Circuit Implementation of a Four-Dimensional Topological Insulator

Topological insulators host insulating bulk spectra with protected boundary states determined by topological invariants of the bulk bandstructure. Their properties depend on dimensionality and symmetry class. While two-dimensional Quantum Hall (2DQH) systems exhibit unidirectional edge modes and three-dimensional topological insulators feature Dirac-like surface states, theoretical classifications predict higher-dimensional phases, including four-dimensional Quantum Hall (4DQH) phases characterized by a second Chern number. Such phases cannot exist in electronic materials constrained to three spatial dimensions. Engineered platforms (cold atoms, photonics, acoustics, mechanics, and electric circuits) can emulate lattice models that enable access to higher-dimensional physics. Prior experimental efforts realized topological pumps that map 4D physics onto lower-dimensional systems, but these do not realize genuine high-dimensional lattices with protected surface states. Here the authors implement a genuine 4D lattice using electric circuits to realize a Class AI (time-reversal-symmetric, spinless, no special spatial symmetry) topological insulator. Class AI is trivial in 1–3D, so 4D is the minimal dimension for a nontrivial insulator. The chosen 4D lattice model exhibits a nonzero second Chern number with vanishing first Chern numbers, enabling observation of surface states intrinsically tied to 4D topology.

The paper situates the work within the tenfold classification of topological phases and the dimensional hierarchy, emphasizing that Class AI systems are trivial in 1–3D but allow nontrivial topology in 4D via the second Chern number. It contrasts the 4DQH effect with the 2DQH effect, which relies on a nonzero first Chern number and broken time-reversal symmetry. Engineered systems (cold atoms, photonics, acoustics/mechanics, circuits) have enabled realizations of diverse topological models, including topolectrical circuits and higher-order topological phases. Synthetic-dimension approaches use internal degrees of freedom to emulate extra spatial dimensions, with experiments mainly limited to 1D and 2D synthetic lattices. Topological pumping maps higher-dimensional invariants onto lower-dimensional parameter scans and has probed Class A 4DQH physics in 2D systems, but pumps access quasi-static trajectories rather than realizing genuine higher-dimensional lattices and often involve nonzero first Chern numbers. The present work instead employs explicit circuit connectivity to construct a genuine 4D lattice implementing a Class AI 4DQH phase with nonzero second Chern number and strictly vanishing first Chern numbers, following a theoretical model previously proposed.

Model and design: The target is a 4D tight-binding lattice with coordinates (x, y, z, w) and four sublattices (A–D), nearest-neighbour hoppings J, and additional long-range hoppings J' and ±J'' within the x–z plane to control pairs of Dirac points. On-site mass terms +m (A,B) and −m (C,D) tune topological transitions at m = ±J^2/J'' and m = ±J''^2/J'. For J' > J'', the lower bands have second Chern number C2 = −2 for |m| < 3|J'|, while all first Chern numbers vanish due to time-reversal symmetry. Parameters chosen: J = 1, J' = J'' = 2, giving bulk topological transitions at m = ±6. Finite 4D lattice: 6 sites along x and z, 2 along y and w, with periodic boundary conditions in y and w (sampling ky = kw = 0). Total sites: 144, with 16 bulk sites (more than two sites away from any surface) and 128 surface sites.

Circuit mapping: Each lattice site maps to a circuit node. Hoppings J_ij are implemented as circuit elements with complex conductance D_ij(f) = iα H_ij(f), where α > 0 is a constant; capacitors realize positive hoppings and inductors realize negative hoppings. Mass terms map to grounding elements with conductance −D (inductors for negative mass and capacitors for positive mass), paralleled by additional grounding elements +D to satisfy the mapping. Kirchhoff’s law yields I_i = Σ_j L_ij V_j, where L is the circuit Laplacian. The circuit is tuned so that at a working frequency f0 the effective Hamiltonian H(f0) reproduces the target lattice Hamiltonian at energy E via I_i = −iα Σ_j (H_ij(f0) − E δ_ij) V_j.

Impedance–LDOS relation: The impedance between node r and ground, Z_r(f) = V_r/I_r, equals the rth diagonal entry of L^{-1}. It is shown that Re[Z_r(f0)] is proportional (up to a scalar) to the local density of states (LDOS) of the target lattice at energy E. With finite resistances, eigenenergies acquire small imaginary parts, smoothing impedance features.

Implementation details: The lattice is built from stacked printed circuit boards (PCBs) with vertical interconnects; each PCB carries a 6×6 array of nodes corresponding to the x–z plane. Component mapping at f0: choose α = 2π f0 C0 with C0 = 1 nF such that nearest-neighbour positive hopping J = 1 maps to capacitance C0 = 1 nF; long-range positive hopping J' maps to C = 2 nF; nearest-neighbour negative hopping −J maps to inductance L = 2 mH; long-range negative hopping −J'' maps to L' = 1 mH. The mass term uses capacitors (positive mass) or inductors (negative mass) to ground with magnitude set by m. The working frequency is set by f0 ≈ 1/(2π√(LC)) ≈ 113 kHz. Each site includes additional grounding components to satisfy the required on-site terms. Component part numbers: 1 nF capacitors (Murata GCM155R71H102KA37D) and 1 mH inductors (Taiyo Yuden LB2518T102K), combined in series/parallel as needed; manufacturer-specified series resistances are included in simulations (≈10 Ω per capacitor, ≈24 Ω per inductor).

Measurement protocol: For each fabricated circuit, an AC voltage source of 1 V at frequency f is applied to a chosen node r relative to common ground. The steady-state voltage V_r and current I_r are measured to obtain Z_r(f) = V_r/I_r. Measurements focus on f = f0 to probe the LDOS at target energy E (E ∈ {0,1}) and also scan f to explore frequency dependence. Samples were fabricated for m ∈ {0,1,...,8} and E ∈ {0,1}. Mean surface and bulk LDOS proxies are computed by averaging Re[Z_r] over designated surface or bulk node sets; a surface-to-bulk LDOS ratio is then formed.

Theory and simulations: Bulk and finite-size spectra are computed for the target tight-binding model, identifying gap closures and surface localization via a surface concentration metric ln(|ψ|_surface/|ψ|_bulk). Finite-size band edges are analyzed for lattices with increased x and z extents (6, 8, 10, 14, 20, 50) and periodic y, w to assess finite-size shifts. SPICE simulations (ngspice) model the full circuit including series resistances and incorporate 10% uniformly distributed disorder in component values to test robustness. Simulations mirror the experimental measurement procedure to generate Re[Z_r(f)].

• The infinite 4D lattice model exhibits a topological bandgap centered at E = 0 with nontrivial second Chern number for |m| < 6; surface states reside in this gap, while first Chern numbers are strictly zero by time-reversal symmetry.
• Finite 144-site lattice (6×6 in x–z, 2×2 in y–w with periodic y, w): finite-size effects split bands into sub-bands and shift the bulk gap closing from m = 6 (infinite) to approximately m ≈ 4. Surface states are most prominent at small |E| and |m|.
• Experimental LDOS maps (via Re[Z_r] at f = f0) show high surface and low bulk LDOS in the topological regime (e.g., m = 0 at E = 0 and E = 1), consistent with surface states on the 3D boundary. Near the gap-closing (E = 0, m ≈ 4) surface and bulk LDOS are comparable. In the trivial regime (E = 0, m ≥ 8) LDOS is low throughout, indicating a trivial bandgap without surface states.
• Quantitative surface enhancement: the mean surface-to-bulk LDOS ratio at E = 0 and f = f0 decreases sharply with m, from around 4.5 in the 4DQH regime to approximately 1 in the conventional insulator regime.
• Frequency dependence: for small m (e.g., m = 0), the surface LDOS remains elevated over a frequency range around f0 that coincides with a bulk gap, indicating topological surface states spanning the corresponding spectral window. As m increases, the gap closes and no distinct surface/bulk features remain.
• Robustness and validation: Circuit simulations with manufacturer-specified series resistances and with up to 10% component tolerances reproduce experimental trends and confirm robustness of the surface states. Topological features persist across frequency variations so long as the gap remains open, consistent with Class AI invariance.
• Scale and implementation: A genuine 4D lattice structure with 144 sites was realized physically via circuit connectivity, not by mapping to lower-dimensional pumps or synthetic dimensions.

The experiments demonstrate a genuine 4D topological lattice implemented via electric circuits, directly addressing the challenge of realizing higher-dimensional topological insulators that are inaccessible in electronic materials. Observed 3D surface states, their frequency extent matching a bulk gap, and their robustness to disorder confirm a 4D Quantum Hall phase protected by a nonzero second Chern number in a time-reversal-symmetric, spinless (Class AI) system. Because first Chern numbers must vanish under time-reversal symmetry, the surface phenomena cannot be ascribed to lower-dimensional topology; instead they are intrinsically tied to 4D band topology. The frequency-parametric response of the effective Hamiltonian remains within Class AI, preserving the topological character while the gap is open, explaining persistence of surface-dominated LDOS near f0. Agreement between measurements and SPICE simulations, including with realistic resistances and disorder, supports the interpretation and indicates practicality of circuits as a platform for exploring exotic higher-dimensional topological phases. Finite-size effects shift the gap-closing point relative to the infinite model but do not obscure the topological transition or surface-state signatures.

This work reports the first experimental realization of a Class AI 4D topological insulator using electric circuits that embody a genuine 4D lattice connectivity. By mapping a 4D tight-binding model with nonzero second Chern number and vanishing first Chern numbers onto an LC network, the authors observe enhanced LDOS on the 3D surface within the topological bandgap, a clear gap closing with varying mass parameter m, and robustness against component disorder, all in agreement with simulations. The demonstration establishes electric circuits as a versatile platform for high-dimensional topological matter. Future research directions include directly probing the predicted detailed structure of the 3D surface states (e.g., two isolated Weyl points of the same chirality), scaling to larger lattices to minimize finite-size effects, exploring other symmetry classes and higher Chern numbers, and implementing dynamical probes and transport analogs in topolectrical 4D systems.

• Finite-size effects: The 144-site lattice (6 sites along x and z; 2 along y and w) leads to sub-band formation and shifts the apparent bulk gap closing from m = 6 (infinite system) to approximately m ≈ 4.
• Boundary conditions and sampling: Periodic connections in y and w effectively sample ky = kw = 0, limiting exploration of the full 4D Brillouin zone.
• Measurement scope: LDOS proxies are derived from impedance at selected frequencies and planes; detailed momentum-resolved surface-state structure was not directly measured.
• Component tolerances and dissipation: Up to 10% variations in capacitances/inductances and finite series resistances introduce disorder and damping; while robustness is demonstrated, these factors broaden features and complicate precise spectral extraction.
• Simulations omit some resistive elements: PCB interconnect resistances and other parasitics were not fully characterized in simulations, potentially contributing to quantitative discrepancies.
• Topological invariant not directly measured: The second Chern number is inferred via surface-state phenomenology rather than being measured through a quantized response.