Demonstration of Hopf-link semimetal bands with superconducting circuits

Hopf-link semimetals exhibit exotic gapless band structures with fascinating topological properties, which have never been observed in nature. Here we demonstrate nodal lines with topological form of Hopf-link chains in artificial semimetal-bands. Driving superconducting quantum circuits with elaborately designed microwave fields, we mapped the momentum space of a lattice to a parameter space of the Hamiltonian for a Hopf-link semimetal. By measuring the energy spectrum, we directly imaged nodal lines in cubic lattices. By tuning the driving fields, we adjusted various parameters of Hamiltonian. Important topological features, such as link-unlink topological transitions and the robustness of the Hopf-link chain structure were investigated. Moreover, we extracted the linking number by detecting the Berry phase associated with different loops encircling nodal lines. This topological invariant clearly reveals the nontrivial topology of the Hopf-link semimetal. Our results provide knowledge for developing new materials and quantum devices.