# Itzykson-Zuber Integral & Degenerate Eigenvalues

1. Apr 16, 2010

### Alamino

It seems that according to the formula, when the eigenvalues of one of the matrices inside the trace of the IZ integral are degenerate, the integral diverges.

Is it correct or the formula is different for this case? For instance, suppose the group is U(N) and I want to calculate

$$\int dU \, \exp\left[\mbox{Tr} (U A U^\dagger \sigma^1_z)\right]$$

where A is Hermitean and $$\sigma^1_z$$ is the spin in the Z direction of the first spin in a 2-qubit system, i.e., the direct product of two Z Pauli matrices. Is the integral divergent in this case?