The quantum anomalous Hall effect (QAHE) describes the response of a material resulting from topological properties of its band structure. These topological properties are often characterized by the Chern number, which can be calculated by calculating the Berry curvature F over the Brillouin zone. According to Ch. 2 of 3, the Berry curvature is the 2-form $$F_{\mu\nu} = dA = \partial_\mu A_\nu - \partial_\nu A_\mu,$$ where the Berry connection of a band is a 1-form defined by $ A = (1/i) < u_-|du_-> $, for states $|u_\pm>$.

Now, when reading Altland and Simons' Condensed Matter Field Theory 2, I noticed in page 16 of the 2nd edition that they formulate a simplified version of Maxwell's theory (inhomogeneous, source-free and no magnetic monopoles) as a variational principle by introducing the 4-potential $A_\mu=(\phi,-A)$, where $\phi$ is the scalar potential and $A$ is the vector potential, such that $E = -\nabla\phi-\partial_t A $ and $B = \nabla\times A$. They then define the EM field tensor: $$ F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu= \begin{bmatrix} 0 & E_1 & E_2 & E_3 \\ -E_1 & 0 & -B_3 & B_2 \\ -E_2 & B_3 & 0 & -B_1 \\ -E_3 & -B_1 & B_1 & 0 \end{bmatrix}. $$

I understand that the mathematical structures are not identical (the EM tensor is a $4\times 4$ matrix whereas the Berry curvature is a $2\times 2$ matrix). However, the field theory notation $F_{\mu\nu}$ is very similar in both cases (QAHE and Maxwell), and they both are related to ideas of EM fields. I get that we use similar mathematical constructs in other fields too (such as GR and high energy), but I felt like the contexts here might overlap more than the others.

In the case of topological insulators, the Berry curvature is often referred to as the 'strength of the magnetic flux'**, and the singularities of Berry curvature are often considered magnetic monopoles (and occur at Dirac points on the Brillouin zone). Maxwell's equations, when written including imaginary magnetic monopoles, can be written as follows 1: $$ \nabla\cdot D = \rho_E, $$ $$ \nabla\cdot B = \rho_M, $$ $$ \nabla\times E = -\frac{\partial B}{\partial t} - J_M, $$ $$ \nabla\times H = \frac{\partial D}{\partial t} - J_E. $$

So, is there a way to relate the two different contexts? I apologize if this is a trivial relationship, but I am just getting started with QFT. If clarification for the relationship between Maxwell's equations and the quantum Hall effect (not just anomalous) will be more intuitive, I would find that useful too. Additionally, please let me know if this question can be improved.

** how does this relate to H or B of Maxwell's equations, if at all? I know they aren't the same because the magnetic flux in topological insulators often results from something abstract (such as next-nearest-neighbor hopping in the Haldane model). But are they analogous?

  1. Forms of Maxwell's Equations
  2. Condensed Matter Field Theory
  3. A Short Course on Topological Insulators

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