Abstract
The Wilson fermion (WF) is a fundamental particle in the theory of quantum chromodynamics. Theoretical calculations have shown that the WF with a half skyrmion profile represents a quantum anomalous semimetal phase supporting a chiral edge current, but the experimental evidence is still lacking. In this work, we report a direct observation of the WF in circuit systems. We find that WFs manifest as topological spin textures analogous to the half skyrmion, half-skyrmion pair, and Néel skyrmion structures, depending on their mass. Transformations of different WF states are realized by tuning the electric elements. We further experimentally observe the propagation of chiral edge current along the domain-wall separating two circuits with contrast fractional Chern numbers. Our work provides experimental evidence for WFs in topolectrical circuits. The nontrivial analogy between the WF state and the skyrmionic structure builds an intimate connection between the two burgeoning fields.
Similar content being viewed by others
Introduction
Lattice quantum chromodynamics is an effective method to study the strong interactions of quarks mediated by gluons1,2. In lattice quantum chromodynamics calculation, quarks are represented by fermionic fields and placed at lattice sites, and gluons play the role of interactions between neighboring sites3. However, when naively putting the fermionic fields on a lattice, we will meet the fermion doubling problem4, i.e., the emergence of 2d−1 spurious fermionic particles for each original fermion (d is the dimension of the spacetime). The origin of the doubling problem is deeply connected with chiral symmetry and can be traced back to the axial anomaly5. To remove the ambiguity, Kenneth Wilson developed a technique by introducing wave-vector-dependent mass, which modifies the Dirac fermions to Wilson ones6. The fermion doubling issue exists in condensed matter physics as well7,8,9,10,11. It prevents the occurrence of quantum anomalies in lattices, such as the quantum anomalous Hall insulator10 and Weyl semimetal with single node11. It is known that Dirac fermions manifest as the low-energy excitations of topological semimetals/insulators (e.g., graphene)12,13,14,15. However, the observation of Wilson fermion (WF) is yet to be realized.
Recent works show that electrical circuits can be used to mimic and explore the properties of both bosons16,17 and fermions18,19,20. In this work, we utilize a lattice model to realize the WF and probe it in topolectrical circuit experiments20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39. We find that the nontrivial state of the WF strongly depends on its mass and can be classified into three categories characterized by different Chern numbers of 0, ±1/2, and ±1, corresponding to the half-skyrmion pair, half skyrmion, and Néel skyrmion, respectively. We propose a circuit method to efficiently manipulate the transport and transformation of the WF states. In this system, the fractional Chern number dictates a quantum anomalous semimetal (QASM) phase with a chiral edge current as suggested in ref. 40. Here, we report a direct observation of the chiral current along the domain wall (DW) separating two circuits with contrast fractional Chern numbers being 1/2 and −1/2. WFs in a three-dimensional (3D) circuit system are constructed as well. They are characterized by 3D winding numbers and accompanied by the emergence of the surface states and DW states at the boundaries. Our work opens the door for realizing the exotic WFs in solid-state systems.
Results and discussion
We begin from the Dirac Hamiltonian \({{{{{\mathcal{H}}}}}}=c{{{{{\bf{k}}}}}}\cdot \alpha +m{c}^{2}\beta \) with c the light speed, k the wave vector, and α, β being the Dirac matrices, which describes a Dirac fermion with the mass m41. Expressing this Dirac Hamiltonian on a lattice of the tight-binding form, we obtain \({{{{{{\mathcal{H}}}}}}}_{{{{{{\rm{D}}}}}}}=\mathop{\sum }\nolimits_{i = 1}^{d}\frac{\hslash v}{a}\sin ({k}_{i}a){\alpha }_{i}+m{v}^{2}\beta \) with a the lattice constant and ℏv the hopping strength (d is the space dimension). It is straightforward to verify that 2d − 1 non-physics fermion doublers appear at the Brillouin zone (BZ) boundaries ki = π/a. Following Wilson’s method, we derive the WF Hamiltonian of the following form \({{{{{\mathcal{H}}}}}}={{{{{{\mathcal{H}}}}}}}_{{{{{{\rm{D}}}}}}}+{{{{{{\mathcal{H}}}}}}}_{{{{{{\rm{W}}}}}}}\) with \({{{{{{\mathcal{H}}}}}}}_{{{{{{\rm{W}}}}}}}=\frac{4b}{{a}^{2}}{\sin }^{2}\frac{{k}_{i}a}{2}\beta \) being the k-dependent Wilson mass term40. Here, the k-independent mass m in Hamiltonian \({{{{{{\mathcal{H}}}}}}}_{{{{{{\rm{D}}}}}}}\) is referred to as the dispersionless mass of WFs. It is noted that the \({{{{{{\mathcal{H}}}}}}}_{{{{{{\rm{W}}}}}}}\) term breaks the parity symmetry in two-dimensional (2D) and chiral symmetry in 3D cases, which can circumvent the fermion doubling problems4 and reproduce the quantum anomaly in the continuum limit. In Supplementary Note I, we show the details how the doublers from Dirac Hamiltonian are removed by introducing the Wilson mass term. Next, we report the realization of the Wilson Hamiltonian in electrical circuits.
Circuit model
We consider a 2D spinful square lattice in Fig. 1a. The circuit is constituted by four types of capacitors ± C1,2 and the negative impedance converters with current inversion (INICs) in Fig. 1b, where A and B parts correspond to the massless Dirac and Wilson mass Hamiltonians, respectively. It is noted that one can utilize inductors to replace negative capacitors because the admittance of the negative capacitor −iωC is equivalent to the inductor \(-i\frac{1}{\omega L}\) for L = 1/(Cω2)42. Here, ω is the working frequency. We implement two sites to imitate a (pseudo-) spin, indicated by the red rectangle in Fig. 1a, b. The INIC is set up by an operational amplifier and three identical resistors R, as shown in Fig. 1c. In Fig. 1d, we show the realization of the staggered on-site potential, which is modeled as the dispersionless mass of WF.
The circuit response is governed by Kirchhoff’s law \(I(\omega )={{{{{\mathcal{J}}}}}}(\omega )V(\omega )\) with I the input current and V the node voltage. \({{{{{\mathcal{J}}}}}}(\omega ,{{{{{\bf{k}}}}}})\) is the circuit Laplacian written as
where \({j}_{11}=4i\omega {C}_{2}-2i\omega {C}_{2}(\cos {k}_{x}+\cos {k}_{y})+4i\omega \Delta {C}_{2},{j}_{12}=2iG\sin {k}_{x}+2\omega {C}_{1}\sin {k}_{y},{j}_{21}=2iG\sin {k}_{x}-2\omega {C}_{1}\sin {k}_{y},\) and \({j}_{22}=-4i\omega {C}_{2}+2i\omega {C}_{2}(\cos {k}_{x}+\cos {k}_{y})-4i\omega \Delta {C}_{2}\), with G = 1/R the conductance and Δ being the mass coefficient of WFs. In the presence of conductance, the time-reversal symmetry (\({{{{{\mathcal{T}}}}}}\)) of the system is broken because of \({{{{{\mathcal{J}}}}}}{(\omega ,{{{{{\bf{k}}}}}})}^{* }\ne -{{{{{\mathcal{J}}}}}}(\omega ,-{{{{{\bf{k}}}}}})\)27.
By expressing \({{{{{\mathcal{J}}}}}}(\omega )=i{{{{{\mathcal{H}}}}}}(\omega )\) with the Dirac matrices, we obtain
where αx, αy and β represent the Pauli matrices σx, σy, and σz, respectively. It is noted that the above Hamiltonian fully simulates the lattice model of WFs. The first three terms in Eq. (2) represent the Hamiltonian of WF, and the last term is the dispersionless mass of WF. Meanwhile, by tuning the electric elements parameters G, C1, and C2, one can conveniently manipulate the shape of Wilson cones (similar to the Dirac cones).
In what follows, we analyze the topological properties of the lattice model.
Topological phase and phase transition
For a \({{{{{\mathcal{T}}}}}}\)-broken 2D two-band system, one can evaluate the Chern number43
to judge its topological properties. Here \({{{{{\bf{A}}}}}}({{{{{\bf{k}}}}}})=i \langle {u}_{{{{{{\bf{k}}}}}}}| {\nabla }_{{{{{{\bf{k}}}}}}}| {u}_{{{{{{\bf{k}}}}}}} \rangle \) is the Berry connection with \(| {u}_{{{{{{\bf{k}}}}}}} \rangle \) the eigenstate of lower band.
In following calculations, we adopt Ci = C = 1 nF (i = 1, 2), f = ω/(2π) = 806 kHz (In experiments, we will use L = 39 μH to replace −C, so we choose \(\omega =1/\sqrt{LC}\)), and G = ωC = 0.005 Ω−1 (R = 200 Ω). Calculations of Chern number as a function of the dispersionless mass parameter Δ are plotted in Fig. 2a. We find the Chern numbers are quantized to five values ± 1, \(\pm \frac{1}{2}\), and 0. It is noted that the Chern number is irrelevant to the value of C1 but becomes opposite if C2 changes its sign. We show the first BZ and typical band structures in Fig. 2b. Combined with the topological index, we classify these topological phases as follows. The band gaps open for the parameter intervals ① and ⑦, where the Chern number is zero. It gives the trivial insulator phase. For parameters in ② and ⑥, the band gaps close at M and Γ points, respectively. Surprisingly, the Chern numbers are quantized to ∓1/2, respectively. This phase is dubbed the QASM40. For parameter zones ③ and ⑤, the band gaps open with the topological number \({{{{{\mathcal{C}}}}}}=\mp 1\), both of which represent Chern insulators with opposite chiralities. For the parameter region ④, the band structure closes at X point with a vanishing Chern number, indicating a normal semimetal phase.
By expressing Eq. (2) as \({{{{{\mathcal{H}}}}}}={{{{{\bf{f}}}}}}({{{{{\bf{k}}}}}})\cdot {{{{{\boldsymbol{\sigma }}}}}}\), one can define a unit spin vector \(\hat{{{{{{\bf{f}}}}}}}({{{{{\bf{k}}}}}})\) as \(\hat{{{{{{\bf{f}}}}}}}({{{{{\bf{k}}}}}})=\frac{{{{{{\bf{f}}}}}}({{{{{\bf{k}}}}}})}{| {{{{{\bf{f}}}}}}({{{{{\bf{k}}}}}})| }\), where f(k) = (fx, fy, fz) is the coefficient of Pauli matrices and \(| {{{{{\bf{f}}}}}}({{{{{\bf{k}}}}}})| =\sqrt{{f}_{x}^{2}+{f}_{y}^{2}+{f}_{z}^{2}}\). Figure 2c displays the spin textures of 2D WFs, which evolve as the increasing of dispersionless mass. The spin textures are reminiscent of the magnetic solitons in the condensed matter system44,45,46. It is observed that the spin textures of trivial insulator, Chern insulator, QASM, and normal semimetal correspond to the ferromagnetic ground state, skyrmion, half skyrmion, and half-skyrmion pair, respectively. By evaluating the topological charge \(Q=\frac{1}{4\pi }{\int}_{{{{{{\rm{BZ}}}}}}}\hat{{{{{{\bf{f}}}}}}}\cdot (\frac{\partial \hat{{{{{{\bf{f}}}}}}}}{\partial {k}_{x}}\times \frac{\partial \hat{{{{{{\bf{f}}}}}}}}{\partial {k}_{y}})d{k}_{x}d{k}_{y}\) in Fig. 2c, we identify an intimate connection with the Chern number as \(Q+{{{{{\mathcal{C}}}}}}=0\). This finding thus establishes an interesting map between WFs and magnetic solitons in electrical circuits.
In the broad spintronics community, the manipulation of skyrmion motion is crucial for the next-generation information industry47. Here, we propose a method to control the circuit skyrmion motion in momentum space. We first consider a skyrmion configuration with Δ = − 0.5. To generate a skyrmion propagation along kx direction over a distance k0, one can modify Eq. (2) to \({{{{{\mathcal{H}}}}}}(\omega )=2G\sin ({k}_{x}-{k}_{0}){\sigma }_{x}+2\omega {C}_{1}\sin {k}_{y}{\sigma }_{y}-4\omega {C}_{2}[\cos ({k}_{x}-{k}_{0})+\cos {k}_{y}+\frac{1}{2}]{\sigma }_{z}\), which can be recast as \({{{{{\mathcal{H}}}}}}(\omega )=2{G}^{{\prime} }\sin {k}_{x}{\sigma }_{x}-2{G}^{{\prime\prime} }\cos {k}_{x}{\sigma }_{x}+2\omega {C}_{1}\sin {k}_{y}{\sigma }_{y}-[4\omega {C}_{2}^{{\prime} }\cos {k}_{x}+4\omega {C}_{2}^{{\prime\prime} }\sin {k}_{x}+4\omega {C}_{2}(\cos {k}_{y}+\frac{1}{2})]{\sigma }_{z}\). Compared with the original Eq. (2), one merely needs to modify two hopping strengths (\({G}^{{\prime} }=G\cos {k}_{0}\) and \({C}_{2}^{{\prime} }={C}_{2}\cos {k}_{0}\)) and to add two extra hopping terms (\(-2{G}^{{\prime\prime} }\cos {k}_{x}{\sigma }_{x}\) and \(4\omega {C}_{2}^{{\prime\prime} }\sin {k}_{x}{\sigma }_{z}\)). Variable resistors and capacitors can be conveniently adopted to realize these operations in circuit systems.
Quantum anomalous semimetals
To show the properties of QASM (Δ = 0 with \({{{{{\mathcal{C}}}}}}=\frac{1}{2}\)), we consider a ribbon configuration with periodic boundary condition along \(\hat{x}\) direction and Ny = 50 nodes along \(\hat{y}\) direction. Figure 3a shows the admittance spectra, where the conduction and valence bands touch at kx = 0. There is no isolated band in this admittance spectra, so the edge state is absent. Considering the Hamiltonian (2), one can define a velocity operator as \(v=1/(i\hslash )[x,{{{{{\mathcal{H}}}}}}]=\frac{\partial {{{{{\mathcal{H}}}}}}}{\partial {k}_{x}}=2G\cos {k}_{x}{\sigma }_{x}+2\omega {C}_{2}\sin {k}_{x}{\sigma }_{z}\). The transverse current density then can be written as
with ϕn(kx, y) being the wave functions of the n-th band and the sum index n indicating the bands below the admittance μ. We plot the current density for different positions in Fig. 3b. It is found that the current density decays from the boundary nodes, and its values are opposite for the top and bottom edges. Consequently, the chiral edge currents constitute the bulk-edge correspondence of QASM40,48. It is noted that the chiral current differs from the one in magnetic systems that originates from the chiral spin pumping effect induced by the topological structures in real space49,50.
Then, we consider a 10 × 10 square lattice to study the finite-size effect. Diagonalizing the corresponding circuit Laplacian, we obtain the admittance spectra shown in Fig. 3c and the wave functions near jn = 0 Ω−1 in the inset of Fig. 3c. In our circuit, the impedance between the node a and the ground is computed by \({Z}_{a,{{{{{\rm{ground}}}}}}}={\sum }_{n}\frac{| {\phi }_{n,a}{| }^{2}}{{j}_{n}}\) with ϕn,a the wave function of node a for nth admittance mode, which reflects the features of wave functions near jn = 0 Ω−1 and can be measured readily25,33. By comparing Fig. 3d and 3c (inset), we find that the impedance of each node against the ground exhibits almost the same spatial distribution to wave functions, and one does not find an edge state.
To demonstrate the bulk-boundary correspondence, we consider a one-dimensional DW with 10 × 11 “spins” (an extra column is set up for DW configuration), as shown in Fig. 3e, where the capacitor C2 has a kink at the center of the sample, i.e., C2 > 0 ( < 0) in the light blue (green) region. In this circumstance, the Chern number varies from 1/2 (left) to −1/2 (right). The eigenvalue and wave function of the bound state can be solved as J = 2ωC1ky and \(\phi (x,y)=\frac{1}{\sqrt{2\pi }}{\chi }_{y}\sqrt{\lambda }\exp (-\lambda | x| +i{k}_{y}y)\) with \({\chi }_{y}=\frac{\sqrt{2}}{2}{(-i,1)}^{{{{{{\rm{T}}}}}}}\) and \(\lambda =\frac{{C}_{1}}{| {C}_{2}| }+\sqrt{{(\frac{{C}_{1}}{{C}_{2}})}^{2}+{k}_{y}^{2}}\). The nonzero eigenvalue means the bound states do not localize at zero admittance. One can obtain the effective velocity of the bound state \({v}_{{{{{{\rm{eff}}}}}}}=\frac{\partial J}{\partial {k}_{y}}=2\omega {C}_{1}\), indicting the DW bound state propagating along \(\hat{y}\) direction (see Supplementary Note 2). In Fig. 3f, we show the admittance spectrum with the insets displaying the wave functions and impedances, from which one can clearly see the bound state confined inside the DW. Here, we use the inverse participation ratio p = lg(∑i∣ϕn,i∣4) of the system to characterize the localization properties of the wave functions51,52, and the green dots indicate the localized state.
Then, we prepare a printed circuit board to verify the numerical results, as shown in Fig. 3g (see Methods for details). Figure 3h shows the experimental impedance distribution. It demonstrates a localized state between two domains, which compares well with the theoretical result in Fig. 3f. The existence of one bound state is closely related to the fact that the topological invariants between the two sides of the DW differ by 1. In addition, one cannot observe the edge states on the rest boundaries of the sample, which confirms that there is indeed no edge state. We also study the influence of the homogeneity of the circuit components on our results, which does not change our conclusion.
To show the chiral propagation of the bound state, we perform the circuit simulation with the software LTSPICE (http://www.linear.com/LTspice). By inputting a Gauss signal close to the DW, we observe the bound mode propagating along the \(\hat{y}\) direction of DW. Finally, the voltage signal becomes a steady bound state inside the DW (see Supplementary Note 2).
Chern insulators and normal semimetal
To characterize Chern insulators (\({{{{{\mathcal{C}}}}}}=\pm 1\)), we compute the admittance for a ribbon configuration, with the results being shown in Fig. S3 of Supplementary Note 3. For the parameter Δ = −0.5 and −1.5, we find two crossing bands in the admittance gaps but with opposite charities. To show the chiral propagation of edge states, we perform the circuit simulation on a finite-size square lattice and observe a chiral voltage propagation (see Supplementary Note 3).
We note that at the phase transition point separating two Chern insulators (Δ = −1), the Chern number vanishes but the spin texture is still non-trivial. We consider a finite-size lattice with 10 × 10 “spins”. The admittance spectrum is plotted in Fig. 4a, showing that a series of localized states lie near jn = 0 Ω−1, and the corner states lying at nonzero admittance is caused by the open of band gap at phase-transition point (see Supplementary Note 4). The wave functions of localized states are displayed in the inset of Fig. 4a, from which we identify a corner state. The origin of the emerging corner states can be interpreted as the convergence of two Chern insulators with opposite chiralities, as shown by the green and black arrows in Fig. 4b. Due to the contrast of the chirality, the one-dimensional edge states can only accumulate at the sample corners, forming the zero-dimensional localized states, i.e., corner states. These corner states can be detected by measuring the distributions of impedance, as shown in Fig. 4c. If we consider circular samples with these parameters, the corner states will disappear for the circle with a large radius (smooth boundary) and the wave functions will concentrate on the convex right angle of the sample for a small circle (see Supplementary Note 5).
Three-dimensional Wilson fermions
As a nontrivial generalization, we consider a 3D hyper-cubic lattice with four sites in each cell, as shown in Fig. 5a. The hopping terms and on-site potentials are shown in Fig. 5b, c, respectively. One can write the circuit Laplacian as \({{{{{\mathcal{J}}}}}}(\omega )=i{{{{{\mathcal{H}}}}}}(\omega )\) with
where αx = σx ⊗ σx, αy = σx ⊗ σy, αz = σx ⊗ σz, and β = σz ⊗ σ0 are Dirac matrices (see Supplementary Note 6).
The topological properties of the 3D system can be characterized by the winding number w353. We find that the topological index w3 can only take five quantized values, i.e., 0, \(\pm \frac{1}{2}\), ± 1. For the topological insulator phase w3 = 1, one can observe surface states. At the border of the two topological insulators with opposite winding numbers, we find the hinge states induced by the overlap of the surface states. For the QASM phase w3 = 1/2, we observe the bounded surface state in a finite-size DW circuit along the \(\hat{x}\) direction (see Supplementary Note 6).
Conclusions
To summarize, we experimentally observed WFs in circuit systems. In addition, we mapped WFs with different masses or configurations to magnetic solitons with different skyrmion charges, which will enable us to study the properties of skyrmions, half skyrmions, and half-skyrmion pairs in electrical circuit platforms. We showed that the nontrivial spin-texture of WFs in momentum space is fully characterized by Chern numbers and winding number in 2D and 3D systems, respectively. The chiral edge current associated with the QASM state dictated by a fractional Chern number was directly detected. Although we have measured the band structures of WFs and their topological boundary states, how to reveal the quasiparticle nature of WFs is still an open question. Our work presents the first circuit realization of WFs, which sets a paradigm for other platforms, such as cold atoms, photonic, and phononic metamaterials, to further explore these fascinating phenomena.
Methods
Experimental details
We implement the circuit experiment on a printed circuit broad shown in Fig. 6a. The circuit is composed of 10 × 11 cells, with each cell containing two nodes. The details of the circuit components are shown in the inset of Fig. 6a. Figure 6b displays the experimental instruments: DC power supply (IT6332A) and impedance analyzer (E4990A), which are used to provide the power for the operational amplifiers and measure the impedance over the sample, respectively.
In Table 1, we list all elements used in our experiments, including the product companies, packages, mean values and their tolerances.
Data availability
Data available upon request from the authors.
Code availability
Code available upon request from the authors.
References
Kogut, J. B. The lattice gauge theory approach to quantum chromodynamics. Rev. Mod. Phys. 55, 775 (1983).
Gattringer, C. & Lang, C. B. Quantum Chromodynamics on the Lattice. Lecture Notes in Physics 788 (Springer, 2010).
DeTar, C. & Gottlieb, S. Lattice quantum chromodynamics comes of age. Phys. Today 57, 45 (2004).
Nielsen, H. B. & Ninomiya, M. A no-go theorem for regularizing chiral fermions. Phys. Lett. B 105, 219 (1981).
Chandrasekharan, A. & Wiese, U.-J. An introduction to chiral symmetry on the lattice. Prog. Part. Nucl. Phys. 53, 373 (2004).
Ginsparg, P. H. & Wilson, K. G. A remnant of chiral symmetry on the lattice. Phys. Rev. D 25, 2649 (1982).
Semenoff, G. W. Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 53, 2449 (1984).
Messias de Resende, B., Crasto de Lima, F., Miwa, R. H., Vernek, E. & Ferreira, G. J. Confinement and fermion doubling problem in Dirac-like Hamiltonians. Phys. Rev. B 96, 161113(R) (2017).
Yang, Z., Schnyder, A. P., Hu, J. & Chiu, C. Fermion doubling theorems in two-dimensional non-hermitian systems for fermi points and exceptional points. Phys. Rev. Lett. 126, 086401 (2021).
Haldane, F. D. M. Model for a quantum Hall effect without Landau Levels: Condensed-matter Realization Of The “Parity Anomaly”. Phys. Rev. Lett. 61, 2015 (1988).
Yu, Z., Wu, W., Zhao, Y. X. & Yang, S. A. Circumventing the no-go theorem: a single Weyl point without surface Fermi arcs. Phys. Rev. B 100, 041118(R) (2019).
Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 84, 109 (2009).
Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Lv, B. Q., Qian, T. & Ding, H. Experimental perspective on three-dimensional topological semimetals. Rev. Mod. Phys. 93, 025002 (2021).
Zhang, W. et al. Observation of flat-band localizations and topological edge states induced by effective strong interactions in electrical circuit networks. Commun. Phys. 4, 250 (2021).
Zhou, X., Zhang, W., Sun, H. & Zhang, X. Observation of flat-band localizations and topological edge states induced by effective strong interactions in electrical circuit networks. Phys. Rev. B 107, 035152 (2023).
Motavassal, A. & Jafari, S. A. Circuit realization of a tilted Dirac cone: platform for fabrication of curved spacetime geometry on a chip. Phys. Rev. B 104, L241108 (2021).
Lu, Y. et al. Probing the Berry curvature and Fermi arcs of a Weyl circuit. Phys. Rev. B 99, 020302(R) (2019).
Song, L., Yang, H., Cao, Y. & Yan, P. Square-root higher-order Weyl semimetals. Nat. Commun. 13, 5601 (2022).
Jia, N., Clai, O., Ariel, S., David, S. & Jonathan, S. Time- and site-resolved dynamics in a topological circuit. Phys. Rev. X 5, 021031 (2015).
Albert, V. V., Glazman, L. I. & Jiang, L. Topological properties of linear circuit lattices. Phys. Rev. Lett. 114, 173902 (2015).
Zhao, E. Topological circuits of inductors and capacitors. Ann. Phys. 399, 289 (2018).
Imhof, S. et al. Topolectrical-circuit realization of topological corner modes. Nat. Phys. 14, 925 (2018).
Lee, C. H. et al. Topolectrical circuits. Commun. Phys. 1, 39 (2018).
Hadad, Y., Soric, J. C., Khanikaev, A. B. & Alú, A. Self-induced topological protection in nonlinear circuit arrays. Nat. Electron. 1, 178 (2018).
Hofmann, T., Helbig, T., Lee, C. H., Greiter, M. & Thomale, R. Chiral voltage propagation and calibration in a topolectrical Chern circuit. Phys. Rev. Lett. 122, 247702 (2019).
Wang, Y., Price, H. M., Zhang, B. & Chong, Y. D. Circuit implementation of a four-dimensional topological insulator. Nat. Commun. 11, 2356 (2020).
Helbig, T. et al. Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747 (2020).
Chen, R., Chen, C.-Z., Gao, J.-H., Zhou, B. & Xu, D.-H. Higher-order topological insulators in quasicrystals. Phys. Rev. Lett. 124, 036803 (2020).
Olekhno, N. A. et al. Topological edge states of interacting photon pairs emulated in a topolectrical circuit. Nat. Commun. 11, 1436 (2020).
Song, L., Yang, H., Cao, Y. & Yan, P. Realization of the square-root higher-order topological insulator in electric circuits. Nano. Lett. 20, 7566 (2020).
Yang, H., Li, Z.-X., Liu, Y., Cao, Y. & Yan, P. Observation of symmetry-protected zero modes in topolectrical circuits. Phys. Rev. Res. 2, 022028(R) (2020).
Zhang, X. X. & Franz, M. Non-Hermitian exceptional Landau quantization in electric circuits. Phys. Rev. Lett. 124, 046401 (2020).
Yang, H., Song, L., Cao, Y., Wang, X. R. & Yan, P. Experimental observation of edge-dependent quantum pseudospin Hall effect. Phys. Rev. B 104, 235425 (2021).
Yang, H., Song, L., Cao, Y. & Yan, P. Experimental realization of two-dimensional weak topological insulators. Nano. Lett. 22, 3125 (2022).
Ventra, M. D., Pershin, Y. V. & Chien, C.-C. Custodial chiral symmetry in a su-schrieffer-heeger electrical circuit with memory. Phys. Rev. Lett. 128, 097701 (2022).
Yang, H., Song, L., Cao, Y. & Yan, P. Observation of type-III corner states induced by long-range interactions. Phys. Rev. B 106, 075427 (2022).
Wu, J. et al. Non-Abelian gauge fields in circuit systems. Nat. Electron. 5, 635 (2022).
Fu, B. et al. Quantum anomalous semimetals. npj Quantum Materials 7, 94 (2022).
Dirac, P. A. M. The quantum theory of the electron. Proc. R. Soc. A 117, 610 (1928).
Zheng, X., Chen, T. & Zhang, X. Topolectrical circuit realization of quadrupolar surface semimetals. Phys. Rev. B 106, 035308 (2022).
Asbóth, J. K. et al. A Short Course on Topological Insulator. Lecture Notes in Physics 919 (Springer, 2016).
Yu, H., Xiao, J. & Schultheiss, H. Magnetic texture based magnonics. Phys. Rep. 905, 1 (2021).
Chen, S. et al. Recent progress on topological structures in ferroic thin films and heterostructures. Adv. Mater. 33, 2000857 (2021).
Tokura, Y. & Kanazawa, N. Magnetic skyrmion materials. Chem. Rev. 121, 2857 (2021).
Fert, A., Reyren, N. & Cros, V. Magnetic skyrmions: advances in physics and potential applications. Nat. Rev. Mater. 2, 17031 (2017).
Zou, J.-Y. et al. Half-quantized Hall effect and power law decay of edge-current distribution. Phys. Rev. B 105, L201106 (2022).
Kim, D.-Y. et al. Quantitative accordance of Dzyaloshinskii-Moriya interaction between domain-wall and spin-wave dynamics. Phy. Rev. B 100, 224419 (2019).
Chen, J., Hu, J. & Yu, H. Chiral emission of exchange spin waves by magnetic skyrmions. ACS Nano 15, 4372 (2021).
Araki, H., Mizoguchi, T. & Hatsugai, Y. Phase diagram of a disordered higher-order topological insulator: a machine learning study. Phy. Rev. B 99, 085406 (2019).
Wakao, H., Yoshida, T., Araki, H., Mizoguchi, T. & Hatsugai, Y. Higher-order topological phases in a spring-mass model on a breathing kagome lattice. Phys. Rev. B 101, 094107 (2020).
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
Acknowledgements
This work was supported by the National Key Research Development Program under Contract No. 2022YFA1402802 and the National Natural Science Foundation of China (Grants No. 12074057, No. 11604041, and No. 11704060).
Author information
Authors and Affiliations
Contributions
P.Y. and H.Y. conceptualized the project. H.Y. developed the theory and wrote the manuscript supervised by P.Y. and Y.C. H.Y. and L.S. designed the circuits and performed the measurements. All authors discussed the results and revised the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Communications Physics thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yang, H., Song, L., Cao, Y. et al. Realization of Wilson fermions in topolectrical circuits. Commun Phys 6, 211 (2023). https://doi.org/10.1038/s42005-023-01326-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005-023-01326-6
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.