Experimental identification of the second-order non-Hermitian skin effect with physics-graph-informed machine learning

Topological phases of matter are conventionally characterized by the bulk-boundary correspondence in Hermitian systems: The topological invariant of the bulk in $d$ dimensions corresponds to the number of $(d-1)$-dimensional boundary states. By extension, higher-order topological insulators reveal a bulk-edge-corner correspondence, such that $n$-th order topological phases feature $(d-n)$-dimensional boundary states. The advent of non-Hermitian topological systems sheds new light on the emergence of the non-Hermitian skin effect (NHSE) with an extensive number of boundary modes under open boundary conditions. Still, the higher-order NHSE remains largely unexplored, particularly in the experiment. We introduce an unsupervised approach -- physics-graph-informed machine learning (PGIML) -- to enhance the data mining ability of machine learning with limited domain knowledge. Through PGIML, we experimentally demonstrate the second-order NHSE in a two-dimensional non-Hermitian topolectrical circuit. The admittance spectra of the circuit exhibit an extensive number of corner skin modes and extreme sensitivity of the spectral flow to the boundary conditions. The violation of the conventional bulk-boundary correspondence in the second-order NHSE implies that modification of the topological band theory is inevitable in higher dimensional non-Hermitian systems.

Recently, the concept of non-Hermiticity has been intertwined with topological phases of matter [26][27][28][29] to yield the non-Hermitian skin effect (NHSE) with an extensive number of boundary modes and the necessity to assess non-Hermitian topological properties beyond Bloch band theory [30][31][32].
Despite a fast-growing number of theoretical predictions for non-Hermitian topological systems [33][34][35][36][37][38][39][40][41][42][43][44][45], experimental explorations are still at an early stage [46][47][48][49][50]. To date, the first-order NHSE has been realized in photonic [51] and in circuitry [46][47][48] environments, whereas the experimental realization of the higher-order NHSE remains open. Although skin corner modes have been observed in very recent research [49,50], the unique features of the higher-order NHSE need to be fully demonstrated, both the extensive number of boundary modes under open boundary conditions and the extreme sensitivity of the spectral flow to the boundary conditions. To analyze the spectral flow in higher dimensions, traditional methodologies are challenged by the large-scale data generated. The data size will grow exponentially with the dimension, and additional boundary conditions make it more difficult to analyze the outcome. Machine learning (ML) is a promising way to process large amounts of data [52][53][54]. The existing approaches, however, are unable to efficiently extract the crucial observables, in particular with a largely unexplored state of matter at hand. There is a pressing need for integrating fundamental physical laws and domain knowledge by teaching ML models the governing physical rules, which can, in turn, provide informative priors, i.e., theoretical constraints and inductive understanding of the observable features. To this end, physics-informed ML, using informative priors for the phenomenological description of the world, can be leveraged to improve the performance of the learning algorithm [55].
In this article, we report two significant advances: (i) The methodology of physics-graphinformed machine learning (PGIML) is introduced to enforce identification of an unrevealed physical phenomenon by integrating physical principles, graph visualization of features, and ML. The informative priors provided by PGIML enable an analysis that remains robust even in the presence of imperfect data (such as missing values, outliers, and noise) to make accurate and physically consistent predictions of phenomenological parameters. (ii) The second-order NHSE, characterized by skin corner modes and the violation of the conventional bulk-boundary correspondence, is realized in a two-dimensional (2D) non-Hermitian topoelectrical circuit. We achieve the first experimental demonstration of the extreme sensitivity of the spectral flow to (fully controlled) boundary conditions (PBCx-PBCy, PBCx-OBCy, OBCx-PBCy, and OBCx-OBCy, where PBC (OBC) represents a periodic (open) boundary condition and x (y) represents direction), and observe corner skin modes under OBCx-OBCy as well as edge skin modes under PBCx-OBCy. Prospectively, the powerful tool of PGIML can be applied more widely to solve digital twin problems [56][57][58], thus bridging the physical and digital worlds by linking the flow of data/information between them [59,60].

Physics-graph-informed machine learning
The PGIML framework is implemented in the context of a circuitry environment. In an electrical circuit, the scattering matrix (S-matrix) relates the voltage of the waves incident to ports to those of the waves reflected from ports (see Supplementary Material Sec. S1), providing a complete description of the circuit [61]. According to graph theory (network topology), a N -port electrical circuit can be converted into a matrix G = (P, S) of complex-weighted directed bipartite graphs G ab = (P ab , S ab ) with the matrix P of positions P ab = (a, b) and the S-matrix S of scatteringparameters (S-parameters) S ab , where a, b ∈ {1, 2, . . . , N } denotes the ports [62]. We define the set of graphs as G = (P, S) = {G ab |a, b ∈ {1, 2, . . . , N }} with the set of positions P and the set of S-parameters S. To identify the characteristic features of the circuit, especially of a large circuit, cluster analysis can be used to detect graphs with similar properties. Here, a K-means G κ . The axiom of choice [65] states that for every indexed G κ we can find a representative graphĜ κ such thatĜ κ ∈ G κ . In a digital twin scenario of simulation and experiment, the set of simulated graphs G sim. is generated to describe the numerical outcome that imitates the set of experimental graphs G exp. . As G sim. and G exp. are isomorphic, the subsets G sim.,κ and G exp.,κ are isomorphic [66]. Therefore, PGIML can be understood in the teacher-student scenario in the sense that the teacher (G sim. ) imparts informative priors (Ĝ sim. ) to the student (G exp. ). The reconstructed experimental S-matrixŜ exp. is retrieved.
We depict the PGIML framework in Fig. 1: (i) A lattice model that embeds the unrevealed physical phenomenon is generated and converted into a matrix of graphs G. (ii) The simulated S-matrix S sim. of the circuit is constructed and a learning set G sim. = (P sim. , S sim. ) is accumulated.
(iii) The set of simulated positions P sim. and the set of simulated S-parameters S sim. are classified into clusters P sim.,κ and S sim.,κ using the K-means method (see Supplementary Material Sec. S3).
(iv) The graph-to-graph mappingĜ sim.,κ →Ĝ exp .,κ is translated into a sampling mask that mirrors the clustering information. (v) The representative experimental S-parametersŜ exp.,κ are measured in the circuit. (vi) The S-matrix is encoded with the measured features K κ=1 {Ŝ exp .,κ } and the reconstructed experimental S-matrixŜ exp. is retrieved. The experimental S-matrix S exp. is then given by where E ab is a single-entry matrix (element ab is one and the other elements are zero) [67]. Compared to conventional measurements of N 2 elements, the PGIML method is N 2 /K times faster, as it filters out redundancies, especially efficient for circuits that are too complex for a human to process.

Second-order non-Hermitian skin effect
We are now set up to explore the second-order NHSE, which gives rise to new types of boundary To realize the second-order NHSE experimentally, we design a topoelectrical circuit that represents a 2D non-Hermitian two-band model. The 10 × 10 circuitry lattice is shown in Fig. 2b and the unit cell is shown in Fig. 2c as photograph and in Fig. 2d as scheme. The tight-binding analog of the circuit is shown in Fig. 2e with intracell couplings γ y , intercell couplings λ y in the y-direction, and intercell non-reciprocal couplings ±λ x in the x-direction.
According to Kirchhoff's laws, any circuit can be described by the block diagonal admittance matrix (circuit Laplacian) J(ω) = iωC + 1 iω W, where C and W are the Laplacian matrices of the  capacitance and inverse inductance, respectively. For a given input current of frequency ω = 2πf , we obtain the non-reciprocal two-band admittance matrix (see Supplementary Material Sec. S1)
As the boundary connections can be customized, we can observe phase transitions through differences in the spectral flow, enabling the study of the topological modes at any choice of boundary conditions. The admittance eigenvalues and eigenstates are accessible by an S-parameter measurement using the PGIML framework. We address the circuit for PBCx-PBCy in Figs. 3a-e, for

CONCLUSION
In times of digital research and measurement, many scientific disciplines produce large amounts of data that by far surpass conventional computational abilities for processing and analyzing.
Hence, we develop the PGIML method by integrating physical principles, graph visualization of features, and ML to enforce the identification of an unrevealed physical phenomenon. At the example of a topoelectrical circuit, we embed the physical principles of the second-order NHSE into the circuit, observe the skin corner modes, demonstrate the violation of the conventional bulk-boundary correspondence, and reveal an intriguing interplay between higher-order topology and non-Hermiticity. Our results suggest that the PGIML method provides a paradigm shift in processing and analyzing data, opening new avenues to understanding complex systems in higher dimensions.
where H(k) is the non-Hermitian Bloch Hamiltonian. The second-order NHSE occurs when w(E) = 0. The non-Hermitian topology of H(k) can also be understood in terms of the extended Hermitian which is topologically nontrivial with a finite energy gap if and only if H(k) is topologically nontrivial with a point gap at E.
Since chirality and inversion symmetry here commute, the non-Hermitian topology ofJ (k, ω 0 ) is characterized by the chiral symmetry C = σ z ⊗ τ z . Thus, the second-order NHSE is characterized by the Z topological invariant [40,71] v 2D = w x w y , with the winding numbers where α = x, y. Thus, w x = 1 as E ∈ (−λ x , λ x ) and w y = 1 as λ y /γ y > 1. Hence, we obtain a nonzero topological invariant v 2D = 1 if and only if E ∈ (−λ x , λ x ) and λ y /γ y > 1. v 2D changes when the edge and bulk modes close the gap, establishing the second-order non-Hermitian topology. where I is the identity matrix and Z 0 is the characteristic impedance. In an S-parameter measurement between two ports, the other ports are connected with 50 Ω load terminators to ensure zero reflection. Note that the impedance matrix obtained by our method is equivalent to that obtained by current probes [72,73], while the measurement is simplified dramatically and the experimental stability is improved.

DATA AVAILABILITY STATEMENT
The datasets generated and analyzed in the current study are available from the corresponding author on reasonable request.