Geometric vs. Topological Properties

The Fundamental Distinction: Local Metrics vs. Global Invariants

At its core, the difference between geometry and topology lies in what properties of a space are considered significant.

• Geometry is the study of local properties that depend on a notion of distance, angle, and curvature. Quantities like the length of a path, the area of a surface, or the value of the Berry phase for a specific loop are all geometric. They are typically continuous and will change if you deform the space or the path within it.
• Topology, on the other hand, studies the global properties of a space that are preserved under any continuous deformation, such as stretching or bending, but not tearing or gluing. Topology is concerned with properties like the number of connected pieces, the presence of holes, and whether the space has a boundary. Topological properties are captured by discrete numbers, often integers, called topological invariants.

A classic example is the difference between a sphere, a torus (the surface of a donut), and a coffee mug. Geometrically, they are all very different—they have different surface areas and local curvatures. Topologically, however, the torus and the coffee mug are equivalent because each has exactly one hole, while the sphere has none. You can smoothly deform a coffee mug into a torus without cutting or pasting, but you cannot turn a sphere into a torus without creating a hole.

The following short text and video provide a clear and intuitive introduction to these foundational ideas.

The Bridge: Integrating Local Geometry to Find Global Topology

How can these abstract ideas of topology manifest in a physical system described by local physics? A powerful theme in physics and mathematics is that integrating a local geometric quantity over a closed global space can yield a global topological invariant.

The classic mathematical example is the Gauss-Bonnet Theorem. This theorem states that if you take a two-dimensional compact surface $S$ (like a sphere or a torus), integrate its local Gaussian curvature $K$ over the entire surface, the result is always an integer multiple of $2\pi$. Specifically:

Here, $K$ is a geometric quantity that varies from point to point on the surface. But the integral over the whole closed surface gives $\chi (S)$, the Euler characteristic, which is a topological invariant—an integer that counts the "holes" in the surface (for a surface with $g$ holes, $\chi =2-2g$). For a sphere, $g=0$, so $\chi =2$, and the integral is $4\pi$. For a torus, $g=1$, so $\chi =0$, and the integral is 0. No matter how you stretch or dent the sphere, the total curvature integral remains $4\pi$.

This provides a profound analogy for the physics of topological materials. The Berry curvature $\Omega (\mathbf{k})$ we studied in the last lesson is a local geometric property of the Brillouin zone. The total Berry phase for a loop depends on the loop's geometry. However, if we integrate the Berry curvature over the entire closed surface of the Brillouin zone (which is topologically a torus), we can obtain a quantized integer known as a Chern number. This number is a topological invariant of the electronic band structure, and it is precisely this integer that determines robust, quantized physical observables like the Hall conductance.

The Physicist's Topological Toolkit

To classify topological properties, physicists primarily use two related mathematical frameworks: cohomology and homotopy. Your mathematics minor will have introduced these, but here we'll focus on their physical intuition.

Cohomology: Closed but Not Exact Forms

This framework is a natural extension of our discussion of Berry curvature. In the language of differential forms, the Berry connection $\mathbf{A}$ is a "1-form" and the Berry curvature $\mathbf{\Omega }$ is its exterior derivative, a "2-form" $\mathbf{\Omega }=d\mathbf{A}$.

• An exact form is a form that is the derivative of another form, like a conservative force field $\mathbf{F}=-\nabla V$. The line integral of an exact form around any closed loop is always zero.
• A closed form is a form whose own derivative is zero, like a curl-free vector field $\nabla \times \mathbf{F}=0$.

Every exact form is automatically closed (since $d(d\varphi )=0$, or $\nabla \times (\nabla V)=0$). The interesting topological features arise when a form is closed but not exact. This happens in spaces with "holes". The vector potential $\mathbf{A}$ in the Aharonov-Bohm effect is a prime example: $\mathbf{B}=\nabla \times \mathbf{A}=0$ everywhere outside the solenoid, so $\mathbf{A}$ is locally closed. However, $\oint \mathbf{A}\cdot d\mathbf{l}\ne 0$ for a loop around the solenoid, so $\mathbf{A}$ is not globally exact. De Rham cohomology is the mathematical tool that classifies a space by counting its "closed but not exact" forms. The topological invariants we compute, like Chern numbers, are elements of these cohomology groups.

Homotopy: Winding and Wrapping

Homotopy is an even more fundamental way to classify spaces. It asks: how many fundamentally different ways can you map a sphere of one dimension into your space?

• The first homotopy group, ${\pi }_1(M)$, classifies closed loops (maps from a 1-sphere, $S^1$) on a manifold $M$. If any loop on $M$ can be continuously shrunk to a point, ${\pi }_1(M)$ is trivial. If $M$ has a hole (like the plane minus a point), loops are classified by an integer winding number, and ${\pi }_1(M)=\mathbb{Z}$.
• Higher homotopy groups, ${\pi }_n(M)$, classify maps from higher-dimensional spheres $S^n$.

In condensed matter physics, homotopy groups are invaluable for classifying topological defects. For instance, in a 2D system where the order parameter lives on a circle (like the phase of a superconductor), a vortex is a point defect. A loop in real space around the vortex corresponds to a map from $S^1$ to the order parameter space $S^1$. The winding number of this map classifies the vortex.

A Final Word from Michael Berry

To conclude our discussion, let's hear the distinction made by the originator of the geometric phase himself. In this brief clip, Sir Michael Berry precisely distinguishes between what makes a phase geometric versus topological.

Conclusion

Today we've drawn a sharp line between two profound concepts that are often used interchangeably. Understanding this difference is a key step in developing the research methodology of a theoretical physicist.

Key Takeaways:

• Geometric properties (e.g., Berry phase for an arbitrary loop) are continuous, depend on local details like metric and curvature, and change smoothly as the system is deformed.
• Topological properties (e.g., winding numbers, Chern numbers) are discrete, quantized, and invariant under continuous deformations. They capture the global "connectedness" of a space.
• A key mechanism in physics is the emergence of topological invariants from the integration of local geometric quantities over a closed space (like the Gauss-Bonnet theorem or integrating Berry curvature over the Brillouin zone).
• Physicists use cohomology (classifying closed but not exact forms) and homotopy (classifying maps and winding) to formalize and calculate these topological invariants.

In the next lesson, we will put these ideas into practice. We will move from the abstract to the concrete by constructing our first topological model, the Su-Schrieffer-Heeger (SSH) model of polyacetylene, and explicitly calculating its topological invariant—the winding number. This will be our first step towards understanding how these mathematical ideas lead to astonishing and robust physical phenomena like protected edge states.