🔬 mesoscale physics

Real-Space Imaging of Band Topology via Wavefunction Zeros

This paper establishes a theoretical framework linking the symmetry-enforced zeros of electronic wavefunctions at high-symmetry momenta to band topology, enabling the direct experimental determination of topological invariants via scanning tunnelling microscopy and providing insights into interaction effects in correlated materials.

Original authors: Julian Ingham, Raquel Queiroz

Published 2026-07-27

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Julian Ingham, Raquel Queiroz

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

Imagine the world of electrons inside a solid material as a bustling city. In this city, electrons don't just sit still; they zip around in waves, forming "bands" of energy that determine whether the material is a metal, an insulator, or something exotic like a topological insulator. For decades, scientists have mapped these energy bands by looking at the electrons' momentum—essentially, how fast and in what direction they are moving. It's like trying to understand a city's layout by only watching the speedometers of cars on a highway; you know the traffic flow, but you can't see the buildings, the parks, or the hidden alleys where the action really happens.

However, there is another way to look at this city: real space. This is the actual physical arrangement of atoms and electrons within the material's tiny repeating unit cell. A powerful tool called Scanning Tunneling Microscopy (STM) acts like a super-powered camera that can take pictures of the electron density in this real space, showing us exactly where the electrons are hanging out. The big question has always been: Can a simple picture of "where the electrons are" (which ignores the complex wave-like phases that usually hold the secrets of topology) actually reveal the deep, hidden topological nature of the material? It's a bit like asking if you can tell the genre of a song just by looking at the silence between the notes, without hearing the melody.

This paper, titled "Real-Space Imaging of Band Topology via Wavefunction Zeros," says yes, you absolutely can. The authors, Julian Ingham and Raquel Queiroz, have discovered a mathematical rule that proves certain electron waves must have "dead zones"—places where the electron density is exactly zero—due to the symmetry of the crystal. They call these the "dark sets." Just as a spinning top has a specific axis it cannot wobble around, an electron wave in a crystal with specific symmetries is forced to vanish at specific spots in the unit cell. The paper proves that the pattern of these dark spots is a unique fingerprint of the material's topology. By simply looking at where the electron density is missing in an STM image, scientists can now deduce complex topological numbers (like the Chern number or $Z_2$ index) that were previously thought to require invisible, phase-sensitive measurements. The authors demonstrate this with simulations on materials like transition metal dichalcogenides (TMDs), the Haldane model, and twisted bilayer graphene, showing that these "dark spots" not only reveal the material's identity but also dictate how electrons interact with each other, reshaping the material's behavior in surprising ways.

The "Ghost" in the Machine

To understand the magic here, imagine you are trying to identify a mysterious dancer in a dark room. Usually, you need to see their full costume and hear their music (the phase of the wave) to know who they are. But this paper suggests that if you just look at the floor, you can still identify the dancer by the specific spots they never step on.

In the world of crystals, electrons are waves. When these waves hit the symmetrical structure of a crystal, they interfere with each other. Sometimes, this interference is destructive, meaning the waves cancel each other out perfectly at certain points. The authors prove that for high-symmetry points in the crystal's momentum space (think of these as specific "speeds" or "directions" the electrons can have), the symmetry of the crystal forces the electron wave to be exactly zero at specific locations in the unit cell. They call these locations the "dark set."

Think of it like a game of musical chairs, but with a twist: the chairs are fixed in place by the rules of symmetry, and for certain "songs" (electron states), specific chairs are legally forbidden. If you are an electron in a specific state, you are physically incapable of sitting in those forbidden chairs. The paper shows that the pattern of these forbidden chairs is unique to the "song" being played. If you see a dark spot at the center of a hexagon but not at the corners, you know exactly which "song" the electrons are singing, even if you can't hear the music.

The Detective Work: Reading the Shadows

The authors applied this idea to several famous models in physics to see if it holds up.

First, they looked at Transition Metal Dichalcogenides (TMDs), specifically a material called WSe $_2$ . In this material, the electrons in the valence band (the "ground floor" of the energy building) are supposed to be centered on the atoms. But the paper shows that due to topology, the electrons are actually "obstructed"—they are pushed away from the atoms to the empty spaces between them. In a standard STM image, this would look like the electron density shifting from the atoms to the hollows. The authors' theory predicts that at a specific energy point (the K point), the electron density must vanish at the atomic sites and the hollows, leaving only the "hollow center" bright. Their simulations confirm this: the "dark set" appears exactly where the math says it should, acting as a smoking gun for this obstructed atomic state.

Next, they tackled the Haldane model, a theoretical playground for understanding the Quantum Hall Effect. Here, the goal is to find the "Chern number," a topological number that tells you how many "twists" the electron wave has. The authors found that if the Chern number is zero, the "dark spots" appear on one specific sublattice of the honeycomb grid. But if the Chern number is non-zero (meaning the material is topological), the dark spots shift, leaving only the very center of the hexagon dark. By simply counting which spots are dark in an STM image, you can determine the Chern number modulo three. It's like solving a puzzle by looking at the empty spaces rather than the filled ones.

They also applied this to the Bernevig–Hughes–Zhang (BHZ) model, which describes a Quantum Spin Hall insulator. Here, the "dark set" reveals the $Z_2$ invariant, another topological switch. The rule is surprisingly simple: if the bonds between atoms are bright at both the center ( $\Gamma$ ) and the corner ( $M$ ) of the energy map, the material is topological. If they are dark at the corner, it's trivial. Again, the presence or absence of electrons at specific bonds tells the whole story.

Beyond the Picture: How Electrons Talk to Each Other

The paper doesn't just stop at taking pictures; it explains why these dark spots matter for how electrons interact. Imagine electrons as people in a crowded room. If two people are standing in the same spot, they might bump into each other (interact strongly). But if the rules of the room force one person to stand in a corner and the other in the center, they never meet, and their interaction is weak.

The authors show that in kagome metals (materials with a lattice shaped like a basket weave), the "dark set" forces electrons at certain energy levels to live on different sublattices. They are effectively in different rooms, so they avoid interacting directly. This avoidance actually helps long-range interactions take over, which can lead to exotic states of matter like loop currents.

In Twisted Bilayer Graphene (TBG), the "dark set" explains a mystery about how the material's energy bands change when electrons interact. The electrons at one specific momentum ( $\Gamma$ ) have a "dark spot" right where the electrons at another momentum ( $K$ ) are brightest. This means the $\Gamma$ electrons can hide from the electric field generated by the $K$ electrons. As a result, the $\Gamma$ electrons don't get pushed up in energy as much as the $K$ electrons do. This "hiding" mechanism, dictated by the symmetry-enforced zeros, is what reshapes the flat bands in twisted graphene, a crucial factor in understanding why this material can become a superconductor.

The Bottom Line

This paper provides a rigorous mathematical proof that the "shadows" cast by electron waves in a crystal are not random; they are a direct map of the material's topological soul. By proving that symmetry forces electrons to vanish at specific, predictable locations, the authors have connected the abstract, invisible world of momentum-space topology to the tangible, visible world of scanning tunneling microscopy. They show that you don't need to measure the invisible phase of a wave to know its topology; you just need to know where the wave refuses to go.

The authors demonstrate this through simulations and theoretical proofs across various models, showing that the "dark set" is a robust, gauge-invariant feature. While the results are currently based on theoretical models and simulations (like the STM images of WSe $_2$ , the Haldane model, and twisted bilayer graphene), the framework offers a new, powerful way to diagnose topological phases and understand electron interactions in real materials. It turns the "negative space" of an electron density map into the most important part of the picture.

Title: Real-Space Imaging of Band Topology via Wavefunction Zeros
Authors: Julian Ingham and Raquel Queiroz
Date: July 27, 2026

Problem Statement

Band topology is traditionally characterized by momentum-space invariants (e.g., Berry phases, Chern numbers) or indirect transport signatures. While real-space descriptions using Wannier functions exist, a fundamental question remains: can the topology of an electronic band be inferred directly from a phaseless, real-space image of the charge density? Scanning tunneling microscopy (STM) measures the local density of states (LDOS), $|\Psi_k(r)|^2$ , which discards the complex phase of the wavefunction. Previous work suggested that shifts in charge density between atomic sites and hollow positions (as seen in monolayer WSe $_2$ ) might signal topological features, but a general theoretical framework connecting these real-space patterns to band topology was lacking. The central challenge is to determine if symmetry-enforced zeros in the wavefunction amplitude—visible as "dark spots" in the LDOS—can serve as a universal diagnostic for topological invariants.

Methodology

The authors develop a group-theoretic framework to derive selection rules for wavefunction zeros at high-symmetry momenta ( $k^*$ ).

Symmetry Analysis: They define the intersection of the little group $G_{k^*}$ (symmetries leaving momentum $k^*$ invariant) and the site symmetry group $G_{r^*}$ (symmetries leaving a real-space position $r^*$ invariant).
Reynolds Operator Construction: By evaluating the symmetry transformation of the Bloch wavefunction at an invariant position $r^*$ , they construct a projector operator (Reynolds operator) $P_{k^*,r^*}$ .
Derivation of the Dark Set Index: They derive a scalar index $m_{k^*,r^*}$ (Eq. 4) based on the characters of the irreducible representation (irrep) $\rho$ at $k^*$ and the local phase factors $e^{-ik^* \cdot t}$ .
$m_{k^*,r^*} = \frac{1}{|H_{k^*r^*}|} \sum_{h \in H_{k^*r^*}} e^{-ik^* \cdot t_h(r^*)} \chi_{\rho}(h)$
If $m_{k^*,r^*} = 0$ , the wavefunction amplitude is symmetry-forced to vanish at $r^*$ . The set of such positions is termed the "dark set."
Gauge Invariance: The authors prove that while the representation matrices depend on gauge choices (origin, rotation axes), the physical locations of the zeros (the dark set) are gauge-invariant observables.
Systematic Tabulation: They apply this formalism to all 17 two-dimensional wallpaper groups, tabulating the dark sets for every irrep at every high-symmetry momentum.

Key Contributions and Results

The paper establishes that the "dark set" of a band at a high-symmetry momentum uniquely determines the band's representation content (up to complex conjugation at time-reversal invariant momenta and modulo nonsymmorphic data). This allows for the direct inference of topological invariants from real-space charge density images.

1. Transition Metal Dichalcogenides (1H-TMDs):
Applied to WSe $_2$ , the theory predicts that the valence band at the $K$ point possesses symmetry-enforced zeros at the metal (1a) and chalcogen (1b) sites, leaving only the hollow site (1c) bright. At the $\Gamma$ point, no such zeros exist. This shift in the LDOS pattern between $\Gamma$ and $K$ confirms the "obstructed atomic limit" nature of the valence band, where Wannier centers are displaced from atomic sites.

2. Haldane Model (Chern Insulator):
For the honeycomb lattice, the dark set distinguishes the Chern number modulo 3.

Trivial Phase ( $C=0$ ): The wavefunctions at $K$ and $K'$ have zeros on the same sublattice.
Topological Phase ( $|C|=1$ ): The zeros at $K$ and $K'$ occur on opposite sublattices. Consequently, the total LDOS at a fixed energy shows a distinct pattern: in the trivial phase, one sublattice is dark; in the topological phase, only the hexagon center is dark. This allows STM to diagnose the Chern number modulo three.

3. Bernevig-Hughes-Zhang (BHZ) Model ( $Z_2$ Topological Insulator):
The dark set provides a real-space diagnostic for the $Z_2$ invariant.

Trivial Phase ( $\nu=0$ ): Bonds connecting atoms are dark at the $M$ point.
Topological Phase ( $\nu=1$ ): Bonds are bright at both $\Gamma$ and $M$ .
The contrast of the bond intensity between $\Gamma$ and $M$ directly maps to the Kane-Mele invariant.

4. Interaction Effects:
The paper demonstrates that these zeros are not merely spectroscopic signatures but impose constraints on electron-electron interactions:

Kagome Metals: The dark sets of Van Hove singularities (VHS) at the $M$ point explain the "sublattice interference effect." Different VHS states reside on distinct sublattices, suppressing onsite interactions and enhancing long-range Coulomb interactions, which stabilizes exotic loop current orders.
Twisted Bilayer Graphene (TBG): The $C_3$ -enforced zero of $\Gamma$ -point states at the AA-stacking regions allows them to avoid the Hartree potential generated by the bright $K$ -point states. This explains the qualitative reshaping of flat bands under interactions, where $\Gamma$ states remain relatively unperturbed while $K$ states shift significantly in energy.

Significance

The paper claims to provide a "group-theoretic answer" to the question of whether phaseless real-space images can reveal band topology. The primary significance is the unification of three previously separated views of quantum materials:

Momentum-space topology: The dark set is a real-space manifestation of momentum-space irreps.
Real-space spectroscopy: It provides a direct method to read topological invariants (Chern numbers, $Z_2$ indices) from STM images without phase retrieval.
Correlated electronic order: The zeros act as non-perturbative constraints on interaction channels, explaining interaction-induced phenomena like band reshaping in TBG and the stabilization of loop currents in kagome metals.

The authors frame this as a "spectral geometry" result: just as one can "hear the shape of a drum," one can "see the band topology" in the charge density because crystal symmetry forces the wavefunction amplitude to vanish at specific, symmetry-determined locations. This represents a new form of topological band spectroscopy where the intracell structure of the bulk wavefunction determines the global topology.