed energy) maintains correct sign assignment at $\ge 99\%$ of sites, even in the presence of significant disorder.

Significance and Claims
The paper claims to provide a route to imaging the locations where the topological character of a material changes in real space, rather than merely identifying where a spectral gap appears. By establishing how to locally determine the sign of the Dirac mass, the work enables the mapping of real-space topological structures, such as networks of chiral channels connecting domains of opposite Hall response.

The authors propose a straightforward experimental protocol involving:

Calibration of tip polarization.
Recording $\pm B$ -averaged spectra on $C_3$ symmetry sites at band edges.
Cross-checking with the zeroth Landau level position.
Imaging domain walls to verify antisymmetry and in-gap modes.

The framework is immediately applicable to established magnetic topological materials like MnBi $_2$ Te $_4$ and doped (Bi,Sb) $_2$ Te $_3$ , as well as kagome magnets. Beyond determining the topological sign of a Dirac mass, the authors suggest that the same spin-resolved tunneling observables contain information about the Berry curvature and quantum metric, opening a route toward resolving quantum geometry and local topological character in real space with atomic-scale access.

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