Topological Phases of Matter and Homotopy Theory

Phases of quantum matter with a nonzero gap for local excitations are commonly referred to as topological phases of matter, because of their long-assumed relation to Topological Quantum Field Theory. While this relation is no longer believed to be true in all cases, topology, and more specifically homotopy theory, are used to define gapped phases rigorously. The connection with the homotopy theory is especially fruitful in the case of invertible gapped phases where a precise conjecture about the classification of such phases has been advanced. The topology of the space of gapped many-body systems on a lattice can be probed using higher-dimensional generalizations of the Berry phase.