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

AI summary

Overview

Research area: Condensed-matter and quantum physics — magnetic topological insulators, surface Dirac fermions, Berry curvature and quantum geometry, scanned-probe (spin-polarized scanning tunneling) spectroscopy.

Technical level: Advanced. The paper is a theory-and-numerics proposal that assumes familiarity with Dirac surface Hamiltonians, Berry curvature, Landau quantization, and tunneling Green's-function formalism.

One-sentence scope: The paper works out how spin-resolved tunneling spectroscopy, symmetry selection rules, and Landau-level spectroscopy can determine the sign (not just the magnitude) of an exchange-generated Dirac mass on a magnetic topological insulator surface, locally and with atomic-scale spatial resolution.

What This Paper Is About

A magnetic topological insulator surface hosts a Dirac cone gapped by an exchange mass M. Standard scanning tunneling microscopy measures only the gap size |M|, which is identical for +M and −M, even though the two signs correspond to opposite Berry curvature, opposite surface Hall response, and opposite chirality of the modes bound to magnetic domain walls. The paper asks how a local probe can distinguish these two spectroscopically identical situations, and what calibration references are needed to assign an absolute topological (Hall) sign.

Key Contributions

1. Complete classification of what tunneling can determine. Using three sign-reversing operations (an antiunitary mirror, time reversal, and a spin rotation), the authors show that tunneling data can access at most three independent sign combinations: sgn(MP_z), sgn(vMP_∥), and sgn(MB). Each requires its own calibrated reference, and no static map can reveal chirality without a time-reversal-odd reference of known sign.
2. A rotation-symmetry selection rule that makes the spin signal equal the Berry-curvature sign. On a C_n axis (n ≥ 2) with magnetization along it, the band-edge state seen by an axial tip is fully spin polarized, so the band-edge tunneling asymmetry equals ±P_z·sgn(Ω_occ) exactly, independent of orbital composition and of the strength of spin–orbit coupling. This removes the usual spin–orbital entanglement ambiguity.
3. An independent, unpolarized-tip route via the zeroth Landau level. The zero-mode energy E_0 = −sgn(eB)(M − β|eB|/ħ) sits at the lower band edge when sgn(eBM) > 0 and at the upper edge otherwise, giving sgn(MB) without a spin-polarized tip.
4. Self-calibrating consistency checks and an experimental error budget. Opposite domains, their exactly antisymmetric spin signal, and the in-plane-polarized chiral domain-wall mode form an internal cross-check; explicit noise, tip-perturbation, and disorder bounds translate this into a concrete measurement protocol.

Main Findings

• Mass sign defined physically: The "mass sign" is defined as the sign of the occupied-band Berry curvature, Ω_occ = v²M/(2E³) for either sign of v, which equals the sign of the half-quantized surface Hall response and fixes domain-wall mode chirality.
• Only three sign combinations are measurable: The three operations Θ_⊥ = σ_x K, Θ_T = iσ_y K, and R_z = iσ_z K form a Klein four-group; the invariant sign monomials are sgn(MP_z), sgn(vMP_∥), and sgn(MB). Transport-type in-plane channels give sgn(vM), not the mass sign.
• Time reversal forces exact cancellation: For any nonmagnetic surface, even with disorder, orbitals, and spin–orbit coupling, Tr_orb[σ_a G^R(R,R;E)] vanishes identically, so a nonmagnetic reference shows zero spin asymmetry; the ±B-averaged signal is exactly odd in the magnetization.
• Clean band-edge limit: For the isotropic cone, A ≡ ρ_z/ρ_0 = M/E at |E| > |M|, and with Lorentzian broadening η, A(±|M|) → ±sgn M as η → 0. Warping and curvature produce no signal on a gapless surface; the apparent "band-inversion" sign change in lattice-regularized models is an artefact of their time-reversal-odd curvature term.
• Selection rule verified numerically: In a C_3v tight-binding model of p orbitals with an s-wave tip, the orbital-traced polarization of the upper edge is +0.16, yet a tip on any C_3 axis sees exactly −1 (the sign of Ω_occ), with M̃/(Jc_z) = 0.9999 at small J. In 9872 random parameter sets the on-axis sign matched the Hall sign in every case, while the orbital-traced sign was opposite in 57% of them (the Fig. 2 caption describes a separate 250-parameter-set panel).
• Finite tip height degrades the signal predictably: The deviation 1 − |c_z^eff| decays as exp(−2Δκz) with Δκ = √(κ² + G²) − κ; for Bi₂Se₃ (G = 1.75 Å⁻¹, κ ≈ 1.1 Å⁻¹) this is about 10⁻⁴ at z = 5 Å. Off-axis and tilted tips produce a Lorentz-boost-like correction to the measured Bloch vector whose fixed points are n_z = ±1, so filtering shifts the signal away from the edge but never at it.
• Landau route confirmed: Lattice calculations confirm the zeroth-Landau-level formula to 3×10⁻⁴. For v = 4×10⁵ m/s the level spacing is 14.5 meV·√(B[T]) and ℓ_B = 25.7 nm/√(B[T]) sets the spatial resolution.
• Domain walls as controls: In the continuum model (M₀ = 0.3, wall width 0.9 v/M₀, η = 0.1M₀), ρ_z/ρ_0 is exactly antisymmetric about the wall (residual 3×10⁻¹⁵); the in-gap Jackiw–Rebbi mode has ρ_x/ρ_0 = −0.82 at E = 0, is dark to a z tip and bright to in-plane tips. Cuts at E = 1.2M₀ give ±0.73 (continuum) and +0.57 / −0.77 (lattice F1, whose asymmetry traces to its time-reversal-odd regulator).
• Decision statistics: For M = 15 meV, 60 energies in [1.3, 4]|M|, and κ = 0.05, the probability of a correct sign at 99% confidence is P = 0.19 at noise σ = 0.1G₀ and P = 0.999 at σ = 0.03G₀.
• Tip-induced artefacts bounded: A tip-induced mass shift is even in energy to first order and cancels to <0.4% when both band edges are compared, but biases one-sided fits by −27% at δM_tip = 0.2|M| for κ = 0.05; the orbital part of a ferromagnetic tip's stray field does not cancel and at κ = 0.2 reverses the near-edge sign above about 1.5 T. Antiferromagnetic tips avoid both effects.
• Disorder robustness: Under 30% mass disorder, the sign read at the band edges and from the field-odd zeroth level is correct at ≥99% of sites; band-edge reading also images patches of reversed sign beyond one Dirac length from their boundaries.

Methodology in Plain English

The authors treat the gapped surface as a two-level (pseudospin) problem and compute what a spin-polarized tip actually measures: a density of states weighted by the tip polarization, expressible as an asymmetry A = P·ρ/ρ₀. They then ask which sign changes of the model parameters leave the tunneling data unchanged, using an exact covariance relation for the Green's function that holds for any defect, any broadening, and the exact T-matrix. This symmetry bookkeeping tells them the maximum information content of any tunneling experiment. To beat the spin–orbital entanglement problem, they use crystalline rotation symmetry: on a rotation axis, the rotation acts only on spin, so components with the wrong spin eigenvalue must vanish at the band edge — a purely symmetry-based argument that survives arbitrarily strong spin–orbit coupling. They check every claim against tight-binding lattice models (including a C_3v p-orbital model and a Chern-insulator lattice model), against exact T-matrix real-space maps, and against Monte Carlo statistics of the sign-decision probability.

Why This Matters

The work converts a qualitative distinction — "gapped versus not gapped" — into a measurable, local, signed quantity, which is what determines the local Hall response and the chirality of edge channels in magnetic topological materials.

• Quantum anomalous Hall devices: Knowing the local sign of M identifies where the Hall response reverses and where chiral channels form, which constrains how domains degrade macroscopic quantization.
• Domain-wall chiral electronics: The predicted in-plane-polarized, unidirectional wall modes are the building blocks of reconfigurable chiral interconnects and racetrack-type concepts.
• Materials screening: The protocol gives a local test for candidate magnetic topological materials (MnBi₂Te₄, Cr- or V-doped (Bi,Sb)₂Te₃, kagome magnets TbMn₆Sn₆ and Fe₃Sn₂) without relying on domain-averaged transport.
• Metrology of quantum geometry: The same spin-resolved observables carry Berry-curvature and quantum-metric information, suggesting a route to mapping quantum geometry in real space.

Industry relevance: The results are directly relevant to groups developing topological quantum electronics and low-power spintronics, and to instrument makers building spin-polarized scanning tunneling microscopes, since the paper specifies concrete requirements on tip polarization stability, tip height, stray field, and noise (for example 0.03G₀ versus 0.1G₀ sensitivity, and antiferromagnetic tip design).

Future Directions

• Experimental realization of the protocol: Step (i) calibrate sgn P_z on a reference magnet and verify the exact vanishing of the spin signal on a nonmagnetic topological insulator; step (ii) record ±B-averaged tip-reversal spectra at atop or hollow sites at both band edges; step (iii) cross-check with the field-odd zeroth Landau level; step (iv) image domain walls and compare the antisymmetry, in-gap mode, and in-plane contrast.
• Tip engineering: Developing antiferromagnetic tips, and characterizing tips with nonzero orbital angular momentum (such as p_x and p_y orbitals of CO-functionalized tips), which couple only to suppressed G ≠ 0 components and can be exposed by a tip-height series.
• Extension to quantum geometry: Using the same spin-resolved tunneling observables to extract Berry curvature and the quantum metric locally, not just the mass sign.
• Broader material classes: Testing whether the symmetry-based readout generalizes to gapped Dirac states in kagome magnets and to other surfaces where the Kramers doublet has nontrivial orbital content.

Target Audience

Experimentalists in scanning tunneling microscopy and spin-polarized tunneling who work on magnetic topological insulators; theorists studying Berry curvature, quantum geometry, and magnetic domain-wall modes; and graduate students or advanced researchers in topological quantum matter who want a rigorous account of what a local spin-resolved measurement can and cannot determine. Readers without background in Dirac surface states and Green's-function tunneling theory will find the derivations demanding, though the core symmetry arguments are stated compactly enough to be followed conceptually.

Authors’ abstract

A Dirac mass is characterized not only by its magnitude but also by its sign, which influences the Berry curvature, Hall response, and the propagation direction of chiral boundary modes. However, local spectroscopy of magnetic topological surfaces has so far accessed only the gap magnitude $|M|$. Here we show how the topological sign of $M$ can be determined locally. We first establish the sign information that spin-resolved tunneling can access and the reference required to obtain an absolute Hall sign. Although spin--orbital entanglement generally obscures the relation between measured spin and Dirac pseudospin, crystalline rotation symmetry alleviates this ambiguity at the band edge. On a $C_3$ axis, an axial tip couples to a fully spin-polarized state whose tunneling asymmetry directly reveals the sign of the occupied-band Berry curvature, independent of orbital composition. We further show that the zeroth Landau level provides an independent mass-sign measurement with an unpolarized tip, while opposite domains and their chiral domain-wall modes provide self-consistency checks. These results pave the way for imaging the locations where the topological character changes within a material, rather than merely identifying where a spectral gap appears.