  The discovery of topological phases in condensed matter systems has changed
the modern conception of phases of matter. The global nature of topological
ordering makes these phases robust and hence promising for applications.
However, the non-locality of this ordering makes direct experimental studies an
outstanding challenge, even in the simplest model topological systems, and
interactions among the constituent particles adds to this challenge. Here we
demonstrate a novel dynamical method to explore topological phases in both
interacting and non-interacting systems, by employing the exquisite control
afforded by state-of-the-art superconducting quantum circuits. We utilize this
method to experimentally explore the well-known Haldane model of topological
phase transitions by directly measuring the topological invariants of the
system. We construct the topological phase diagram of this model and visualize
the microscopic evolution of states across the phase transition, tasks whose
experimental realizations have remained elusive. Furthermore, we developed a
new qubit architecture that allows simultaneous control over every term in a
two-qubit Hamiltonian, with which we extend our studies to an interacting
Hamiltonian and discover the emergence of an interaction-induced topological
phase. Our implementation, involving the measurement of both global and local
textures of quantum systems, is close to the original idea of quantum
simulation as envisioned by R. Feynman, where a controllable quantum system is
used to investigate otherwise inaccessible quantum phenomena. This approach
demonstrates the potential of superconducting qubits for quantum simulation and
establishes a powerful platform for the study of topological phases in quantum
systems.
