Inner skin effects on non-Hermitian topological fractals

Abstract

Non-Hermitian (NH) crystals, quasicrystals, and amorphous network display an accumulation of a macroscopic number of states near one of its specific interfaces with vacuum, such as edge, surface, hinge, or corner. This phenomenon is known as the NH skin effect, which can only be observed with open boundary condition. In this regard self-similar fractals, manifesting inner boundaries in the interior of the system, harbor a novel phenomenon, the inner skin effect (ISE). Then the NH skin effect appears at the inner boundaries of the fractal lattice with periodic boundary condition. We showcase this observation by implementing prominent models for NH insulators and superconductors on representative planar Sierpinski carpet fractal lattices. They accommodate both first-order and second-order ISEs at inner edges and corners, respectively, for charged as well as neutral Majorana fermions. Furthermore, over extended parameter regimes ISEs are tied with nontrivial bulk topological invariants, yielding intrinsic ISEs. With the recent success in engineering NH topological phases on highly tunable metamaterial platforms, such as photonic and phononic lattices, as well as topolectric circuits, the proposed ISEs can be observed experimentally at least on fractal metamaterials with periodic boundary condition.

A brief review of hybrid skin-topological effect

“…(i) The various mechanisms of HSTE discussed in this review already call for a systematic classification of these phases, and distinct properties of different classes of HSTE are to be explored. (ii) As a boundary phenomena, variations of HSTE may also arise in higher-dimensional non-Hermitian systems with different defects or impurities, while investigations along this route has only focused on bulk properties so far [58][59][60][61][62]. (iii) The destructive interference of non-reciprocity (which generates HSTE in [63]) has also been shown to induce some anomalous phenomena in 1D, e.g.…”

“…Manifesting as a boundary localization of eigenstates, NHSE becomes more sophisticated in higher dimensional systems, where richer structures of boundaries emerge from different geometries [51][52][53][54][55][56][57] and defects [58][59][60][61][62] of lattices. In this topical review, we provide an overview of the discovery and developments of HSTE [63], a novel phenomenon arising from the interplay of NHSE and topological boundary states in two-or higher-dimensions.…”

Higher-order topological Anderson insulator on the Sierpiński lattice

“…For example, topological phases in fractals do not possess a well-defined bulk like their crystalline counterparts, but they are able to support topologically protected states on the boundary. [22] Topological phases have been widely investigated in different fractal systems, [23][24][25] such as the Chern insulator, [26][27][28][29][30][31] higher-order topological insulator, [32][33][34] non-Hermitian topological insulator, [35] and topological superconductor. [36,37] The second-order topological insulator (SOTI) in fractals exhibits unique inner corner modes.…”