Localized low - frequency Neumann modes in 2 d - systems with rough boundaries

– We compute the relative localization volumes of the vibrational eigenmodes in two-dimensional systems with a regular body but irregular boundaries under Dirichlet and under Neumann boundary conditions. We find that localized states are rare under Dirichlet boundary conditions but very common in the Neumann case. In order to explain this difference, we utilize the fact that under Neumann conditions the integral of the amplitudes, carried out over the whole system area is zero. We discuss, how this condition leads to many localized states in the low-frequency regime and show by numerical simulations, how the number of the localized states and their localization volumes vary with the boundary roughness. Introduction. – The problem of localization in disordered systems [1] has been subject to intense research for several decades. Localization of vibrational or electronical eigenmodes has turned out to change the physical properties, as e.g. the electric properties [1–3], the damping of acoustic resonators [4] and cavities [5], the band gap in semiconductors [6], the wave transport [7] and the density of vibrational states [8, 9]. The reasons for localization have been discussed for long and it is generally accepted that localization always involves some geometrical or structural irregularity of the system. This irregularity can be present as bulk irregularity or as boundary roughness. The first case has been widely investigated and standard models are e.g. the Anderson model [1, 3] and the percolation model [10]. Less works however addressed the problem of localization in systems with an ordered bulk material and an irregular boundary. One model for this problem is the model of fractal drums, where localized states have been found under Dirichlet [9, 12] as well as under Neumann [8, 9] boundary conditions. Recently [11], localized states have also been found in 2d-systems with non-fractal boundary roughness under Neumann boundary conditions, while under Dirichlet conditions they are rare and seem to be linked to the existence of confined regions (which indeed occur in the case of fractal drums [12] or in systems with hard scatterers [13]). In this Letter we focus on this phenomenon and consider several types of systems with non-fractal irregular surfaces. Under Neumann boundaries we find many localized states in all systems with arbitrarily shaped irregular boundaries. We elucidate this behavior by numerical simulations and show how it can be explained by using a sum-rule for the Neumann case.