Classical bosonic systems may be tailored to support topological order and unidirectional edge transport exploiting gauge fields. Here, we theoretically explore how synthetic gauge fields may be used to induce higher-order topological phases and zero-energy boundary states. We demonstrate these principles in two types of three-dimensional topolectrical circuits with synthetic gauge fields threading through their reduced two-dimensional lattices, leading to a half-quantized quadrupole charge within a region of the momentum space. We theoretically show the emergence of nodal line rings and Weyl points in the bulk dispersion, whose projected surfaces and hinges support surface Fermi arcs and flat hinge Fermi arcs emanating from the nodal line ring and Weyl points, representing the spectral signature of higher-order topological semimetals. These analogs of higher-order semimetals realized in electric circuits using synthetic gauge fields may be extended to various photonic platforms and find applications in photonic crystals, nano-optics, and cold atom research.

Topological matter, including topological insulators (TIs)1–4 and topological semimetals,5–10 has widely been investigated in the past two decades because of the elegant theoretical framework that describes its response and the inherent robustness of its response. These concepts, born in the field of condensed matter physics and electronic materials, have been extended to classical wave systems,11–15 where synthetic gauge fields play the essential role of endowing neutral particles with the dynamics of charged particles subject to real gauge fields, inducing topological order in photonics,16–21 cold atoms,22–24 mechanics,25,26 and acoustics.27–29 Approaches to create synthetic gauge fields have included symmetry-based engineering of the geometrical features of metamaterials and active spatiotemporal modulation of the material properties; both approaches are more experimentally achievable in classical systems than in condensed matter.18–20 As a result, many topological phenomena proposed in condensed matter have been successfully demonstrated in classical platforms via artificial gauge fields, such as topological edge states with unidirectional propagation,17–19 the Aharonov–Bohm effect for photons,30 reconfigurable light guiding,31,32 and gauge-induced topological lasing.33 

In the field of topological matter, the concept of higher-order TIs has recently emerged and raised substantial interest, extending the family of TIs.34–39 They support the localization of d − 2 or lower-dimensional gapless topological boundary states, while both d-dimensional bulk states and d − 1 surface states have gapped spectra. Soon after the discovery of higher-order TIs, higher-order topological semimetals have also been proposed, which not only support symmetry-protected Weyl points/nodal lines and surface Fermi arcs bridging the Weyl points in the momentum space, such as conventional (first-order) semimetals, but also allow the existence of hinge Fermi arcs, which are gapless in the energy spectrum, evanescently decay in two dimensions, and are protected by higher-order topological features.40–44 

Here, we study theoretically and numerically the role of synthetic gauge fields in the generation of higher-order topological physics to enable 3D higher-order topological semimetals for classical waves. We consider a topolectrical circuit platform due to its built-in flexibility and reconfigurability, but similar principles can be extended to other classical wave and photonic platforms. Electrical circuits have already been widely explored to realize first-order TIs, higher-dimensional TIs, and higher-order TIs,45–55 including our recent work implementing topolectrical higher-order Chern insulators.56 

We explore two circuit lattice models in which gauge fields can be differently introduced. In our first topolectrical model, the synthetic gauge field threads through every plaquette of its reduced 2D lattice, which is realized by engineering the interlayer connections such that a Bloch phase arises in the third dimension. We discover that such a nontrivial synthetic gauge field causes a 1/2 quantization of the quadrupole charge, inducing higher-order topology in the lattice. Our calculations show that the 3D circuit model not only supports the nodal line ring closely touching the dispersionless surface Fermi arcs but also displays flat hinge Fermi arcs across the entire momentum space, which is a direct consequence of the higher-order topology. In our second design, the synthetic gauge field is induced by a negative coupling in one direction of the reduced 2D model. By stacking such 2D lattices to form a 3D metamaterial and introducing complex hopping coefficients between layers, we construct a higher-order Weyl semimetal, which supports bulk Weyl points emanating both surface Fermi arcs and hinge Fermi arcs. Finally, we implement first-principles simulations of these topolectrical higher-order Weyl semimetals, revealing hinge Fermi arcs in a finite momentum range. The excitation of these circuits demonstrates the coexistence of bulk, surface, and hinge states propagating in one dimension in the same frequency range, consistent with the features of higher-order Weyl semimetals.

Nodal line semimetals characterized by first-order topological phases have been extensively studied in various topological configurations and experimentally realized in photonic crystals and metamaterials.44,57–63 In contrast, higher-order nodal line semimetals, which possess higher-order topological features and support both surface Fermi arcs and hinge Fermi arcs in the projected momentum space, have not been explored in any classical platform to the best of our knowledge. Here, we explore the design of a higher-order topological nodal line semimetal supporting one-dimensional degenerate rings of bulk dispersion in the momentum space. A one-to-one correspondence between the tight-binding model and electrical circuits can be realized using standard circuit elements,56 and throughout the text, all tight-binding model Hamiltonians are transformed into the admittance form of their equivalent circuit model. The equivalent of 2D insulators is implemented with the square lattice in each layer of Fig. 1(a), and it supports corner states when the intercell hopping λ is larger than the intracell hopping γ, causing the bulk polarization to be quantized as 1/2, which supports first-order topological order protected by the time-reversal symmetry and lattice symmetries.38 We form a 3D metamaterial by stacking an array of such 2D insulators, and we replace half of the hoppings in the y direction by next-nearest neighbor hoppings in the z direction with a zigzag pattern, as shown in Fig. 1(a). The corresponding admittance matrix reads as

Yω=iωdII4×4+d1Γ1kz+d2Γ2kz+d3Γ3+d4Γ4,
(1)

where d1=Cλsinky, d2=CλcoskyCγ, d3=Cλsinkx, d4=CλcoskxCγ, and dIω=1ω2L0+C0+2(Cγ+Cλ), where capacitor C0 and inductor L0 are the elements of each resonator and four resonators are connected to form a loop via capacitors with capacitance Cγ, which represent the intracell hoppings described by blue lines in Fig. 1(a). Capacitors with capacitance Cλ connect unit cells within the same layer (intercell hoppings), which are denoted by the red lines; k = (kx, ky, kz) is the momentum vector in the Brillouin zone (BZ); I4 × 4 is a four-by-four identity matrix; and the gamma matrices are defined as Γi = −τ2τi, i = 1, 2, 3 and Γ4 = τ1τ0, in which τi s (i = 1, 2, 3) are the Pauli matrices and τ0 is a two-by-two identity matrix; Γ1kz=000i;00ieikz0;0ieikz00;i000; and Γ2kz=0001;00eikz0;0eikz00;1000.

FIG. 1.

Higher-order topological phase of a 2D square lattice induced by a synthetic gauge field. (a) Schematic illustration of the 3D structure constructed from stacked 2D square lattices. (b) Schematic illustration of the effective square lattice with the synthetic gauge field θ corresponding to the 3D metamaterial in (a), where θ is determined by the out of plane momentum kz. Red lines refer to the intercell hopping λ, and blue lines refer to the intracell hopping γ.

FIG. 1.

Higher-order topological phase of a 2D square lattice induced by a synthetic gauge field. (a) Schematic illustration of the 3D structure constructed from stacked 2D square lattices. (b) Schematic illustration of the effective square lattice with the synthetic gauge field θ corresponding to the 3D metamaterial in (a), where θ is determined by the out of plane momentum kz. Red lines refer to the intercell hopping λ, and blue lines refer to the intracell hopping γ.

Close modal

A synthetic gauge field θ is generated in every plaquette of the square lattice, as shown in Fig. 1(b), via the Bloch phase factor eikz, which arises from any Bloch wave propagating in the z direction. Since θ = kz, θ varies periodically between 0 and 2π. We note that in the special case, when θ = π, the model defined in Eq. (1) projected into the xy plane becomes the well-known quadrupole insulator, realized in a few recent experiments in which the gauge flux π was introduced through negative couplings.52,64–69 Therefore, the longitudinal momentum kz in our metamaterial acts as the bridge connecting a conventional insulator response with quadrupole insulators, as verified in the following paragraphs.

We calculate the band structure of the tetragonal lattice in the 3D Brillouin zone [Fig. 2(a)] by substituting detY=0 in Eq. (1). The calculated bulk bands shown in Fig. 2(b) reveal double degenerate nodal lines in the kz = 0(2π) plane. The nodal line ring follows the high symmetric paths Γ → M in the Brillouin zone and is protected by the mirror symmetry Mx of the lattice. To calculate the higher-order topological invariant of our topolectrical circuit, we write Kirchhoff equations in the Schrödinger form as

iddt|ψ=H|ψ,
(2)

where H has the matrix form

H=i0Î4Ĉ1L̂10,
(3)

where Ĉ=(C0+2(Cγ+Cλ))I4×4+d1Γ1kz+d2Γ2kz+d3Γ3+d4Γ4 and L̂1=L01Î4×4. The eigenstate ψt is defined as |ψt=[V(t);V̇(t)], and V(t) is the vector consisting of eigenvoltages at various nodes in the circuit. Once we obtain ψ from the eigenvalue equation, we can use it to examine the quadrupole topology of the system by calculating the Wannier bands and the nested Wannier bands based on the weighted inner product of the eigenstates such that the orthogonality is preserved in the Hilbert space ψn|ψmw=12Vn|ĈVm+12ĈV̇n|L̂ĈV̇m=δn,m.38,39 Interestingly, the quadrupole charge qxy characterizing the second-order topology is always quantized as 1/2 in the momentum space kz, except at kz = 0, where qxy is not well defined because of the nodal line degeneracy [see the inset of Fig. 2(b)]. Therefore, we regard our 3D metamaterial circuit as the simplest model to realize higher-order nodal line semimetals.

FIG. 2.

Topolectrical higher-order nodal line. (a) The BZ for a tetragonal lattice and the projected BZ for surface (001). In the projected BZ, the red line is the bulk nodal line and blue lines depict surface Fermi arcs. (b) Band structure of the tetragonal lattice with Bloch periodic boundary conditions applied in three directions; the inset shows the distribution of the quadrupole charge qxy as a function of momentum kz. (c) Band structure projected in the (001) surface and local density of states of the corresponding surface Fermi arcs. The strip geometry consists of 20 unit cells, terminated by open boundaries in the z direction and periodic in x and y directions. (d) Band structure projected along hinge [001] overlapped with local density of states of the hinge Fermi arcs. The square geometry consists of 5 × 5 unit cells in the xy plane; it has open boundaries in both x and y directions and is periodic in the z direction. The field distribution of hinge states is displayed in the right panel. The circuit parameters are C0 = 200 μF, L0 = 1/2 μH, Cλ = 4 μF, and Cγ = 2 μF.

FIG. 2.

Topolectrical higher-order nodal line. (a) The BZ for a tetragonal lattice and the projected BZ for surface (001). In the projected BZ, the red line is the bulk nodal line and blue lines depict surface Fermi arcs. (b) Band structure of the tetragonal lattice with Bloch periodic boundary conditions applied in three directions; the inset shows the distribution of the quadrupole charge qxy as a function of momentum kz. (c) Band structure projected in the (001) surface and local density of states of the corresponding surface Fermi arcs. The strip geometry consists of 20 unit cells, terminated by open boundaries in the z direction and periodic in x and y directions. (d) Band structure projected along hinge [001] overlapped with local density of states of the hinge Fermi arcs. The square geometry consists of 5 × 5 unit cells in the xy plane; it has open boundaries in both x and y directions and is periodic in the z direction. The field distribution of hinge states is displayed in the right panel. The circuit parameters are C0 = 200 μF, L0 = 1/2 μH, Cλ = 4 μF, and Cγ = 2 μF.

Close modal

According to the bulk-edge correspondence, we expect both surface and hinge Fermi arcs to coexist on the boundaries of a finite sample of our circuit. We stress that the zigzag coupling along the z direction is not the only way to induce higher-order topology; the chiral coupling has been shown in recent works to also demonstrate higher-order Weyl semimetal responses.40,42 To verify this prediction, we study the boundary spectra of a finite strip, according to the schematic in Fig. 1, with periodic boundaries in x and y directions and a finite width along the z direction. The boundaries in the z direction are suitably connected to ground, and the strip contains 20 unit cells. The surface Fermi arcs are clearly revealed by the surface local density of states in the projected Brillouin zone; their flat dispersion follows the high symmetric lines Γ̄X̄M̄ and connects to the bulk nodal lines projected in the same Brillouin zone plane [Fig. 2(a)]. The signature of higher-order topology in such 3D nodal line semimetals is the existence of long hinge Fermi arcs.41,43 In our circuit model, the hinge Fermi arcs are disclosed by the hinge local density of states and show flat bands extending in the whole momentum space kz, as seen in Fig. 2(d) for a finite sample consisting of 5 × 5 unit cells in the xy plane. These hinge Fermi arcs are exponentially localized at the corners of the square lattice in the reduced xy plane. The behavior of these hinge states is consistent with the expected response stemming from qxy and the bulk-edge correspondence.

An alternative simulation of the higher-order topological semimetal in a topolectrical circuit is based on the theoretical model recently introduced in Ref. 44. We apply this model to investigate the effects of synthetic gauge fields on its higher-order topological features. We start from a quadrupole insulator realized with the 2D square lattice in Fig. 3(a) and arrange an array of them in the z direction. The artificial gauge field π is created in each plaquette through negative couplings between sites 2 and 3, which provides a different form of synthetic gauge field compared to the previous example. By stacking these 2D quadrupole insulators into a 3D metamaterial, with interlayer couplings according to the schematic illustrated in Fig. 3(b), we achieve the circuit analog of a higher-order Weyl semimetal supporting type-I Weyl points in the bulk dispersion. We notice that in an earlier work, a first-order Weyl semimetal was proposed based on an ultracold gas;70 however, the method of achieving negative coupling and the topological phase in that work are distinct from ours. In addition, in our model, two types of interlayer couplings are introduced: the first type is the hopping between next-nearest neighbors in the z direction, which is denoted by the yellow links in Fig. 3(b), and the second type is the hopping between third-nearest neighbors in the z direction, which is represented by the green links in Fig. 3(b). We refer to the shaded box region in Fig. 3(a) as the quadrupole cell, and the interlayer hopping occurs only between quadrupole cells in different layers and shares the same projected coordinates in the xy plane. Thus, the corresponding admittance matrix is given by

Y4×4ω=iωdII4×4+n=15dnΓn,
(4)

where d1=Cλsinky, d2=CλcoskyCγ2Czcoskz, d3=Cλsinkx,d4=CλcoskxCγ2Czcoskz,d5=2Cdsin(kz),dIω=2ω2L0+C0+2(Cγ+Cλ+2Cz+Cd), and Γ5 = iΓ1Γ3, and other parameters are the same as in Fig. 2.

FIG. 3.

Circuit model for the higher-order Weyl semimetal. (a) Schematic of the model in the xy plane, which represents a quadrupole insulator. (b) Schematic illustrating the 3D structure formed by an array of quadrupole insulators. Two types of links connect quadrupole cells in different layers stacked in the z direction: the yellow links represent next-nearest neighbor interlayer hoppings λz, and the green links represent third-nearest neighbor interlayer hoppings λd. Dotted lines indicate negative couplings.

FIG. 3.

Circuit model for the higher-order Weyl semimetal. (a) Schematic of the model in the xy plane, which represents a quadrupole insulator. (b) Schematic illustrating the 3D structure formed by an array of quadrupole insulators. Two types of links connect quadrupole cells in different layers stacked in the z direction: the yellow links represent next-nearest neighbor interlayer hoppings λz, and the green links represent third-nearest neighbor interlayer hoppings λd. Dotted lines indicate negative couplings.

Close modal

The band structure of the tetragonal lattice is obtained from Eq. (4), and it supports Weyl points carrying different topological charges located somewhere between the high symmetric path M → A, as shown in Figs. 4(a) and 4(b). Their projected coordinate in the momentum kz is referred to as k1 and k2, where k1 < k2. Based on Eq. (2), in which the admittance form is now replaced by Eq. (4), we evaluate the quadrupole charge qxy as a function of kz, as shown in Fig. 4(c). When kz<k2, qxy ≠ 1/2, indicating that in this scenario, the system is trivial and no flat hinge Fermi arcs exist in this range. When kz>k2, however, qxy = 1/2 and dispersionless topological hinge Fermi arcs appear. Therefore, we conclude that the Weyl points at k1 and k2 carry different topological charges. Specifically, Weyl points at k1 possess first-order topological charges, while the Weyl points at k2 carry second-order topological charges, and the hinge states appear only between Weyl points with second-order topological charges. If surface states also exist, they must be gapped between the higher-order topological Weyl points.

FIG. 4.

Topolectrical higher-order Weyl semimetal. (a) The BZ for a tetragonal lattice and the projected BZ for surface (010). In the projected BZ, the red and blue dots represent higher-order Weyl points carrying positive and negative monopole charges, respectively. The green line is the surface Fermi arc, and orange lines depict hinge Fermi arcs along kz. (b) Band structure of our higher-order Weyl semimetal model with periodic boundaries in three directions. (c) Quadrupole charge qxy as a function of kz. (d) Band structure projected in surface (010) and the local density of states of the corresponding surface Fermi arcs. The strip geometry consists of 20 unit cells, terminated by open boundaries in the y direction and periodic in x and z directions. (e) Band structure projected in hinge [001] overlapped with the local density of states of hinge Fermi arcs. The square geometry consists of 5 × 5 unit cells in the xy plane, has open boundaries in both x and y directions, and is periodic in the z direction. Circuit parameters are C0 = 200 μF, L0 = 1 μH, Cλ = 2 μF, Cγ = 2 μF, Cz = 1 μF, and Cd = 1 μF.

FIG. 4.

Topolectrical higher-order Weyl semimetal. (a) The BZ for a tetragonal lattice and the projected BZ for surface (010). In the projected BZ, the red and blue dots represent higher-order Weyl points carrying positive and negative monopole charges, respectively. The green line is the surface Fermi arc, and orange lines depict hinge Fermi arcs along kz. (b) Band structure of our higher-order Weyl semimetal model with periodic boundaries in three directions. (c) Quadrupole charge qxy as a function of kz. (d) Band structure projected in surface (010) and the local density of states of the corresponding surface Fermi arcs. The strip geometry consists of 20 unit cells, terminated by open boundaries in the y direction and periodic in x and z directions. (e) Band structure projected in hinge [001] overlapped with the local density of states of hinge Fermi arcs. The square geometry consists of 5 × 5 unit cells in the xy plane, has open boundaries in both x and y directions, and is periodic in the z direction. Circuit parameters are C0 = 200 μF, L0 = 1 μH, Cλ = 2 μF, Cγ = 2 μF, Cz = 1 μF, and Cd = 1 μF.

Close modal

To verify the above prediction, we again study a 20-unit cell strip geometry with periodic boundary conditions in x and z directions, which is grounded in the y direction. The band structure, as well as the surface local density of states, is obtained and plotted in Fig. 4(d). Surface states are clearly revealed in the local density of states and connect higher-order Weyl points located at k1 and k2, and their energies are gapped between Weyl points with second-order topological charges. Next, we construct a 5 × 5 square lattice of finite size in the xy plane but periodic boundaries in the z direction. The resulting band structure and the hinge local density of states are overlapped in Fig. 4(e), revealing topological hinge Fermi arcs only for kz>k2 that connect the projected higher-order Weyl points carrying the same topological charge.44 Therefore, higher-order Weyl points, surface Fermi arcs, and hinge Fermi arcs are interconnected in the momentum space as shown in the projected plane of Fig. 4(a), and the behavior of hinge Fermi arcs exactly matches the prediction from higher-order topology correspondence.

To test whether the higher-order topolectrical semimetal can be experimentally implemented and whether it allows the coexistence of bulk states, surface Fermi arcs, and hinge Fermi arcs at the same frequency, we design the higher-order Weyl semimetal model in a circuit layout, as a finite topolectrical circuit array composed of 5 × 5 × 5 unit cells, and carry out full-wave simulations based on commercially available Advanced Design System (ADS). The first step is to construct the quadrupole cell with the flux π threading into it. We employ a capacitor-inductor-capacitor (CLC) resonator composed of two capacitors with capacitance C0 and one inductor with inductance L0 such that it supports both antisymmetric and symmetric eigenmodes due to mirror symmetry, and we only consider the antisymmetric case in the following simulations. Four CLC resonators are connected into a plaquette via capacitors with capacitance Cγ; the negative coupling is realized by connecting v2(v3) to v7(v6), while the other three couplings are positive since the connections occur between the nodes with positive (negative) voltages [Fig. 5(a)]. To obtain the 2D circuit version of our quadrupole insulators, the quadrupole cells are connected via capacitors with capacitance Cλ in the xy plane so that every plaquette sustains a synthetic flux π, as shown in Fig. 5(b). Next, the 2D circuit quadrupole insulators are connected by capacitors with capacitances Cz and Cd. The first type capacitors (shown in orange) realize the next-nearest neighbor interlayer hoppings, whereas the second type capacitors (shown in green) represent the third-nearest neighbor interlayer hoppings [Fig. 5(c)]. Based on the circuit lattice constructed in Fig. 5, we introduce Bloch boundary conditions in all directions, and the Kirchhoff equations of the circuit arrays read as

I=Y8×8V,
(5)
Y8×8=iωY1,Yint;Yint,Y2,
(6)

where the admittances have the following matrix form:

Y1=Y2=C0+2Cγ+Cλ+2Cz+Cd1ω2L0I4×4+0CdeikzCγCλeikx2CzcoskzCγCλeiky2CzcoskzCdeikz00CγCλeikx2CzcoskzCγCλeikx2Czcoskz00CdeikzCγCλeiky2CzcoskzCγCλeikx2CzcoskzCdeikz0,Yint=1ω2L0I4×4+0Cdeikz00Cdeikz0CγCλeiky2Czcoskz00CγCλeiky2Czcoskz0Cdeikz00Cdeikz0,

where I=i1,i2,,i8T,V=v1,v2,,v8T. Since we consider antisymmetric eigenmodes supported by each CLC resonator, the voltages at various nodes satisfy the relations vi = −vi+4, i = 1, 2, 3, 4. Due to the symmetry constraint, Eq. (5) is split into two decoupled equations Y8×8=iωY4×4,0;0,Y4×4, where Y4 × 4 is the admittance in Eq. (4). As a consequence, the full equations of the microscopic circuit array are simplified and mapped to the ones in Sec. III.

FIG. 5.

Schematic of the topolectrical circuit realizing a higher-order Weyl semimetal. (a) Schematic of the circuit design for the quadrupole cell (QC). Resistor R0 is replaced with an ideal wire in the lossless case. (b) Schematic of the circuit design for connecting QCs in the xy plane. (c) Schematic of the circuit design for connecting quadrupole cells in the z direction.

FIG. 5.

Schematic of the topolectrical circuit realizing a higher-order Weyl semimetal. (a) Schematic of the circuit design for the quadrupole cell (QC). Resistor R0 is replaced with an ideal wire in the lossless case. (b) Schematic of the circuit design for connecting QCs in the xy plane. (c) Schematic of the circuit design for connecting quadrupole cells in the z direction.

Close modal

In our first configuration, periodic boundaries are applied to the circuit lattice in the z direction by connecting the first unit cell to the fifth one, and x and y boundaries are grounded to emulate open boundaries. The simulation is carried out in the frequency domain, and we select a current source placed at the midpoint of the hinge along the z direction to drive the circuit. We calculated the voltages at various nodes and present their spectra as a function of frequency without and with realistic losses considered in the circuit lattice [gray curves and red curves in Fig. 6(a)]. Interestingly, although the bulk states and surface states are significantly affected by the presence of losses, both scenarios show prominent peaks near 15.39 kHz, which corresponds to the eigenfrequency of the hinge Fermi arcs. The resonance peaks at the hinge states are less suppressed compared to the ones of bulk and surface state resonances when losses are present, indicating the more local nature of topological hinge states. Next, the spectrum is calculated based on the spatial Fourier transform of the extracted data, including amplitude and phase information collected at the nodes along the z hinge, shown in Fig. 6(b), showcasing flat bands (in the dashed box) for a specific range of momentum kz, corresponding to the predicted dispersion of the hinge states and consistent with our theoretical calculations in Fig. 4(d). Note that we adopt a finite size lattice in the simulation, with only five unit cells and exhibiting periodicity in the z direction, and the band structure is calculated based on the data from this reduced lattice, leading to a coarse mesh in momentum kz. As a consequence, the dispersive bands of both bulk and surface states look flat in the spectrum. These effects can be alleviated by increasing the number of unit cells in the z direction.

FIG. 6.

First-principles simulations of the higher-order Weyl semimetal consisting of 5 × 5 × 5 unit cells, with periodic boundaries in the z direction and open boundaries in other dimensions. (a) Voltage spectra for all nodes as a function of frequency [gray curves represent the voltage responses without the loss considered in the circuit lattice, and red curves represent the voltage responses with loss]. (b) Fourier spectrum for the nodes at the z hinge when the loss is included. The current source is placed in the middle of the z hinge. The effect of loss is simulated by introducing a single resistor R0 in each QC of the lattice, where R0 = 10 Ω.

FIG. 6.

First-principles simulations of the higher-order Weyl semimetal consisting of 5 × 5 × 5 unit cells, with periodic boundaries in the z direction and open boundaries in other dimensions. (a) Voltage spectra for all nodes as a function of frequency [gray curves represent the voltage responses without the loss considered in the circuit lattice, and red curves represent the voltage responses with loss]. (b) Fourier spectrum for the nodes at the z hinge when the loss is included. The current source is placed in the middle of the z hinge. The effect of loss is simulated by introducing a single resistor R0 in each QC of the lattice, where R0 = 10 Ω.

Close modal

At the Weyl point frequency, surface and hinge states have zero group velocity between the projected positions of Weyl points along the kz direction, while they possess small and nonzero group velocities near the Weyl points due to their dispersive bands, as shown in Figs. 4(d) and 4(e). We can excite them separately by choosing different input ports because of the decay nature of surface and hinge states. To verify the above speculation, we carried out a second simulation in which the circuit lattice is grounded in all directions, forming a finite sample of the 3D topolectrical metamaterial, which we drive at fs = 15.39 kHz. In Fig. 7(a), the source is placed at the center of the 3D lattice, dominantly exciting bulk modes due to the existence of bulk Weyl points at the chosen frequency. Next, we move the source to the center of the xz surface of the tetragonal lattice, as shown in Fig. 7(b). The excited states are dominated by surface states representing the surface Fermi arcs in the xz surface. Finally, in Fig. 7(c), the source is shifted to the middle of the z hinge, clearly exciting the hinge states localized exponentially. These simulation results evidently demonstrate the coexistence of bulk states and surface and hinge Fermi arcs consistent with our theoretical models in Fig. 4. To demonstrate the nonzero group velocity of the respective modes, we provide in the videos of the supplementary material time-resolved simulations of the bulk, surface, and hinge states propagating in the 3D space. Note that the simulation of the higher-order nodal line circuit can be performed in a similar fashion and a simpler configuration, for example, just based on a CL resonator, can be used as the site in the lattice since the negative coupling is not required in the higher-order nodal line model.

FIG. 7.

First-principles simulation of a higher-order Weyl semimetal consisting of 5 × 5 × 5 unit cells, with open boundaries in all dimensions. (a)–(c) Field distribution when the current source is placed at (a) the center of the tetragonal lattice, (b) the center of the surface (101), and (c) the midpoint of the z hinge, as indicated by the yellow star, with frequency fs ≈ 15.39 kHz.

FIG. 7.

First-principles simulation of a higher-order Weyl semimetal consisting of 5 × 5 × 5 unit cells, with open boundaries in all dimensions. (a)–(c) Field distribution when the current source is placed at (a) the center of the tetragonal lattice, (b) the center of the surface (101), and (c) the midpoint of the z hinge, as indicated by the yellow star, with frequency fs ≈ 15.39 kHz.

Close modal

In this paper, we have introduced synthetic gauge fields in 3D topolectrical circuits to emulate the physics of higher-order semimetals, demonstrating higher-order topological phases induced by synthetic gauge fields. We studied two different circuit models, supporting the nodal line ring and Weyl points in their bulk dispersion and boundaries supporting not only surface states but also hinge states connecting the projected nodal lines/Weyl points, which represent the unique features of higher-order topological semimetals. These results have been validated by first-principles simulations of realistic circuit lattices formed by arrangements of capacitors and inductors, which can be easily implemented experimentally.

In a broader context, our introduced platforms offer opportunities beyond electrical circuits to implement higher-order topological physics for photonics and acoustics. Similar implementations can be realized using metamaterials made of coupled resonators,71 evanescently coupled optical waveguides,72 and optical lattices73 with the ability of tuning the phase in the hoppings and emulating the tight-binding dynamics. The synthetic gauge field in our introduced first model is more experimentally feasible in photonics since it does not require negative couplings. We envision that this approach for synthetic gauge fields to emulate higher-order nodal lines can be implemented over on-chip synthetic optics and photonic crystal platforms. The near-zero group velocity of topological hinge states arising in these higher-order topological semimetals opens interesting opportunities for several slow-light applications in nanophotonics, including topologically robust delay lines and enhanced nonlinearities.

To support our claim that surface and hinge states at the Weyl point frequencies propagate with nonzero group velocities, we carried out time-domain simulations for the excitation of a 5 × 5 finite circuit lattice (the same structure as the one studied in this paper in the frequency domain) and show the temporal dynamics of bulk, surface, and hinge states in the supplementary material videos.

This work was supported by the Office of Naval Research under Grant No. N00014-19-1-2011 and the National Science Foundation EFRI program.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material