Topological circuits of inductors and capacitors

TL;DR

This paper proposes a design framework for topological circuits composed of inductors and capacitors, demonstrating their connection to topological insulators and quantum lattice models through a unifying Lagrangian approach.

Contribution

It introduces a systematic method to construct and analyze topological LC circuits using a unifying Lagrangian and $H$-matrix framework, including higher-dimensional examples.

Findings

Design of topological circuits with protected edge modes

Connection between circuit topologies and quantum topological phases

Extension to higher-dimensional topological circuit models

Abstract

Alternating current (ac) circuits can have electromagnetic edge modes protected by symmetries, analogous to topological band insulators or semimetals. How to make such a topological circuit? This paper illustrates a particular design idea by analyzing a series of topological circuits consisting purely of inductors (L) and capacitors (C) connected to each other by wires to form periodic lattices. All the examples are treated using a unifying approach based on Lagrangians and the dynamical $H$-matrix. First, the building blocks and permutation wiring are introduced using simple circuits in one dimension, the SSH transmission line and a braided ladder analogous to the ice-tray model also known as the $\pi$-flux ladder. Then, more general building blocks (loops and stars) and wiring schemes ($m$-shifts) are introduced. The key concepts of emergent pseudo-spin degrees of freedom and synthetic gauge fields are discussed, and the connection to quantum lattice Hamiltonians is clarified. A diagrammatic notation is introduced to simplify the design and presentation of more complicated circuits. These building blocks are then used to construct topological circuits in higher dimensions. The examples include the circuit analog of Haldane's Chern insulator in two dimensions and quantum Hall insulator in four dimensions featuring finite second Chern numbers. The topological invariants and symmetry protection of the edge modes are discussed based on the $H$-matrix.