Imaging nodal knots in momentum space through topolectrical circuits

Ching Hua Lee,Lay Kee Ang,Ronny Thomale,Yuhan Liu,Xiao Zhang,Amanda Sutrisno,Tobias Helbig,Martin Greiter,Tobias Hofmann,Yee Sin Ang

doi:10.1038/s41467-020-17716-1

Abstract

Knots are intricate structures that cannot be unambiguously distinguished with any single topological invariant. Momentum space knots, in particular, have been elusive due to their requisite finely tuned long-ranged hoppings. Even if constructed, probing their intricate linkages and topological "drumhead” surface states will be challenging due to the high precision needed. In this work, we overcome these practical and technical challenges with RLC circuits, transcending existing theoretical constructions which necessarily break reciprocity, by pairing nodal knots with their mirror image partners in a fully reciprocal setting. Our nodal knot circuits can be characterized with impedance measurements that resolve their drumhead states and image their 3D nodal structure. Doing so allows for reconstruction of the Seifert surface and hence knot topological invariants like the Alexander polynomial. We illustrate our approach with large-scale simulations of various nodal knots and an experiment which maps out the topological drumhead region of a Hopf-link.

Highlights

Knots are intricate structures that cannot be unambiguously distinguished with any single topological invariant
In the pursuit of ever more exotic topological states, contemporary research has witnessed a shift from established topological insulator platforms photonic, mechanical, and acoustic mwietthamZateorrialZs12–3tothpaotlomgyimtioc topological nodal semimetals[4,5,6,7,8,9,10]
Features of the 3D Brillouin zone (BZ), and great finesse is required in imaging them. We show how these challenges can be overcome via (i) a special scheme for designing nodal knots circuits with mirror-image partners, (ii) a new robust impedance measurement approach for imaging nodal knots and their accompanying drumhead surface states, and (iii) an instructive experimental demonstration of how the topological drumhead region of a nodal knot can imaged

Summary

Introduction

Knots are intricate structures that cannot be unambiguously distinguished with any single topological invariant. This circuit is physically implemented with an array of connected printed circuit boards (PCBs), each representing one unit cell, which can be adjusted to accurately correspond to different (ky, kz) points by tuning the inductors (Fig. 6 of Methods).

Results

Conclusion