Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits

Abstract

Robust boundary states epitomize how deep physics can give rise to concrete experimental signatures with technological promise. Of late, much attention has focused on two distinct mechanisms for boundary robustness topological protection, as well as the non-Hermitian skin effect. In this work, we report the first experimental realizations of hybrid higher-order skin-topological effect, in which the skin effect selectively acts only on the topological boundary modes, not the bulk modes. Our experiments, which are performed on specially designed non-reciprocal 2D and 3D topolectrical circuit lattices, showcases how non-reciprocal pumping and topological localization dynamically interplays to form various novel states like 2D skin-topological, 3D skin-topological-topological hybrid states, as well as 2D and 3D higher-order non-Hermitian skin states. Realized through our highly versatile and scalable circuit platform, theses states have no Hermitian nor lower-dimensional analog, and pave the way for new applications in topological switching and sensing through the simultaneous non-trivial interplay of skin and topological boundary localizations. Much of contemporary condensed matter physics has been dominated by theoretical and experimental investigations into robust boundary phenomena. Originally formulated in the context of anomalies, they revolutionized the field of condensed matter physics in the form of topological insulators and semimetals [1-3]. More recently, they aroused much attention again as non-Hermitian skin states, which demonstrated how unbalanced non-Hermitian gain/loss can challenge well-held tenets of bulk-boundary correspondence [4-12]. While skin and topological effects are already by themselves conceptually deep, with contrasting implications on the bulk-boundary correspondence, their simultaneous nontrivial interplay has been particularly intriguing. Introduced in Ref.[13], hybrid skin-topological states represent a hierarchy of novel higher-dimensional new states without analogs in Hermitian or non-topological settings. Characterized by scenarios where topological localization dynamically allows the non-Hermitian skin effect to act only on certain topological modes, they greatly augment the richness of higher-dimensional robustness beyond higher-order topological phenomena, and provokes the re-formulation of higher-order topological bulk-boundary correspondence to accommodate various avenues of higher-order skin and topological interplay [14, 15]. Their non-triviality and richness is also apparent in the context of applications: this simultaneous interplay of the skin effect and higher-order topology has been proposed to be used as a topological switch [16]. In these advances, theory has always preceded experiments. The reason is because most tight-binding models proposed are rather artificial for realization in conventional materials or metamaterials. While higher-order topological phenomena [17-23] and the non-Hermitian skin effect [9-12] have been separately realized in several high-profile experiments, their combined interplayed has so far remained a theoretical fantasy due to concomitant challenges of high dimensionality, artificial sublattice structure and non-Hermitian instabilities. In this work, we demonstrate the first experimental realization of hybrid higher order skin-topological states in 2D and 3D through a topolectrical circuit platform. Electrical circuits are ideally suited for transcending the abovementioned challenges, since electronic components, which have benefitted from industrial refinement over the decades, can accommodate almost any desired features like arbitrary long-range connectivity, dimensionality, non-Hermiticity gain/loss as well as non-reciprocity. Recently, simulating topological states with electric circuits has attracted lots of interests based on the similarity between circuit Laplacian and lattice Hamiltonian [23-30]. Some topological states have been observed in circuit networks [31-35]. Through a combination of z-direction non-Hermitian INICs and 2D topological circuits, here we managed to achieve a 3D realization of not just the non-Hermitian skin effect, but also a network of competing such effects that conspire to result in higher-order hybrid states, much beyond the scope of previous circuit demonstrations of 1D skin or higher-order topological robustness individually [10, 18]. Hybrid higher-order topological skin effect in 2D topolectrical circuits Hybrid skin-topological phenomena represent not just the simultaneous presence of non-Hermitian skin as well as topological localizations. They are special scenarios where topological localization in one direction dynamically “switches on” the skin effect in another direction, thereby allowing the skin effect to be felt only by topological modes. Given the wide variety of possible types of topological modes, as well as various nontrivial ways whereby the skin effect modifies topological properties, even in 1 dimension, hybrid modes thus comes in a vast array of possibilities. This is especially interesting in 2 dimensions or higher, which we also experimentally probe, because even topological localization per se is subject to higher-order topological effects, which can be modified by skin localization even before they interplay as hybrid phenomena. We firstly provide the theoretical design of 2D electric circuits to observe skin-topological (ST) and skin-skin (SS) modes, and then give the experimental results to demonstrate such a design. Voltage measurements are particularly sensitive to the spectrum of the circuit Laplacian. In matrix form, Kirchhoff’s law is expressed as I V J \uf03d , where J is the circuit Laplacian. The electrical potentials V at each node can be obtained by inverting this expression to obtain 1 1 V I I J \uf06d \uf06d \uf06d \uf06d \uf065 \uf079 \uf079 \uf02d \uf02d \uf03d \uf03d\uf0e5 (1) Notably, the potentials V are most sensitive to small eigenvalues \uf06d \uf065 , especially if they are vanishing. In the case of skin-topological hybrid modes, the hybrid mode with zero eigenvalue thus dominates voltage measurements, leading the voltage profile V proportional to the corresponding hybrid eigenmode. Indeed, we experimentally observe that the voltage profile is topologically localized in the y-direction, and skin-localized in the x-direction, even though the bulk modes are not supposed to even experience the skin effect. The designed 2D electric circuit network is shown in Fig.1(a). The sample contains 6×6 units and each unit cell contains four sublattices (a, b, c, d). In the network, different circuit unit cells can be used to construct systems with different functions. For example, if we use the unit cell as shown in Fig. 1(b), the hybrid second-order ST modes appear. In contrast, if the unit cell as shown in Fig. 1(c) is used, the SS modes can be observed. Two kinds of unit cell are composed of capacitances, inductances and three different kinds of negative impedance converter through current inversion (INIC). Operational amplifiers arranged as INIC allow the type of nonreciprocity in the circuit to be precisely tuned, where their detail structures are shown in Fig.1(d). For the unit cell as shown in Fig.1(b), we make the directions of INIC be opposite along x and y directions. This can result in vanished net nonreciprocity since the nonreciprocities cancel along x and y directions, but local nonreciprocity among four sublattices in each unit cell still exists, which corresponds to the 2D hybrid lattice model revealed in Ref. [13]. If the directions of INIC are the same along x and y directions as shown in Fig.1(c), the nonreciprocities along both directions of the electric circuit do not interfere destructively, which accomplishes 2D skin lattice model. Details of the theoretical analysis for the lattice models are provided in Methods. No matter what kind of circuit, we can derive circuit Laplacian \uf028 \uf029 2D J \uf077 in the moment space at the resonance frequency based on Kirchhoff’s current law. It can be written as