Can there be exceptional points in the real axis?

Question

Suppose a matrix $H$ that depends on some parameter $\lambda$ such that, when $\lambda=0$, the eigenvalues of $H$ are real (for instance you can assume that $H$ is Hermitian). By cranking up $\lambda$, the eigenvalues can be shifted into the complex plane (see for a $2\times 2$ matrix example. There are instances in which two eigenvalues coalesce and the matrix has degeneracy in the eigenvalues and the eigenvectors. This is usually termed as an exceptional point "EP" in the literature.

I only have seen such EP where two eigenvalues coalesce in the complex plane. Can there be EP that occur in the real axis, when $H(\lambda=0)$ is Hermitian?

Answer 1

You have asked two different questions: Can exceptional points (EPs) occur when the tuning parameter is real? And can EPs occur when the Hamiltonian is Hermitian?

For the first question, This reference answers with yes.

Since the Hamiltonians considered in this class can be non-Hermitian, the EPs can occur for real parameter values.

For the second question, the answer is no because for a Hermitian matrix always has a set of eigenvectors that can span the whole Hilbert space. (EPs occur when eigenvectors coalesce a.k.a. when the geometric multiplicity of eigenvalues is less than the algebraic mulitplicity.)