🔬 mesoscale physics

Explicit equivalence between the spectral localizer and local Chern and winding markers

This paper explicitly establishes the equivalence between momentum-space topological invariants and real-space markers (such as the local Chern and winding markers) in disordered systems by demonstrating that these markers emerge as leading-order terms in a systematic perturbative expansion of the spectral localizer using only Clifford algebra.

Original authors: Lucien Jezequel, Jens H. Bardarson, Adolfo G. Grushin

Published 2026-05-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lucien Jezequel, Jens H. Bardarson, Adolfo G. Grushin

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

The Big Picture: Two Different Maps for the Same Territory

Imagine you are trying to describe a very strange, bumpy landscape (a "topological material"). In physics, we often want to know if this landscape has a special "knot" or "twist" in its structure. This twist is called a topological invariant. It's a number that tells us the material is special, like a donut having one hole and a sphere having zero.

For a long time, scientists had two different ways to count these knots:

The "Perfect Grid" Method (Chern/Winding Markers): This works great if the landscape is perfectly smooth and repeating, like a tiled floor. You can count the twists by looking at the whole pattern at once. But if the floor is broken, messy, or has random holes (disorder), this method gets confused and stops working.
The "Local Compass" Method (Spectral Localizer Index): This is a newer tool designed for messy landscapes. Instead of looking at the whole floor, it uses a special "compass" (a mathematical operator) that checks the local area to see if the ground is twisted. It works even if the floor is broken or chaotic.

The Problem: Scientists knew both methods usually gave the same answer for the number of knots, but they didn't have a simple, step-by-step proof showing why they were the same. The connection was hidden behind very complex, abstract math (like "K-theory") that is hard for most people to understand.

The Solution: Zooming In with a "Microscope"

This paper provides a clear, simple bridge between the two methods. The authors used a mathematical technique called perturbation expansion, which you can think of as using a microscope to zoom in on the "Local Compass" method.

Here is how they did it:

The Tuning Knob ( $\kappa$ ): The "Local Compass" has a dial or a tuning knob called $\kappa$ (kappa). This knob controls how much weight the compass gives to the "position" of the material versus its "energy."
- Analogy: Imagine you are trying to find a specific house in a city. If you turn the knob one way, you focus on the street address (position). If you turn it the other way, you focus on the building's height (energy). The compass needs a balance between the two to work.
The "Small Knob" Trick: The authors decided to turn the knob to a very small value (close to zero). In math terms, they treated the knob as a tiny "perturbation."
The Expansion (Unfolding the Box): When they expanded the math for this small knob, they found something magical. The complex "Local Compass" formula didn't just look like a random mess; it unfolded into a series of simpler terms.
- The first term in this series (the "leading order") turned out to be exactly the formula for the "Perfect Grid" method (the Chern or Winding marker).
- The subsequent terms were so small they could be ignored.

The Analogy: The Foggy Window

Imagine you are looking at a painting through a foggy window.

The Spectral Localizer is the view through the fog. It's a bit blurry and complex, but it shows the whole picture clearly even if the painting is damaged.
The Local Chern Marker is the view when the window is perfectly clean and you are standing right next to the painting. It's sharp and easy to understand, but only works if the painting is intact.

The authors showed that if you slowly wipe away the fog (by turning the knob $\kappa$ down to zero), the blurry view doesn't just disappear; it transforms directly into the sharp, clean view. They proved mathematically that the "foggy" view is just the "clean" view plus a tiny bit of extra noise that vanishes when you look closely enough.

What They Proved

The paper claims to have explicitly demonstrated that:

In even dimensions (like a flat sheet), the "Local Compass" index is mathematically identical to the Chern marker.
In odd dimensions (like a line or a 3D block), it is identical to the Winding marker.

They did this without using the heavy, abstract machinery that usually connects these ideas. Instead, they used basic algebra and the specific rules of how these mathematical "compasses" are built (Clifford algebra).

Why This Matters (According to the Paper)

Simplicity: It proves the connection using simple, direct math that is accessible to a broader audience of physicists, not just topologists.
Validation: It explains why scientists have been getting the same results using both methods in computer simulations. It confirms that the "Local Compass" is a reliable tool for messy, disordered materials because it is fundamentally the same as the trusted "Perfect Grid" method when you look at it the right way.
The "Knob" Mystery: It helps explain how to choose the value of the tuning knob ( $\kappa$ ). The math shows that as long as the knob is small enough, the two methods will agree.

Summary

The authors took a complex, modern tool for measuring twisted materials (the Spectral Localizer) and showed that, when you look at it through a specific mathematical lens (a small tuning knob), it reveals itself to be the same old, trusted tool (the Chern/Winding marker) that everyone already understood. They provided the missing "instruction manual" that explains exactly how the two are the same.

Technical Summary: Explicit Equivalence Between the Spectral Localizer and Local Chern and Winding Markers

Problem Statement
Topological band insulators are traditionally classified using momentum-space invariants (e.g., Chern or winding numbers) that rely on translational symmetry. However, many physically relevant systems—such as disordered, amorphous, or quasicrystalline materials—lack this symmetry. To characterize topology in these systems, real-space definitions known as "local markers" have been developed. Two prominent approaches are the local Chern marker (and its winding counterpart) and the spectral localizer index.

While the equivalence between various local markers (such as the Kitaev invariant, Bott index, and scattering invariants) has been established, the explicit connection between the spectral localizer index and the local Chern/winding markers has remained obscure. Previous demonstrations of their relationship relied on abstract algebraic topology (K-theory, spectral flow), which obscures the physical intuition and mathematical simplicity. Furthermore, the spectral localizer requires the selection of a scalar parameter $\kappa$ , which weights the position operator relative to the Hamiltonian. The criteria for choosing $\kappa$ are often guided by rigorous mathematical bounds, yet numerical simulations suggest the method remains robust even when these bounds are violated. There is a need for a derivation that explicitly connects these two methods using simpler mathematics to clarify the regime of validity for the spectral localizer.

Methodology
The authors provide a systematic perturbative expansion of the spectral localizer index in powers of the parameter $\kappa$ . The derivation relies exclusively on the Clifford algebra structure inherent to the spectral localizer's construction, avoiding heavy topological machinery.

Definitions: The spectral localizer operator $\hat{L}$ is defined using the Hamiltonian $\hat{H}$ , position operators $\hat{x}_k$ , and auxiliary Clifford matrices $\hat{\sigma}_k$ . The index $I_{SL}$ is defined as half the signature of $\hat{L}$ , or equivalently, the trace of the flattened localizer $\hat{L}_F = \hat{L}(\hat{L}^2)^{-1/2}$ .
Perturbative Expansion: The authors perform a Taylor expansion of $(\hat{L}^2)^{-1/2}$ for small $\kappa$ . They utilize a flattened Hamiltonian $\hat{H}_F$ (where eigenvalues are mapped to $\pm 1$ ) to simplify the algebra.
Algebraic Truncation: The expansion is truncated at order $n=d$ (where $d$ is the spatial dimension). Terms are analyzed using the trace properties of the Clifford algebra. Crucially, the trace over the auxiliary degrees of freedom is non-zero only if the product of $\sigma$ matrices is proportional to the identity. This constraint forces specific terms to vanish and dictates that only terms containing every $\sigma$ matrix at least once survive.
Dimensional Analysis: The authors treat even dimensions (Class A, Chern insulators) and odd dimensions (Class AIII, chiral insulators) separately. They demonstrate that the leading-order term in the expansion corresponds exactly to the local Chern marker (for even $d$ ) or the local winding marker (for odd $d$ ).
Vanishing Terms: A rigorous proof is provided in the appendices showing that the second-order term in the expansion (and higher-order corrections that do not match the marker structure) vanishes due to the cyclic property of the trace and the anti-commutation relations of the operators.

Key Contributions and Results

Explicit Equivalence: The paper establishes the direct equality $I_{SL} = C_{d/2}$ for even dimensions and $I_{SL} = W_{\lceil d/2 \rceil}$ for odd dimensions. This proves that the spectral localizer index reduces to the local Chern and winding markers as the leading-order contribution in the small- $\kappa$ limit.
Simplification of Derivation: By relying on the Clifford algebra and asymptotic analysis of the resolvent, the authors bypass the need for abstract K-theory or spectral flow analysis, offering a derivation accessible to a broader physics audience.
Numerical Validation: A numerical example using the Qi-Wu-Zhang model with disorder demonstrates that while the spectral gap of the localizer is lost during the reduction to the local marker (as the operator becomes un-gapped in the final step), the quantized value of the index remains stable. This confirms that the topological information is preserved throughout the perturbative reduction.

Significance and Claims
The authors claim that this work provides a rigorous foundation for the use of local markers in disordered systems. Specifically:

It justifies why spatial averaging of the local Chern marker yields quantized values in the presence of a bulk gap, a result widely used in numerical studies.
It explains the numerical observation that phase diagrams computed via the spectral localizer and the local Chern marker coincide for small values of $\kappa$ .
The derivation clarifies the relationship between the two methods, suggesting that the spectral localizer is a robust proxy for the local Chern marker in the small- $\kappa$ regime.

The paper remains modest regarding future implications, noting that while the current work focuses on $\mathbb{Z}$ -classified phases (Classes A and AIII), extending this methodology to $\mathbb{Z}_2$ phases (such as time-reversal symmetric systems) remains an open question. The work does not propose new experimental setups but rather solidifies the theoretical underpinnings of existing numerical tools for topological characterization in non-periodic systems.