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 demonstrated nodal lines with topological form of Hopf-link chain in artificial semimetal-bands. Driving superconducting quantum circuits with elaborately designed microwave fields, we mapped the momentum space of a lattice to the parameter space, realizing the Hamiltonian of 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 transition and the robustness of Hopf-link chain structure are investigated. Moreover, we extracted linking number by detecting Berry phase associated with different loops enclosing or disclosing nodal lines. The topological invariant clearly reveals the scenery of the connection between two nodal rings. Our simulations provide foremost knowledge for developing new materials and quantum devices.