Circuit implementation of a four-dimensional topological insulator

The classification of topological insulators predicts the existence of high-dimensional topological phases that cannot occur in real materials, as these are limited to three or fewer spatial dimensions. We use electric circuits to experimentally implement a four-dimensional (4D) topological lattice. The lattice dimensionality is established by circuit connections, and not by mapping to a lower-dimensional system. On the lattice’s three-dimensional surface, we observe topological surface states that are associated with a nonzero second Chern number but vanishing first Chern numbers. The 4D lattice belongs to symmetry class AI, which refers to time-reversal-invariant and spinless systems with no special spatial symmetry. Class AI is topologically trivial in one to three spatial dimensions, so 4D is the lowest possible dimension for achieving a topological insulator in this class. This work paves the way to the use of electric circuits for exploring high-dimensional topological models.

The gap closes at m = 6 with the Dirac point lying at k y = k w = 0.
When the lattice is truncated, surface states appear along the surface in the topologically nontrivial phase (m ≤ 6), as shown in Fig. 1d-e. In this case, the truncation occurs along the x direction, and the surface state cones are centered at k y = k w = 0, k z = ±2π/3.

SUPPLEMENTARY NOTE 2: CIRCUIT DESIGN DETAILS
Here, we provide additional details on the mapping between the electric circuits and tightbinding Hamiltonians. Let I i be the external current injected into node i, V j the voltage (relative to ground) on node j, and D ij the conductance between nodes i and j for i = j.
Moreover, let the conductance between node i and ground be By Kirchhoff's laws, Note that in Eq. (3), the sum can be taken either over all j, or equivalently over j = i. We now adjust D ii so that, at a reference working frequency f 0 , for each node i, with some constant α and target energy E. At f = f 0 , Eq. (5) then becomes We require H ij (f 0 ) to match the target tight-binding Hamiltonian, which has parameters J = 1, J = −J = 2. For real α, positive (negative) real values of H ij correspond to capacitances (inductances). As described in the main text, by choosing α and f 0 we can assign the following circuit elements to the lattice model's hopping terms: For each node, we determine the grounding conductance required to satisfy Eq. (6).
Suppose node i is connected to other nodes by p i type-C 0 capacitors, q i type-L 0 inductors, p i type-C capacitors, and q i type-L inductors (these connections depend on which sublattice the node lies on, and whether it lies in the bulk or on the surface). Then Taking f = f 0 and plugging into Eq. (6) gives The on-site mass term is H ii (f 0 ) = ±m, depending on whether the node is on the A,B or C,D sublattices. Hence, the grounding conductance must satisfy To achieve this in the experiment, we connect each node i to ground with 6 − p i type-C 0 capacitors, 3 − q i type-L 0 inductors, 4 − p i type-C capacitors, and 4 − q i type-L inductors.
Additionally, (i) we connect each node to ground by an extra inductor L g , and (ii) if node i belongs to sublattice C or D, we connect it to ground by an extra capacitor C m = 2mC 0 .
As a result, the grounding conductance of node i at an arbitrary frequency f is where ∓ refers to sublattice A,B or C,D respectively. At f = f 0 , this satisfies Eq. (13) if we Hence, Returning to Eq. (5), define the quantity in the parentheses-which gives rise to the E term in Eq. (7)-as Eq. (7) then generalises to Now observe that in Eq. (18), the first term D We are therefore free to give D ii (f ) any frequency dependence, consistent with its value at With this choice, E i (f ) becomes i-independent, and Eq. (19) simplifies to This can be interpreted as a family of response equations with an f -dependent Hamiltonian and fixed energy E. For general f , the Hamiltonian's hopping terms are determined by the circuit elements summarised in Eq. (9), and its on-site mass term is determined by Eq. (20); for f = f 0 , it reduces to the target Hamiltonian.
Suppose E is in a topological gap of the target Hamiltonian, so that topological surface states exist at frequency f 0 . As we vary f away from f 0 , the Hamiltonian varies smoothly, deviating from the form of the target Hamiltonian (e.g., the positive and negative nearest neighbor hoppings become unequal in magnitude). Throughout this variation, so long as E lies in a gap, the topological properties are unchanged and the topological surface states continue to exist. Thus, the f -dependent response of the circuit behaves like a bandstructure.
For small m, the circuit exhibits a finite-width topological bandgap in f -space. Tuning m closes this bandgap, and causes the surface states to disappear.