Unfolding a diabolic point: a generalized crossing scenario

Abstract

The typical avoided crossings for Hermitian quantum systems depending on parameters, the diabolic crossing scenario, are generalized to the non-Hermitian case, e.g. for resonances. Two types of crossings appear: for type I, the real parts show an avoided and the imaginary parts a true crossing of the eigenenergies, and for type II the opposite is found. A simple symmetric non-Hermitian twostate matrix Hamiltonian is analysed in detail. The diabolic point bifurcates into two exceptional ones on exceptional lines where the matrices are defective. The adiabatic transport of eigenvectors and eigenstates in parameter space is discussed in this generalized diabolic crossing scenario, in particular the geometric Berry phases for a cyclic variation of system parameters, depending on the topology of the closed curves with respect to the exceptional lines.

Topological invariance and global Berry phase in non-Hermitian systems

“…For a general nonHermitian system, H(α) = H † (α). The state vectors |Ψ(α) and |Λ(α) within Hilbert space, |Ψ(α) ∈ H and its dual Hilbert space |Λ(α) ∈ H † satisfy the Schrödinger equation, [15] …”

“…Neglecting the off-diagonal terms in the adiabatic approximation, the coefficient can be expressed as [15] c µ (t) = c µ (0)e…”

Encircling an exceptional point

“…Two-state systems described by such a Hamiltonian have been studied both analytically and numerically in [15]. Moreover, subtracting E from the diagonal elements of H given in [3] and performing a transformation of the type defined in equation (3.6) of [3], the Hamiltonian (1) can be brought to the form given in equation (6.2) of [3] with G=0, whose eigenvalues and eigenvectors provide the refractive indices and the associated polarization vectors of dichroic, non-chiral crystals.…”

“…The parameterisation in terms of B is similar to the one discussed in [15]. However, the authors of [15] did not exactly specify the path chosen for encircling a DP.…”

Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation

“…In modern problems of quantum physics, crystal optics, physical chemistry, acoustics and mechanics it is important to know how the diabolic point bifurcates under arbitrary complex perturbations forming topological singularities of eigenvalue surfaces like a double coffee filter with two exceptional points or a diabolic circle of exceptional points, see e.g. [Mondragon and Hernandez (1993)], [Mondragon and Hernandez (1996)], [Shuvalov and Scott (2000)], [Keck et al (2003)], [Berry and Dennis (2003)], [Korsch and Mossman (2003)], [Berry (2004)]. …”