Interference of Non-Hermiticity with Hermiticity at Exceptional Points

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