Special Colloquium- Daniel Spiegel, Harvard

The Topology of Pure States on Quantum Spin Chains

 

Consider a quantum system defined by a Hamiltonian that depends on some variables in a parameter space. In 1983, Michael Berry discovered that as these variables are changed, the wavefunction of the system is influenced by the geometry of the parameter space. Today, conjectures by Alexei Kitaev and our ever-improving understanding of topological phases of quantum matter have spurred research into parametrized quantum many-body systems, resulting in generalizations of Berry’s discovery. An important ongoing objective in this field is to understand the topology of the space of all quantum many-body ground states of a given spatial dimension. I will present progress on this objective from both purely abstract and more physical points of view. In particular, I will discuss our calculation of the homotopy groups of both the whole pure state space and the space of matrix product states on a quantum spin chain.