The goal of this seminar series is to learn about the classification of symmetry-protected topological (SPT) phases and identify interesting research level problems in the area.

There will be two seminars running in parallel. Lectures will be prepared by the volunteered participants and uploaded to this wiki page.

Symmetry-Protected Topological Phases

Quantum states can be partitioned into collections, called “phases”, under continuous deformation respecting some features of the system. If the system has an energy gap above the ground state and has a certain symmetry then the phases that are obtained by deformations preserving the gap and the symmetry are called symmetry-protected topological (SPT) phases. Classification of SPT phases uses various cohomology theories such as group cohomology, topological $K$-theory, cobordism theory,… In algebraic topology such a (generalized) cohomology theory is described by a space-like object called a spectrum. The associated cohomology theory is obtained by taking the stable homotopy groups of the spectrum. Given this correspondence Kitaev proposed (Kit13, Kit15) a direct approach to the classification problem by constructing a spectrum from physically available data. The resulting spectrum comes in two flavors: (1) $bGP^\times$ for bosonic theories and (2) $fGP^\times$ for fermionic theories. The generators of stable homotopy groups correspond to physically realizable quantum phases. For example, Kitaev’s $E_8$ phase generates the homotopy group $\pi_{-3} bGP^\times =\mathbb{Z}$ and the Majorana chain generates $\pi_{-2} fGP^\times = \mathbb{Z}/2$.

For the list of talks see the individual seminar pages: