Circuit implementation of a four-dimensional topological insulator

Abstract

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.

Four-dimensional semimetals with tensor monopoles: From surface states to topological responses

“…We note that the parity magnetic effect in Eq. ( 16) could be experimentally studied in 3D quantum-engineered setups extended by a synthetic dimension [53], as could be realized for cold atoms in optical lattices [68], for photons in arrays of ring resonators [54], or in electric circuits [69]. The timevarying component b w (t ) could be induced through a periodic driving of an on-site coupling, as proposed in Ref.…”

Four-dimensional topological lattices through connectivity

“…Note Added: In preparation of this manuscript, we became aware of a recent proposal for an eight-band 4D crystalline topological insulator, which has bosonic TRS [58], but which is instead topologically-protected by reflection symmetry and which relies on spin-orbit couplings. Since this proposal was put on arXiv, it has been experimentally implemented in electric circuits [59]. Theoretical proposals have also been made for electric circuits to realise a different spinless (Class AI) 4DQH model [60], to simulate nth-Chern-number insulators [61] and to image nodal boundary Seifert surfaces in 4D circuits [62].…”

Observation of antichiral edge states in a circuit lattice

“…Circuit metamaterials have been the subject of recent theoretical and experimental interest [28][29][30][31][32][33][34][35][36][37][38][39][40] due to the ease with which they can be designed and fabricated to realize different topological phases, as well as unusual lattice configurations that are hard to achieve on other platforms. Circuits have been used to demonstrate nonlinear topological boundary states [33,34], topological corner modes [35][36][37][38], and four-dimensional topological insulators [39,40]. Most notably, Jia et al [7] have shown how a Haldane-type Chern insulator phase can be accessed using a lattice of capacitors (C) and inductors (L) with braided interconnections.…”