Visualizing the quantum evolution of simple spin systems

Even in the ground state, interacting many-body systems are a fluctuating soup of their constituent degrees of freedom. These fluctuations are quantum rather than thermal, and they are related to how the system evolves under the action of the quantum Hamiltonian. The rate of
evolution, which can be fast or slow, encodes important information about the physics of the system (and typically controls the computational cost of numerical simulations).

It is often instructive to develop alternative representations of the state of the system that are geometric or pictorial in character. Then the quantum evolution can be understood as the repeated reconfiguration of easy-to-visualize graphical elements. This approach often gives us better intuition for the quantum dynamics and can suggest effective simulation algorithms or analytical lines of attack. For concreteness, I will discuss these ideas in the context of simple models of quantum magnetism.