Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator

This paper introduces an efficient real-space formulation of the Chern number via a supercell framework that coincides with the Bott index, using it to demonstrate that topological phases in a disordered Chern insulator remain robust against polarized disorder but undergo a transition to a trivial phase under normal disorder.

The Gist

Imagine you are trying to understand the shape of a complex, invisible object. In the world of quantum physics, this object is a material called a Chern Insulator. These materials are special because they act like perfect electrical highways on their edges while being perfect insulators in their middle. This "magic" is caused by something called topology—a mathematical property that describes how a shape is connected, much like how a coffee mug and a donut are topologically the same (both have one hole), but different from a sphere (which has no holes).

For a long time, scientists could only calculate this "hole count" (called the Chern number) if the material was perfectly ordered, like a pristine crystal lattice. But real-world materials are messy. They have impurities, defects, and disorder. Trying to use the old "perfect crystal" math on a messy material is like trying to measure the circumference of a crumpled piece of paper using a ruler meant for a flat sheet—it just doesn't work.

Here is a simple breakdown of what Kiminori Hattori and Shinji Nakata achieved in this paper:

1. The New Tool: A "Real-Space" Map

The authors invented a new way to calculate this topological "hole count" that works even when the material is messy.

• The Old Way (Momentum Space): Imagine trying to understand a city by looking at a map of its traffic patterns from a satellite. It works great if the city is perfectly grid-like, but if the streets are chaotic and broken, the map becomes useless.
• The New Way (Real Space): Instead of looking at the traffic from above, the authors decided to walk the streets. They created a method that looks directly at the atoms and electrons in the material, regardless of how messy the arrangement is.

They call this a "Supercell Framework." Think of it like taking a giant, zoomed-out photo of a messy neighborhood. Even if the houses are slightly crooked or the trees are scattered randomly, you can still count the total number of "loops" or "holes" in the neighborhood's layout by looking at how the corners of the photo connect to each other.

2. The "Wilson Loop" Analogy

To count these holes, the authors use a mathematical trick called a Wilson Loop.

Imagine you are walking around the edge of a giant, invisible park (the material). You start at the North-West corner, walk to the North-East, then South-East, then South-West, and back to the start.

• In a perfect world, when you return to the start, you are exactly where you began.
• In a quantum world, the "direction" you are facing might have twisted slightly during your walk due to the material's magnetic properties.
• The authors measure this total "twist" accumulated during the walk. If the total twist is a whole number of full circles (360 degrees, 720 degrees, etc.), the material has a non-zero Chern number (it's a topological insulator). If the twist cancels out to zero, it's just a normal insulator.

Their big breakthrough was showing that you can calculate this twist using only the positions of the atoms (real space) without needing to know the perfect, idealized structure of the material.

3. The Experiment: Disorder vs. Polarized Disorder

To test their new tool, they simulated a Chern insulator and then "messied it up" by adding random noise (disorder). They tested two types of noise:

• Scenario A: Normal Disorder (The "Equal Opportunity" Mess)
  Imagine a classroom where students are randomly assigned to sit in either the left or right seat, but the noise affects everyone equally.

  • Result: As the noise got louder, the "topological magic" disappeared. The material lost its special edge currents and became a normal, boring insulator. The "hole" in the donut got filled in.

• Scenario B: Polarized Disorder (The "Biased" Mess)
  Imagine the same classroom, but this time the noise only affects students sitting on the left side. The students on the right side are untouched.

  • Result: Surprisingly, the topological magic survived! Even with massive amounts of noise, the material kept its special properties.
  • Why? The authors found that the "edge states" (the electrons flowing on the boundary) were protected. Because the noise was biased, it couldn't destroy the specific connection between the left and right sides that keeps the "hole" open. It's like having a bridge that is only attacked on one side; the other side remains strong enough to hold the structure together.

4. Why This Matters

This paper is a big deal for two reasons:

1. Efficiency: Their new formula is much faster and easier for computers to calculate than previous methods. It's like switching from a hand-drawn map to a GPS app.
2. Robustness: They discovered that topological materials might be much more durable against disorder than we thought, if the disorder is specific (polarized). This suggests that we might be able to build better, more reliable quantum devices that don't break down easily when exposed to real-world imperfections.

In a nutshell: The authors built a new, faster "ruler" to measure the shape of quantum materials. They used it to prove that while some types of messiness destroy a material's special powers, other types of messiness leave the magic completely intact. This brings us one step closer to building real-world quantum computers that can handle the chaos of the real world.

Technical Summary

1. Problem Statement

Topological invariants, such as the Chern number, are traditionally defined in momentum space using integrals over the Brillouin zone (BZ) of the Berry connection. This formulation relies heavily on translational symmetry, making it inapplicable to disordered or inhomogeneous systems where momentum is not a good quantum number. While several real-space approaches exist (e.g., the Bott index, local Chern markers, scattering matrix methods), they often involve complex projection operators or suffer from numerical inefficiencies. The authors aim to:

1. Derive a robust, efficient real-space formulation of the Chern number applicable to both ordered and disordered systems.
2. Demonstrate the equivalence of this new formulation with existing methods (specifically the Bott index).
3. Investigate the robustness of topological phases in a disordered Chern insulator, specifically distinguishing between "normal" disorder and "polarized" disorder.

2. Methodology

A. Theoretical Framework: Real-Space Chern Number

The authors propose a formulation based on a supercell framework where a disordered system is treated as a large unit cell with periodic boundary conditions (PBC).

• Hamiltonian Diagonalization: The real-space Hamiltonian $H$ is diagonalized to obtain eigenstates $|u_n\rangle$ and eigenvalues $\epsilon_n$.
• Overlap Matrices: Instead of integrating Berry connections, the method constructs overlap matrices between eigenstates at the four corners of the BZ ($k_0, k_1, k_2, k_3$). The overlap matrix between corners $\alpha$ and $\beta$ is defined as:
  $$q_{nn'}^{\alpha\beta} = \langle u_n(k_\alpha) | u_{n'}(k_\beta) \rangle = \langle u_n | e^{i(k_\alpha - k_\beta) \cdot r} | u_{n'} \rangle$$
  In matrix notation, $q^{\alpha\beta} = U^\dagger e^{i(k_\alpha - k_\beta) \cdot r} U$, where $U$ contains the occupied (or unoccupied) eigenstates.
• Wilson Loop Construction: A path-ordered product of these overlap matrices is formed around the BZ boundary:
  $$Q = q^{03} q^{32} q^{21} q^{10}$$
  This product $Q$ represents a Wilson loop.
• Chern Number Definition: The real-space Chern number $C$ is defined via the determinant of $Q$:
  $$C = \frac{1}{2\pi i} \ln \det Q = \frac{1}{2\pi} \sum_p \arg(\lambda_p)$$
  where $\lambda_p$ are the eigenvalues of $Q$.
• Analytical Proof: The authors prove that for sufficiently large systems, $Q$ becomes unitary, ensuring $C$ is quantized to an integer. They further demonstrate that this formulation is mathematically equivalent to the Bott index (specifically the stabilized version used in recent literature) but avoids the computationally expensive projection operator $P = UU^\dagger$, thereby improving numerical efficiency.

B. Model System: Disordered Chern Insulator

• Base Model: The system is a 2D Chern insulator constructed by stacking 1D Rice-Mele (RM) models (a generalized Su-Schrieffer-Heeger model with staggered potential) along the $y$-axis. The Hamiltonian includes intracell hopping ($v$), intercell hopping ($w$), interwire hopping ($t$), and a staggered sublattice potential ($m$).
• Disorder Implementation: A random onsite potential $H_{rand}$ is added. Two types of disorder are studied:
  1. Normal Disorder: Random potential applied equally to both sublattices ($A$ and $B$) with strength $W$.
  2. Polarized Disorder: Random potential applied to only one sublattice (e.g., $W_A \neq 0, W_B = 0$).

C. Numerical Verification

To validate the topological nature of the phases, the authors calculate:

• Real-space Chern Number: Using the derived formula.
• Linear Conductance ($G$): Using recursive Green's functions for a strip geometry with open boundaries to observe edge state transport.
• Density of Bulk States ($D$): Calculated for a cylindrical geometry (periodic boundaries) to observe spectral gap closing and state localization.

3. Key Contributions

1. Efficient Real-Space Formulation: The paper derives a simplified real-space Chern number formula based on the Wilson loop of overlap matrices. It is analytically proven to be equivalent to the Bott index but computationally superior because it eliminates the need for explicit projection operators, reducing execution time significantly for large systems.
2. Equivalence Proof: The authors rigorously show the equality between the Wilson-loop formulation, the stabilized Bott index, and the Chern number of the Fermi projection.
3. Discovery of Disorder-Resistant Topology: The study reveals a stark contrast between normal and polarized disorder. While normal disorder destroys the topological phase, polarized disorder preserves the nontrivial topology even at very high disorder strengths.

4. Key Results

A. Numerical Efficiency and Accuracy

• The new formula (Eq. 14) yields results identical to the momentum-space Chern number for ordered systems and matches the Bott index.
• Accuracy: The numerical error is on the order of $10^{-14}$ for moderate system sizes ($N \ge 4$).
• Efficiency: Compared to the Bott index (Eq. 17) and single-point formulas, the proposed method shows significantly lower execution times for large $N$ because it avoids matrix projections.

B. Phase Transitions in Disordered Systems

• Normal Disorder: As disorder strength $W$ increases, the system undergoes a nontrivial-to-trivial phase transition. The Chern number $C$ drops from $\pm 1$ to $0$ at a critical disorder strength ($W \approx 3$). This is accompanied by the merging of the two bands, the closing of the spectral gap, and the localization of all states (levitation-annihilation process).
• Polarized Disorder: The phase diagram remains unchanged even for extremely high disorder strengths ($W_A$ up to 100). The Chern number remains quantized at $\pm 1$.
  • Mechanism: The authors attribute this robustness to the existence of nontrivial band-edge states that persist against polarized disorder.
  • Edge Modes: In the polarized case, one of the chiral edge modes (specifically the one polarized on the clean sublattice) remains delocalized and protected, preventing the mixing and annihilation of the Chern number.

C. Corroborating Evidence

• Conductance: For normal disorder, edge conductance vanishes as $W$ increases. For polarized disorder, edge conductance remains quantized ($e^2/h$) regardless of disorder strength.
• Density of States: In normal disorder, the spectral gap closes and bands merge. In polarized disorder, sharp peaks at the band edges persist, indicating the survival of topological edge states and the preservation of the bulk gap structure.

5. Significance

• Methodological Advancement: The proposed real-space formula provides a highly efficient tool for calculating topological invariants in complex, disordered, or non-crystalline materials, bridging the gap between theoretical definitions and practical numerical simulations.
• Physical Insight: The discovery that polarized disorder does not destroy topological phases challenges the conventional understanding that disorder generally leads to topological triviality. It suggests that specific symmetries or sublattice structures can protect topology against specific types of perturbations.
• Material Design: These findings imply that topological devices could be engineered to be robust against specific types of impurities (e.g., impurities affecting only one sublattice), potentially leading to more stable topological electronics.

In conclusion, the paper successfully establishes a computationally efficient real-space method for calculating Chern numbers and uncovers a novel mechanism of topological protection where polarized disorder fails to destroy the nontrivial phase, supported by robust edge transport and spectral properties.