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 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 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 applications in topological switching and sensing through the simultaneous non-trivial interplay of skin and topological boundary localizations.

Recent theoretical studies have extended the Berry phase framework to account for higher electric multipole moments, quadrupole and octupole topological phases have been proposed.Although the two-dimensional quantized quadrupole insulators have been demonstrated experimentally, octupole topological phases have not previously been observed experimentally.Here we report on the experimental realization of classical analog of octupole topological insulator in the electric circuit system. Three-dimensional topolectrical circuits for realizing such topological phases are constructed experimentally. We observe octupole topological states protected by the topology of the bulk, which are localized at the corners. Our results provide conclusive evidence of a form of robustness against disorder and deformation, which is characteristic of octupole topological insulators. Our study opens a new route toward higherorder topological phenomena in three-dimensions and paves the way for employing topolectrical circuitry to study complex topological phenomena.

Recently, a new family of symmetry-protected higher-order topological insulators has been proposed, and was shown to host lower-dimensional boundary states. However, with the existence of the strong disorder in the bulk, the crystal symmetry is broken, and the associated corner states are disappeared. It is well known that the emergence of robust edge states and quantized transport can be induced by adding sufficient disorders into a topologically trivial insulator, that is the so-called topological Anderson insulator. The question is whether disorders can also cause the higher-order topological phase. This is not known so far, because the interaction between disorders and the higher-order topological phases is completely different from that with the first-order topological system.Here, we demonstrate theoretically that the disorder-induced higher-order topological corner state can appear in a modified Haldane model. In experiments, we construct the classical analog of such higher-order topological Anderson insulators using electric circuits, and observe the disorder-induced corner state through the voltage measurement.Our work defies the conventional view that the disorder is detrimental to the higher-order topological phase, and offers a feasible platform to investigate the interaction between disorders and the higher-order topological phases.

Recently, the theory of quantized dipole polarization has been extended to account for electric multipole moments, giving rise to the discovery of multipole topological insulators (TIs). Both two-dimensional (2D) quadrupole and three-dimensional (3D) octupole TIs with robust zero-dimensional (0D) corner states have been realized in various classical systems. However, due to the intrinsic 3D limitation, the higher dimensional multipole TIs, such as four-dimensional (4D) hexadecapole TIs, are supposed to be extremely hard to construct in real space, although some of their properties have been discussed through the synthetic dimensions. Here, we theoretically propose and experimentally demonstrate the realization of classical analog of 4D hexadecapole TI based on the electric circuits in fully real space. The explicit construction of 4D hexadecapole circuits, where the connection of nodes is allowed in any desired way free from constraints of locality and dimensionality, is provided. By direct circuit simulations and impedance measurements, the in-gap corner states protected by the quantized hexadecapole moment in the 4D circuit lattices are observed and the robustness of corner state is also demonstrated. Our work offers a new pathway to study the higher order/dimensional topological physics in real space.The exploration of topological physics in various systems 1-4 has become one of the most fascinating frontiers in recent years. Based on the bulk-boundary correspondence principle, the conventional topological phase is always featured by the boundary states with one-dimensional (1D) lower than the bulk that hosts them [5][6][7][8][9][10][11] , e.g., 1D edge states of 2D quantum Hall systems and 2D spin surface states of 3D TIs.Recently, a novel class of symmetry-protected higher-order TIs that possess lowerdimensional boundary states have been proposed by generalizing the fundamental relationship between the Berry phase and quantized polarization, from dipole to multipole moments [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] . The most prominent examples are given by the 2 th -order/2D quadrupole and 3 th -order/3D octupole TIs, which possess robust 0D corner states protected by the quantized quadrupole/octupole polarizations. Motivated by the novel property, many experimental implementations of quadrupole and octupole TIs have been realized in various types of classical systems, including mechanics 13 , acoustics [14][15][16] , photonics 17,18 , and electrical circuits [19][20][21] . However, owing to the limitation of three

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