Topology-Informed Circuit Inference

• Topology-informed circuit inference is a method that uses topological band theory to map protected edge states and quantum invariants onto classical circuits.
• Engineered circuit lattices employ cyclic wiring and discrete Fourier transforms to simulate spin Chern numbers and effective magnetic flux, mirroring quantum spin Hall insulators.
• The approach ensures robustness to symmetry-preserving perturbations and enables experimental simulation of non-Abelian effects in classical circuit settings.

Topology-informed circuit inference refers to the process of deducing, controlling, and exploiting the structure and connectivity of electrical circuits based on principles from topological band theory and topological phases of matter. In the context of the seminal work “Topological properties of linear circuit lattices” (Albert et al., 2014), circuits composed of inductors and capacitors are engineered to host protected edge modes and simulate nontrivial topological invariants, closely paralleling phenomena in quantum spin Hall insulators and Hofstadter models. The framework leverages a unitary equivalence between the circuit's normal mode frequency matrix and quantum tight-binding Hamiltonians, allowing one to map topological invariants and robust transport properties onto classical circuit implementations.

1. Topological Mapping of Circuit Lattices

The foundational strategy is to design a lattice of circuit nodes, each site comprising multiple subnodes (e.g., a triangle with 3 nodes per site in the minimal construction). The wiring between these nodes is carefully arranged to implement cyclic permutations—these generate Peierls-like phases and produce an effective magnetic flux through the circuit lattice.

By performing a site-internal discrete Fourier transform on the voltage variables, the circuit equations are block-diagonalized, isolating sectors corresponding to different “spin” components. Each sector mimics the Hamiltonian of a quantum spin Hall insulator (QSHI) or a time-reversal invariant Hofstadter model with fractional flux (e.g., $1/3$ flux per plaquette). The key topological invariant, the spin Chern number $\mathcal{C}_{sc} = \mathcal{C}_1 - \mathcal{C}_2$, classifies the bulk gaps and determines the presence of protected edge modes.

This architecture is underpinned by the following normal mode equation in the transformed basis: $$\ddot{\zeta}_{m,n}^{(j)} = -4\zeta_{m,n}^{(j)} + \zeta_{m+1,n}^{(j)} + \zeta_{m-1,n}^{(j)} + e^{i\frac{2\pi}{3}mj}\zeta_{m,n+1}^{(j)} + e^{-i\frac{2\pi}{3}mj}\zeta_{m,n-1}^{(j)},$$ where the $j = 1,2$ sectors together simulate a “spin-doubled” Hofstadter lattice.

2. Topological Invariants and Robust Edge Transport

The mapping to a block-diagonalized Hamiltonian allows direct computation of topological invariants for the circuit:

• Spin Chern number discriminates between topologically distinct phases and predicts the existence of counterpropagating boundary modes.
• Kramers degeneracy–like protection arises in the circuit due to the specific block structure, reflecting time-reversal symmetry in the original quantum model.

Edge modes manifest in the form of boundary-localized normal modes that are robust to certain classes of perturbations, analogous to the helical edge states in QSHIs.

3. Circuit Design Strategies for Nontrivial Topology

The proposed platform reduces component complexity compared to earlier designs by utilizing only three subnodes per site and relying on a single permutation matrix $y$ for cyclic wiring. This structure ensures that:

• Capacitive couplings within a site (internal triangle) and
• Inductive links between sites (across different triangles)

together generate the effective hopping amplitudes and Peierls phases necessary for nontrivial topology.

The generalization to $d$-fold symmetry (sites with $d$ nodes, using a $d$-gon) allows simulation of arbitrary flux configurations and higher-dimensional synthetic gauge fields. Superpositions of permutation patterns, controlled by inverse inductances, further expand the accessible space of “gauge” topologies (including U(2) non-Abelian terms).

4. Perturbation Analysis and Symmetry Protection

A rigorous symmetry analysis identifies the perturbations that preserve the topological protection of edge modes:

• Perturbations that commute with the circuit symmetry operator $S$ (derived from $y$ and related to $C_3$ symmetry) conserve the block-diagonal structure and do not induce elastic backscattering of edge modes.
• Uniform variations in inductance or capacitance across all nodes within a link (i.e., $M_{11} \propto I_3$ and $M_{12} \propto y$, or functions thereof) leave the edge modes robust.
• Nonuniformities that break site symmetry (distinguishing among internal node connections) generally reduce topological protection and can induce mode mixing or backscattering, in analogy to TR-breaking perturbations in QSHIs.

5. Simulation of Non-Abelian Aharonov-Bohm Effect

The framework enables the simulation of non-Abelian Aharonov-Bohm (AB) effects in classical circuits:

• By constructing parallel paths (Path A with repeated $y$ permutations, Path B with alternate $P$ permutation mixing pseudo-spin sectors), interference effects depending on noncommuting operations emerge.
• The output signal intensity after path recombination is predicted to follow

$$|\phi_A + \phi_B|^2 \propto \cos^2\Big\{ t - \frac{2\pi}{3}\left[(N-1) \bmod 3\right]\Big\}$$

for even $N$-site paths, directly reflecting the non-Abelian phase accumulation.

This capability demonstrates how topological properties and nontrivial gauge structures can be directly inferred and engineered via classical circuit design.

6. Generalizations and Experimental Implications

The methodology extends naturally to:

• Exotic lattice geometries: Honeycomb, Kagome, Möbius strip–like circuits, and interfaces between different topological phases can be realized by adjusting the permutation wiring.
• Gauge field engineering: Abelian and non-Abelian synthetic gauge fields arise from weighted superpositions and permutations, tunable at the component level.
• Nonlinear/interacting extensions: Incorporating nonlinear elements, e.g., Josephson junctions, will enable simulation of interacting topological models and exploration of new many-body phenomena.

The compactness and flexibility of the design make it a promising experimental platform for probing topological physics in highly controlled, locally accessible, and tunable classical settings.

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Summary Table: Key Elements in Topology-Informed Circuit Inference

Feature	Circuit Realization	Quantum Analogy
Site symmetry (e.g., $C_3$)	Triangle with 3 nodes per site	Spin/orbital degrees of freedom
Cyclic wiring (matrix $y$)	Imposes Peierls phases, flux	Simulates magnetic field in Hofstadter/QSH models
Topological invariant	Spin Chern number ($\mathcal{C}_{sc}$)	Classifies topological phase, edge mode protection
Block-diagonalization	Site-wise discrete Fourier transform	Spin sector separation, Kramers degeneracy
Perturbation robustness	Symmetry-conserving parameter changes	TR-preserving disorder in QSHIs
Non-Abelian path interference	Path permutations with $y$ and $P$	Non-Abelian Aharonov-Bohm effect

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In conclusion, topology-informed circuit inference, as formalized in this work, enables systematic design, analysis, and generalization of classical circuits with engineered topological properties. By exploiting circuit symmetry, permutation-induced Peierls phases, and a direct mapping onto topological tight-binding models, the approach provides a versatile and experimentally tractable route to realizing, probing, and inferring topological phenomena in artificial electrical lattices. This bridging of abstract topological invariants and concrete circuit design lays the foundation for further exploration of exotic states and novel functionalities in both classical and quantum simulators.