Interference of non-Hermiticity with Hermiticity at exceptional points

A family of non-Hermitian but $\mathcal{P}\mathcal{T}-$symmetric $2J$ by $2J$ toy-model tridiagonal-matrix Hamiltonians $H^{(2J)}=H^{(2J)}(t)$ with $J=K+M=1,2,\dots$ and $t<J^2$ is studied, for which a real but non-Hermitian $2K$ by $2K$ tridiagonal-submatrix component $C(t)$ of the Hamiltonian is assumed coupled to its other two complex but Hermitian $M$ by $M$ tridiagonal-submatrix components $A(t)$ and $B(t)$. By construction, (i) all of the submatrices get decoupled at $t=t_M=M(2J-M)$ with $M=1,2,\dots ,J$; (ii) at all of the parameters $t=t_M$ with $M=J-K=0,1,\dots ,J-1$ the Hamiltonian ceases to be diagonalizable exhibiting the Kato’s exceptional-point degeneracy of order $2K$; (iv) the system’s $\mathcal{P}\mathcal{T}-$symmetry gets spontaneously broken when $t\le t_{J-1}=J^2-1$.