Experimental observation of non-Hermitian higher-order skin interface states in topological electric circuits

: The study of topological states has developed rapidly in electric circuits, which permits flexible fabrications of non-Hermitian systems by introducing non-Hermitian terms. Here, nonreciprocal coupling terms are realized by utilizing a voltage follower module in non-Hermitian topological electric circuits. We report the experimental realization of one-and two-dimensional non-Hermitian skin interface states in electric circuits, where interface states induced by non-Hermitian skin effects are localized at the interface of different domains carrying different winding numbers. Our electric circuit system provides a readily accessible platform to explore non-Hermitian-induced topological phases, and paves a new road for device applications.

In this work, we experimentally demonstrate higher-order skin interface states in a non-Hermitian topological circuit by utilizing the Hatano-Nelson model based on the theoretical proposal [39].The nonreciprocal coupling terms are realized by utilizing a voltage follower module [60], and adjusted by capacitors.The circuit Laplacian directly corresponds to the Hamiltonian of the physical system [18][19][20][21][22], where the voltages are obtained by simulations and experiments.1D, 2D and high-order Non-Hermitian topological circuits are realized through steady-state and time-domain simulations.
Experiments are carried out to verify the existence of interface state and corner mode in non-Hermitian circuits.Our experimental results are in good agreement with simulations in frequency spectra and voltage distributions.Our designs might pave a road to new designs of non-Hermitian topological circuits.

Ⅱ. 1D non-Hermitian circuit
We start with an interface made of the 1D Hatano-Nelson model as shown in Fig. 1(a)， which is a most typical nonreciprocal system.The asymmetric hopping amplitudes between adjacent sites are indicated by rightward coupling tr and leftward coupling tl.
The topological phase of a non-Hermitian system is characterized by the winding  ( 2 and the matrix H denotes the Hamiltonian of the circuit structure, where vn is the complex voltage on the lattice site n, is the characteristic frequency of the circuit, The energy spectra with the periodic boundary condition (PBC) is：  The designed 1D nonreciprocal circuit reveals a noteworthy non-Hermitian topological feature.This feature is characterized by a distinctive skin effect, which is localized on the edge or interface of a circuit.To observe the skin effect, we implement a closed loop circuit using a FR4 printed circuit board (PCB) as pictured in Fig. 3 Let us construct a non-Hermitian electric circuit with four domains, which possesses four different topological phases.The skin corner mode emerges in the central part.This is the second-order non-Hermitian skin interface state.

Ⅳ. Conclusion
With the help of Op-Amp components, nonreciprocal hopping terms are materialized in non-Hermitian circuits.Based on the Kirchhoff law possessing one-to-one correspondence with the tight-binding Hamiltonian, we have theoretically derived the real and imaginary parts of the energy spectrum of the circuit under PBC and OBC.We have simulated the time-domain propagation of a pulse signal of non-Hermitian skin interface states.Furthermore, we have experimentally demonstrated skin interface states in 1D and 2D non-Hermitian topological circuits.The influencing factors of experimental results in electric circuits are also analyzed.For example, the randomness and resistance give rise to a frequency shift and a small Q factor, which provide a reference for future circuit experiments.Topological circuits provide convenient platforms with a good vision to experimentally demonstrate topological physics.Our work presents fascinating phenomena of non-Hermitian skin effects in different dimensions of topological circuits, which indicates that topological circuit is an effective platform for future Non-Hermitian skin effect is very sensitive to boundary perturbations, so that many non-Hermitian topological devices, for example, the sensor is theoretically proposed [67,68] and experimentally realized in electric circuits [69], which has a high-level sensitivity and a strong robustness.Therefore, the in-depth study of non-Hermitian circuit provides a new idea to realize potential device applications by combing non-Hermitian physics and integrated circuits.
Combining Eq. (A1) and Eq.(A2) to eliminate In, we obtain the following equation, For an open chain circuit with one region, we take the left boundary into consideration, and we obtain The Hamiltonian for a closed-loop circuit with two regions is the region II ( rl tt  ) .All bulk eigenstates are localized at the interface of the two regions with inverted winding numbers, and decay exponentially.The winding directions are indicated by arrows in Fig.1(a).This model can be realized in an electric circuit as shown in Fig. 1(d).Each lattice site is composed of a LC resonant tank with inductance L0 and capacitance C0.The nonreciprocal coupling amplitude is realized by the capacitance Cg with a voltage follower, where current flows unidirectionally.The voltage follower is combined with an Op-Amp, resistors Ra, Rb and capacitance Ca.

FIG. 1
FIG. 1 (a) Schematic diagram of a nonreciprocal non-Hermitian model with two topological distinct regions.(b) The winding number 1 w = for the region I ( . We also consider a closed loop circuit, in which the left side of the region I connects with the right side of the region II, where it has two interfaces.The Hamiltonian for the closed loop circuit is By solving the eigenvalue of the Hamiltonian matrix, we obtain the frequency spectra of the designed circuit.The eigenfrequency is shown as a function of tr/tl in Figs.2(a) and 2(b), corresponding to the open chain circuit and the closed loop circuit, respectively.The hopping amplitude tl is fixed when we choose C0 = 470 pF and Ct = 680 pF, while tr varies with Cg accordingly.All intersection of a red line corresponds to edge (interface) states in the circuit, induced by the non-Hermitian skin effect.The case with Cg = 470 pF corresponding to tr = 2.45 and tl = 1.45 is shown in Figs.2(c), (d) and (e).
detailed derivation in Appendix A).As shown in Fig. 2(c), the energy spectrum is elliptical in the complex plane in the case that tr = 2.45 and tl = 1.45.When the ratio tr/tl in the region Ⅰ and tl/tr in the region Ⅱ are identical, two elliptical energy spectra are degenerated and encircle along inverted directions, which give rise to the winding number 1( 1) w = − + .The energy spectra in the open boundary condition (OBC) is shown in Figs.2(d) and 2(e), which collapse to a line with the zero imaginary part.

FIG. 2
FIG. 2 (a) and (b) Frequency spectra of the nonreciprocal non-Hermitian in the open-chain circuit and the closed-loop circuit, respectively.Parameters are set as C0 = 470 pF, L0 = 47 μH, and Ct = 680 pF.The hopping ratio is / 1 / r l g t t t C C =+ .(c) Elliptical energy spectra with the PBC.(d) and (e) Complex energy spectra of the open-chain circuit and the closed-loop circuit with the OBC respectively, which corresponds to red dashed lines in (a) and (b) with Cg = 470 pF i.e. tr = 2.45 and tl = 1.45.
FIG. 3 (a) Photograph of a fabricated closed loop circuit containing 20 nodes with C0 = 470 pF, Cg = 470 pF, L0 = 47 μH, Ct = 680 pF.The circuit has two interfaces at nodes 11 and 20.(b) and (d) Voltage amplitude at different frequencies for node 11 in experimental measurements and simulations.Different curves in (b) presents 10-times simulated results under 5% randomness in circuit components.(c) and (e) Voltage distribution at different sites in experimental and simulated results, showing strong interface localization at 418, 452 and 475 kHz in experiments and 385, 420 and 460 kHz in simulations.The location of an excited source is placed at node 10 both in simulations and experiments.

Figure 6 (
FIG. 6 (a) Schematic and fabricated sample of the corner mode realized in the 2D nonreciprocal non-Hermitian circuit.The circuit parameters are C0 = 470pF, Cg = 470pF, L0 = 47μH, Ct = 680pF.(b) Voltage amplitude at different frequencies of a 2D nonreciprocal non-Hermitian circuit in experiments.The red and blue curves represent the corner and bulk site of the circuit, respectively.(c) Experimental measurements of the voltage amplitude distribution of the corner mode in 2D non-Hermitian circuit at 308 kHz corresponding to the dashed line in (b).
above derivation shows the Hamiltonian with the OBC, and the dispersion relation with the PBC is deduced as follows: By combining Eq. (A4) and the Bloch theorem 1