Local topological markers for Chern insulators in ribbon geometry

This paper investigates local topological markers in Chern insulators with partial translational symmetry, demonstrating their effectiveness in characterizing boundary behaviors and disorder in ribbon geometries while successfully extracting critical scaling exponents for the Kibble-Zurek mechanism.

The Gist

Imagine you are a cartographer trying to map a mysterious, magical island. This island isn't made of land and water, but of electrons dancing in a grid. Some parts of this island are "topological," meaning they have a special, unbreakable knot in their structure (like a Möbius strip), while other parts are just "normal" flat ground.

In physics, we call these special islands Chern insulators. For a long time, scientists could only measure the "knot-ness" of the whole island at once. But what if you wanted to know the knot-ness of just one specific spot on the island, especially near the edge or where the terrain is messy?

This paper introduces a new, super-smart tool called the Local Topological Marker. Think of it as a "knot-o-meter" that you can hold in your hand and walk around the island with, telling you exactly how "knotted" the ground is right under your feet.

Here is the breakdown of their discovery, using simple analogies:

1. The Problem: The "Half-Island" Dilemma

Most previous maps of these islands assumed the island was either:

• A tiny, finite island: Surrounded by water on all sides (Open Boundaries).
• A giant, repeating pattern: Like a wallpaper that goes on forever in every direction (Periodic Boundaries).

But real life is often in between. Imagine a long, narrow ribbon of land. It stretches on forever in one direction (like a highway) but has a distinct edge on the other side. This is called a "ribbon geometry."

The old "knot-o-meters" got confused at the edges of these ribbons. They would give weird readings that didn't match the math. The authors of this paper realized, "Hey, we need a new way to measure that respects the fact that one side is endless and the other is cut off."

2. The Solution: The Hybrid Map

The authors created a new formula that mixes two ways of looking at the world:

• The Position View: Looking at specific spots (like "I am standing at the edge").
• The Momentum View: Looking at the flow of traffic (like "How fast are the electrons moving?").

By combining these, they built a Hybrid Map. This allows them to calculate the "knot-ness" (Chern number) locally, even on a ribbon. It's like having a GPS that knows both your exact street address and the traffic patterns of the whole city simultaneously.

3. The Edge Effect: The "Cliff" vs. The "Beach"

When they tested this on a famous model (the Haldane model), they found something surprising about the edges of the ribbon.

• In a fully finite island: The "knot-ness" at the edge usually cancels out the "knot-ness" in the middle, like a seesaw balancing perfectly.
• On a ribbon: The edge behaves differently! The "knot-ness" at the edge doesn't cancel out the middle. It's like standing on a cliff; the drop-off is permanent and doesn't balance out the flat ground behind you. The authors showed that while the edge looks weird, if you average the whole ribbon, the math still works out perfectly.

4. The Double-Check: Two Different Rulers

To make sure their new "knot-o-meter" was accurate, they compared it to a second tool called the Local Středa Marker.

• The Chern Marker is like a theoretical calculation based on the shape of the electron waves.
• The Středa Marker is like a physical experiment: you wiggle a magnet near the island and see how the electrons react.

The Result: In the middle of the ribbon (the "bulk"), both rulers agreed perfectly. At the edges, they disagreed slightly, but only because the island wasn't infinitely big yet. As they made the ribbon longer, the two rulers agreed even more. This proved their new method is solid.

They also tested what happens if the island is messy (disordered). Imagine throwing rocks on the ground. As long as the rocks aren't too big, the two rulers still agree. If the rocks get too huge, the whole island's structure breaks, and the rulers disagree—but that's expected because the island is broken!

5. The Time-Lapse: The "Freeze" Mechanism

Finally, the authors used their new tool to watch what happens when the island changes its shape very slowly (a "quench").

Imagine the island is slowly shifting from a "flat" state to a "knotted" state.

• The Kibble-Zurek Mechanism: This is a fancy way of saying that when things change too fast, the system gets confused and freezes in a messy state.
• The Discovery: By using their ribbon method, they could simulate much larger islands than ever before. They watched the "knots" form and grow. They found that the size of these messy patches grows in a very specific, predictable way as the change gets slower. It's like watching ice crystals form in water: the slower you cool the water, the bigger the ice crystals get.

Why Does This Matter?

This paper is like upgrading from a blurry, low-resolution photo of a topological island to a high-definition, 3D map.

1. It fixes the edges: It gives us a clear way to measure topological properties on realistic, ribbon-like materials (which are common in nanotechnology).
2. It saves time: By using the "hybrid" math, they could simulate huge systems that were previously impossible to compute.
3. It predicts the future: It helps us understand how these materials behave when they are changing states, which is crucial for building future quantum computers.

In short, the authors gave us a better ruler for measuring the invisible, knotted world of quantum materials, especially when those materials have edges and are a little bit messy.

Technical Summary

1. Problem Statement

Local topological markers, specifically the Local Chern Marker (LCM), are essential tools for characterizing topological phases in systems lacking translational symmetry (e.g., disordered or finite systems). While the LCM is well-established for fully open boundary conditions (fully-OBC) and fully periodic systems, its application to systems with partial translational symmetry (such as ribbons, which are infinite in one direction and finite in the other) has received less attention.

The paper addresses two main gaps:

1. Methodological Gap: There is a need for a robust numerical formulation of the LCM in a hybrid position-momentum basis suitable for ribbon geometries (Open Boundary Conditions in $x$, Periodic Boundary Conditions in $y$).
2. Physical Gap: The behavior of local markers at the boundaries of ribbons is not fully understood, particularly regarding their relationship with the Local Středa marker (an experimentally measurable quantity related to density response to magnetic flux) and their stability under disorder.

2. Methodology

The authors develop a theoretical framework and numerical implementation based on the following approaches:

• Hybrid Position-Momentum Basis:

  • They derive an expression for the LCM, $c(x)$, by decomposing the Fermi projector $P$ into blocks indexed by the conserved momentum $k_y$ (due to translational invariance in the $y$-direction).
  • The formula utilizes a spectral derivative to calculate the momentum derivative $\partial_{k_y} P(k_y)$ via Fourier transforms.
  • For the position commutator $[\hat{x}, P]$, they introduce a Toeplitz matrix of distances ($\Delta x$) to handle the commutator in the presence of Open Boundary Conditions (OBC) in the $x$-direction, ensuring the formula works for both OBC and Periodic Boundary Conditions (PBC).
  • The resulting expression (Eq. 3) allows for efficient calculation of the LCM by reducing the computational cost from $O(N^3)$ to a more manageable scale by exploiting the block-diagonal structure in $k_y$.

• Models Studied:

  • Haldane Model: Used to analyze the LCM and Local Středa marker in a ribbon geometry with zigzag boundaries.
  • Qi-Wu-Zhang (QWZ) Model: Used to study non-equilibrium dynamics (Kibble-Zurek mechanism) in the presence of weak disorder.

• Disorder and Quench Simulations:

  • Disorder is introduced as uncorrelated on-site energy fluctuations that preserve translational symmetry in the $y$-direction (striped disorder).
  • Kibble-Zurek Mechanism (KZM): The authors simulate slow linear quenches across topological phase transitions to extract scaling exponents for the correlation length.

3. Key Contributions

1. Derivation of LCM in Hybrid Basis: The paper provides a rigorous derivation and numerical recipe for calculating the LCM in systems with partial translational symmetry, bridging the gap between fully local and fully momentum-space methods.
2. Boundary Behavior Analysis: The authors demonstrate that the LCM behavior at the boundaries of a ribbon is qualitatively different from fully-OBC systems. In ribbons, the boundary deviations do not compensate for the bulk value in the same way; instead, the average LCM approaches the Chern number $C$ as $\sim 1/N_x$.
3. Comparison with Středa Marker: A systematic comparison between the theoretical LCM and the experimentally accessible Local Středa marker ($c_S$) is performed.
   • They show that $c_S$ converges to the LCM in the bulk.
   • At boundaries, $c_S$ requires a finite magnetic flux $\delta \phi$ to converge, a finite-size effect that scales as $\delta \phi \sim 1/N_y$.
   • The two markers agree closely in the bulk and remain consistent under moderate disorder, provided the disorder does not induce a topological phase transition (Anderson localization effects).
4. Scaling Exponent Extraction: By leveraging the computational efficiency of the hybrid basis, the authors simulate significantly larger systems than previously possible. This allows for the accurate extraction of critical scaling exponents for the KZM in disordered Chern insulators.

4. Key Results

• Haldane Ribbon:
  • In the trivial phase, the LCM is zero in the bulk with small boundary deviations.
  • In the topological phase ($C=1$), the LCM equals 1 in the bulk. Unlike fully-OBC systems, the boundary contribution does not cancel the bulk deviation to yield an exact integer for finite systems; the average converges to $C$ only in the thermodynamic limit.
  • The Local Středa marker matches the LCM in the bulk. At boundaries, $c_S$ converges to the LCM value as the system size increases and $\delta \phi$ is adjusted.
• Disorder Effects:
  • For disorder amplitudes $\delta \lesssim 3$ (in units of hopping), the LCM and Středa markers remain in close agreement.
  • For strong disorder ($\delta \gtrsim 3$), the bulk average of the LCM deviates from the quantized Chern number due to Anderson physics, and the agreement between the two markers breaks down.
• Kibble-Zurek Mechanism:
  • The study confirms that the correlation length $\xi$ scales with the quench duration $\tau$ as $\xi \sim \tau^{1/(1+\nu z)}$.
  • By simulating large systems ($N \approx 500$), the authors extract the scaling exponent $\nu \approx 1$, which matches the analytically predicted value for the QWZ model. This validates the KZM predictions in disordered topological systems with high precision.

5. Significance

• Computational Efficiency: The hybrid basis approach significantly reduces computational complexity, enabling the study of larger systems and more precise finite-size scaling analysis than was previously feasible for local topological markers.
• Experimental Relevance: By validating the Local Středa marker against the LCM in ribbon geometries, the paper strengthens the link between theoretical topological invariants and experimentally measurable quantities (like orbital magnetization or density response to magnetic fields).
• Fundamental Physics: The work clarifies the nature of topological markers at interfaces and boundaries in partially periodic systems, which are common in real-world materials (e.g., nanoribbons, domain walls).
• Non-Equilibrium Dynamics: The successful application of these markers to the Kibble-Zurek mechanism in disordered systems provides a robust method for studying non-equilibrium topological phase transitions.

In conclusion, this paper establishes a powerful numerical framework for analyzing topological order in inhomogeneous systems with partial symmetry, confirming the robustness of local markers and their utility in probing both equilibrium and non-equilibrium critical phenomena.