Probing chiral symmetry with a topological domain wall sensor

Chiral symmetry is a fundamental property with profound implications for the properties of elementary particles, that implies a spectral symmetry (i.e. E => -E ) in their dispersion relation. In condensed matter physics, chiral symmetry is frequently associated with superconductors or materials hosting Dirac fermions such as graphene or topological insulators. There, chiral symmetry is an emergent low-energy property, accompanied by an emergent spectral symmetry. While the chiral symmetry can be broken by crystal distortion or external perturbations, the spectral symmetry frequently survives. As the presence of spectral symmetry does not necessarily imply chiral symmetry, the question arises how these two properties can be experimentally differentiated. Here, we demonstrate how a system with preserved spectral symmetry can reveal underlying broken chiral symmetry using topological defects. Our study shows that these defects induce a spectral imbalance in the Landau level spectrum, providing direct evidence of symmetry alteration at topological domain walls. Using high-resolution STM/STS we demonstrate the intricate interplay between chiral and translational symmetry which is broken at step edges in topological crystalline insulator Pb$_{1-x}$Sn$_x$Se. The chiral symmetry breaking leads to a shift in the guiding center coordinates of the Landau orbitals near the step edge, thus resulting in a distinct chiral flow of the spectral density of Landau levels. This study underscores the pivotal role of topological defects as sensitive probes for detecting hidden symmetries, offering profound insights into emergent phenomena with implications for fundamental physics.

💡 Research Summary

The paper addresses the longstanding problem of distinguishing chiral symmetry breaking from mere spectral (particle‑hole) symmetry in condensed‑matter systems where the latter can survive even when chiral symmetry is lost. Using the topological crystalline insulator Pb₁₋ₓSnₓSe as a platform, the authors demonstrate that step edges on the (001) surface act as highly sensitive “topological domain‑wall sensors” capable of revealing hidden chiral symmetry breaking. In the bulk, a rhombohedral lattice distortion gaps two of the four surface Dirac cones, giving them finite masses ±m and thereby breaking chiral symmetry, while the overall Landau‑level (LL) spectrum retains E→−E symmetry. The key insight is that a half‑unit‑cell step (height ≈ 3 Å) breaks translational symmetry and introduces a π‑shift in the crystal potential, whereas a full‑unit‑cell step (≈ 6 Å) does not. By performing low‑temperature (1.4 K) scanning tunneling microscopy/spectroscopy (STM/STS) under a 12 T magnetic field, the authors map the differential conductance dI/dU and its second derivative across terraces and step edges. They observe that the 0th LL associated with the massless Dirac cone remains fixed, while the massive LLs (E*⁻ and E*⁺) shift in energy as they cross a half‑unit‑cell step, producing a pronounced asymmetry in the local density of states (LDOS). This “chiral flow” manifests as a gradual fading of one massive LL’s intensity on one side of the step and its re‑emergence on the opposite side, while the opposite‑mass LL stays essentially unchanged. In contrast, across a full‑unit‑cell step the LL spectrum is continuous and symmetric, confirming that translational symmetry breaking is essential for exposing the hidden chiral asymmetry.

To rationalize these observations, the authors construct a k·p Hamiltonian for the X₁ and X₂ surface valleys, incorporating velocity terms, a mass term m, a mixing term δ, and a chiral‑symmetry‑breaking term Δ that originates from the rhombohedral distortion. In a magnetic field, the Landau gauge yields guiding‑center coordinates x₀ = k_y ℓ² (ℓ = 1/√eB). The step edge imposes a boundary condition that effectively flips the sign of the translation operator, leading to opposite shifts of the guiding‑center expectation ⟨x⟩ for states at +E and –E. Consequently, the LDOS becomes asymmetric near the edge, even though the bulk spectrum remains particle‑hole symmetric. Numerical calculations of the density of states from the Landau‑level spectrum of the full Hamiltonian reproduce the experimental chiral flow and confirm that the asymmetry disappears far from the edge where translational symmetry is restored.

The work thus establishes topological defects—specifically half‑unit‑cell step edges—as powerful probes for detecting hidden symmetry breaking that is invisible to conventional spectroscopies. By coupling chiral and translational symmetry breaking, the authors reveal a direct experimental signature of chiral symmetry loss in a system that otherwise preserves spectral symmetry. This methodology opens new avenues for exploring subtle symmetry‑topology interplay in a broad class of quantum materials, including other topological insulators, Dirac/Weyl semimetals, and unconventional superconductors.