# TKNN formula for general Hamiltonian

## TL;DR

This paper proves a general relation between topological invariants and effective action levels in odd-dimensional topological insulators with complex Hamiltonians, using explicit calculations and Ward-Takahashi identities.

## Contribution

It extends the TKNN formula to a broad class of Hamiltonians with non-minimal U(1) gauge couplings, connecting topological numbers to Berry curvature.

## Key findings

- Established the relation between Chern-Simons level and winding number.
- Reduced the winding number to the Chern character of Berry curvature.
- Validated the relation through explicit calculation for general Hamiltonians.

## Abstract

Topological insulators in odd dimensions are characterized by topological numbers. We prove the well-known relation between the topological number given by the Chern character of the Berry curvature and the Chern-Simons level of the low energy effective action for a general class of Hamiltonians bilinear in the fermion with general U(1) gauge interactions including non-minimal couplings by an explicit calculation. A series of Ward-Takahashi identities are crucial to relate the Chern-Simons level to a winding number, which could then be directly reduced to Chern character of Berry curvature by carrying out the integral over the temporal momenta.