Complex symmetric Hamiltonians and exceptional points of order four and five

A systematic elementary linear-algebraic construction of non-Hermitian Hamiltonians H = H(gamma) possessing exceptional points gamma = gamma((EP)) of higher orders is proposed. The implementation of the method leading to EPs of orders K = 4 and K = 5 is described in detail. Two distinct areas of applicability of our user-friendly benchmark models are conjectured: (1) in quantum mechanics of non-Hermitian systems and (2) in their experimental simulations via classical systems (e.g., coupled waveguides).