Microwave Studies of Three Chiral Ensembles in Chains of Coupled Dielectric Resonators

Abstract

Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the Gaussian orthogonal, Gaussian unitary, and Gaussian symplectic one. With a further particle-antiparticle symmetry the chiral variants of these ensembles, the chiral orthogonal, unitary, and symplectic ensembles (the BDI, AIII, and CII in Cartan's notation) appear which are the main point of interest in this paper. Following a recently published work on chiral random matrix ensembles and their experimental realizations, Phys. Rev. Lett.124, 116801 (2020), this is achieved by using dielectric cylinders placed between two parallel aluminium plates. These cylinders act as microwave resonators which are used to create tight-binding chains of finite length up to N = 5. The different ensembles are achieved by using different types of couplings: (i) for the orthogonal case spatial proximity is used, for the unitary case microwave circulators are used, and (ii) for the symplectic case a combination of circulators and cables is used to create the necessary symmetry. In all cases the predicted repulsion behavior between positive and negative eigenvalues for energies close to zero are verified by a comparison with theory taking the finite size of the systems into account. We will show that the difference to the expected universal behavior is given by logarithmic corrections only. These corrections stem from the Hamiltonians having zero entries in their off-diagonal blocks. topics: wave chaos, random matrix theory, chiral systems, topological materials

Microwave studies of the spectral statistics in chaotic systems

“…Some hundred resonators arrangements have been measured by taking Gaussian distributions for the coupling constants. For details it is referred to the original papers (Rehemanjiang et al 2020, Richter et al 2021. Figures 7(a)-(c) shows the resulting ensemble averaged density of states ρ(ν − ν 0 ) for linear chains of lengths N = 2, 3, and 4.…”

“…For N 5 the zeros of the characteristic polynomial of Hamiltonian (13) can be obtained as roots of quadratic equations. The ensemble averaged density of states is obtained by a Gaussian average over the coupling constants, which too, can be done analytically (Richter et al 2021). The analytical calculations reproduce the predicted repulsion behavior (12) correctly, but there are additional logarithmic terms having their origin in the zero elements in the off-diagonal blocks of Hamiltonian (13).…”

“…Resonators 1 and 2, and resonators 1 and 2 are coupled by cables with a length difference corresponding to a phase difference of π at ν = ν 0 . For more details see the original papers (Rehemanjiang et al 2020, Richter et al 2021. Though four resonators are involved, the configuration corresponds to N = 2, where the resonators with and without bar mimic the two spin components.…”