Chern number formula

Explore the concept of the Chern number in quantum physics, its calculation, and significance in topological phase transitions.

The Chern Number: An Overview

The Chern number is a fundamental concept in the field of topological quantum physics and differential geometry. Named after the mathematician Shiing-Shen Chern, it acts as an invariant that characterizes certain classes of topological spaces.

For a two-dimensional system with periodic boundary conditions, the Chern number is defined as an integral over the first Brillouin zone, i.e., the momentum space of the system. It is the integral of Berry curvature over the whole momentum space, divided by 2π.

Conceptually, the Chern number measures the amount of quantum ‘twist’ or topological order in a system. More precisely, it captures the topological invariant of a two-dimensional gapped system, indicating how many times the system’s wavefunction wraps around the Bloch sphere as the momentum varies across the Brillouin zone.

Calculation of Chern Number

Chern number, C, can be calculated using the formula:

1. C = 1/(2π) ∫ _BZ F(k) dk

Here, the integral runs over the Brillouin Zone (BZ), and F(k) represents the Berry curvature, a quantity derived from the Berry connection which describes the geometric phase acquired over a cycle in parameter space.

The Chern number takes integer values, and it remains invariant under continuous changes in Hamiltonian parameters unless the energy gap at some point in momentum space closes. This property makes the Chern number a robust indicator of topological phase transitions in quantum systems.

Significance of Chern Number in Physics

The Chern number has wide applications in several areas of physics. For instance, it plays a crucial role in quantum Hall effect systems, where it equals the Hall conductivity in units of the conductance quantum (e²/h).

In the domain of topological insulators and superconductors, the Chern number helps to distinguish between topologically trivial and nontrivial phases. The presence of a non-zero Chern number in these systems is a signature of chiral edge states and indicates the presence of robust current-carrying modes that are immune to disorder and backscattering.

Example of Chern Number Calculation

Let’s consider a two-dimensional, time-reversal-invariant topological insulator described by the following Bloch Hamiltonian:

• H(k) = d(k) . σ

Here, σ are the Pauli matrices, and the vector function d(k) is given by:

• d(k) = (sin(k_x), sin(k_y), m + cos(k_x) + cos(k_y))

The Berry connection for the occupied band is obtained from the Bloch wavefunctions, and the Berry curvature is the curl of the Berry connection in momentum space:

1. F(k) = ∇_k × A(k)

Where A(k) is the Berry connection.

Once we have the Berry curvature, we can integrate over the first Brillouin zone to compute the Chern number:

1. C = 1/(2π) ∫ _BZ F(k) dk

For the above Hamiltonian, the Chern number calculation will give us C = 1 for m < -2 or m > 2, indicating a nontrivial topological phase. For -2 < m < 2, C = 0, indicating a trivial phase. Hence, a transition between topologically distinct phases occurs at m = ±2, demonstrating how the Chern number can signal topological phase transitions.