Probing chiral symmetry with a topological domain wall sensor

This study demonstrates that topological defects, specifically step edges in the topological crystalline insulator Pb$_{1-x}$Sn$_x$Se, can serve as sensitive probes to experimentally differentiate broken chiral symmetry from preserved spectral symmetry by inducing a distinct spectral imbalance and chiral flow in the Landau level spectrum.

The Gist

The Big Idea: Finding the Invisible by Breaking the Rules

Imagine you are in a perfectly symmetrical room. If you look in a mirror, your reflection looks exactly like you. In physics, this is called symmetry. Sometimes, a system has a special kind of symmetry called Chiral Symmetry. Think of this like a perfect dance where every move to the left has an identical, perfect move to the right. If you have this symmetry, the energy of the particles (the dancers) is perfectly balanced: for every dancer moving fast forward, there is one moving fast backward with the exact same energy.

The Problem:
In the material studied in this paper (a special crystal called $\text{Pb}_{1-x}\text{Sn}_x\text{Se}$), the scientists suspected that this perfect "dance" was broken deep inside the crystal. The crystal had a slight distortion (like a floor that isn't perfectly flat), which should have ruined the symmetry.

However, there was a catch: Even though the "dance" was broken, the energy balance still looked perfect. It was as if the dancers were stumbling, but the music still sounded perfectly in tune. The scientists couldn't tell if the symmetry was truly broken just by listening to the music (measuring the energy) because the "broken" part was hidden.

The Solution: The "Step Edge" Sensor
To find the hidden broken symmetry, the scientists didn't look at the flat floor; they looked at the stairs.

Imagine a long, flat hallway (the crystal surface). Suddenly, there is a step up.

• The Big Step: If the step is exactly the height of one full floor tile, the hallway looks the same on both sides. The symmetry is preserved.
• The Half-Step: If the step is only half a tile high, the floor pattern gets "out of sync." The tiles on the top level don't line up with the tiles on the bottom level. This is called a "structural $\pi$-shift."

The scientists discovered that this half-step acts like a super-sensitive detector. When the electrons (the dancers) hit this half-step, the hidden broken symmetry reveals itself.

What They Actually Saw

The researchers used a super-powerful microscope (Scanning Tunneling Microscope) to watch electrons in a strong magnetic field. In a magnetic field, electrons don't move in straight lines; they spin in circles, like cars on a racetrack. These tracks are called Landau Levels.

1. On the flat ground: The electrons spin in perfect circles. The energy levels look balanced and symmetrical. You can't tell if the symmetry is broken.
2. At the Half-Step: When the electrons approach the half-step, something weird happens.
   • The "cars" trying to spin in one direction get pushed to the left.
   • The "cars" trying to spin in the opposite direction get pushed to the right.

This creates a spectral flow. Imagine a river of water flowing toward a waterfall. Usually, the water flows straight down. But here, the water splits: the "left-flowing" water goes one way, and the "right-flowing" water goes the other way, creating a distinct, swirling pattern right at the edge.

This swirling pattern is the "smoking gun." It proves that the underlying symmetry was indeed broken, even though the energy levels looked balanced before.

The Analogy: The Broken Clock

Think of the crystal as a giant clock face.

• Chiral Symmetry means the clock hands move perfectly: if the minute hand moves 1 minute forward, the second hand moves 1 minute backward in a perfect mirror image.
• The Distortion is like a tiny dent in the clock gear. The hands are actually moving slightly off-rhythm, but because the clock is so big, you can't see the dent just by looking at the time.
• The Step Edge is like running the clock over a bump in the road. The bump (the half-step) jolts the gears. Suddenly, the off-rhythm movement becomes obvious because the hands start to wobble in opposite directions.

Why This Matters

This paper is important because it gives scientists a new tool to find "hidden" symmetries.

• In Physics: It helps us understand how particles behave in the universe. Sometimes, the laws of physics look symmetrical, but they aren't. This method helps us spot those subtle breaks.
• In Technology: Understanding these hidden symmetries is crucial for building future quantum computers and ultra-fast electronics. If we can detect these tiny breaks in symmetry, we can design materials that are more stable and efficient.

In short: The scientists found that by looking at the "stairs" (defects) in a crystal, they could see a hidden "dance break" (symmetry breaking) that was invisible when looking at the flat floor. The step edge acted as a sensor, turning a hidden problem into a visible, swirling pattern.

Technical Summary

1. Problem Statement

In condensed matter physics, chiral symmetry is a fundamental property that implies a spectral symmetry where the energy dispersion satisfies $E \to -E$. While chiral symmetry is often associated with massless Dirac fermions (e.g., in graphene), it can be broken by crystal distortions or external perturbations.

• The Core Challenge: A system can possess spectral symmetry ($E \to -E$) even when chiral symmetry is broken. In such cases, the bulk energy spectrum remains symmetric, making it experimentally difficult to distinguish between a system with intact chiral symmetry and one where chiral symmetry is broken but spectral symmetry is preserved.
• The Specific Case: The topological crystalline insulator (TCI) Pb$_{1-x}$Sn$_x$Se undergoes a rhombohedral lattice distortion. This distortion gaps out two of the four surface Dirac cones, breaking chiral symmetry and giving the fermions a mass. However, the Landau level (LL) spectrum retains an overall spectral symmetry ($E \to -E$), obscuring the broken chiral symmetry in standard bulk measurements.

2. Methodology

The authors employed a combination of high-resolution experimental techniques and theoretical modeling to detect the hidden symmetry breaking:

• Sample Preparation: Monocrystals of Pb$_{1-x}$Sn$_x$Se (with $x=0.33$) were grown using the self-selecting vapor growth (SSVG) method. Samples were cleaved in ultra-high vacuum to expose the (001) surface.
• Experimental Technique:
  • Scanning Tunneling Microscopy/Spectroscopy (STM/STS): Measurements were conducted at 1.4 K under magnetic fields up to 12 T.
  • Focus on Defects: Instead of analyzing the bulk terrace, the study focused on step edges (one-dimensional defects) on the surface. Specifically, they compared steps with a height of one unit cell (preserving translational symmetry) versus half-unit cell (breaking translational symmetry with a structural $\pi$-shift).
  • Data Processing: They analyzed $dI/dU$ spectra and their second derivatives to precisely resolve the positions of Landau levels, particularly the zeroth Landau level (0th LL).
• Theoretical Framework:
  • A $k \cdot p$ model was developed to describe the Dirac fermions at the $X_1$ and $X_2$ points of the surface Brillouin zone.
  • The Hamiltonian included terms for rhombohedral distortion ($\epsilon$) and chiral symmetry breaking ($\Delta$).
  • The model incorporated a step edge via a translation operator ($T_{t3} = -\tau_z$) and simulated the system in a magnetic field using the Landau gauge to calculate the Local Density of States (LDOS).

3. Key Contributions

• Identification of a "Sensor": The paper demonstrates that topological domain walls (specifically half-unit cell step edges) act as sensitive sensors for broken chiral symmetry.
• Mechanism of Detection: The authors show that while the bulk spectrum remains symmetric, the guiding center coordinates of the Landau orbitals shift in opposite directions for positive and negative energy states when chiral symmetry is broken. In a translationally invariant system, this shift is unobservable. However, at a step edge that breaks translational symmetry, this shift manifests as a distinct spectral flow asymmetry.
• Differentiation of Symmetries: The work provides a rigorous experimental method to differentiate between preserved spectral symmetry and broken chiral symmetry, a distinction that is often ambiguous in standard spectroscopy.

4. Key Results

• Landau Level Spectroscopy:
  • In the bulk (away from steps), the LL spectrum shows the expected symmetry: a central 0th LL at the Dirac point and symmetric satellite peaks ($E^*_-$ and $E^*_+$) corresponding to massive Dirac fermions.
  • At one-unit cell steps, the spectral flow of LLs remains continuous and symmetric.
  • At half-unit cell steps (where translational symmetry is broken), a unique pattern emerges:
    • Higher-order LLs vanish due to the nucleation of 1D edge modes.
    • The 0th LLs (both massless and massive) exhibit a "chiral" spectral flow.
• Spectral Flow Asymmetry:
  • As the spectral density crosses the half-unit cell step, the intensity of the massive LL with negative mass (lower energy) fades out and re-emerges at a shifted position.
  • Conversely, the massive LL with positive mass (higher energy) maintains almost constant intensity.
  • This results in a non-uniform, asymmetric spectral flow density near the step edge, which is the direct signature of broken chiral symmetry.
• Theoretical Confirmation:
  • The $k \cdot p$ model confirmed that the rhombohedral distortion breaks chiral symmetry ($S$) but preserves a spectral symmetry.
  • The model predicted that the guiding center coordinate $\langle x \rangle$ shifts oppositely for states at $+E$ and $-E$ near the step edge. This theoretical prediction perfectly matched the experimental LDOS maps, explaining the observed spectral flow.

5. Significance

• Fundamental Physics: The study resolves a long-standing epistemological challenge in quantum systems: how to observe "hidden" symmetries that do not manifest in conserved quantities or bulk spectral symmetries. It proves that topological defects can reveal latent symmetry breaking.
• Material Characterization: It offers a precise diagnostic tool for identifying crystal distortions and symmetry breaking in topological materials without requiring bulk-sensitive probes that might average out local effects.
• Broader Implications: The concept of using domain walls as "symmetry sensors" could be extended to other systems involving emergent Dirac fermions, superconductors, and topological phases, providing new avenues for investigating the interplay between chiral, translational, and time-reversal symmetries in condensed matter.

In summary, the paper establishes that chiral symmetry breaking in Pb$_{1-x}$Sn$_x$Se, though invisible in the bulk spectrum due to preserved spectral symmetry, is unambiguously revealed by the asymmetric spectral flow of Landau levels at half-unit cell step edges. This phenomenon arises from the interplay between broken chiral symmetry (due to lattice distortion) and broken translational symmetry (due to the step edge).