A Classifying Space for Phases of Matrix Product States

Alexei Kitaev has conjectured that there should be a loop spectrum consisting of spaces of gapped invertible quantum spin systems, indexed by spatial dimension d of the lattice. Motivated by Kitaev’s conjecture, I will detail a concrete construction of a topological space B consisting of translation invariant injective matrix product states (MPS) of all physical and bond dimensions, which plays the role Kitaev’s space in dimension d = 1. Having such a space is a useful tool in the discussion of parametrized phases of MPS; in fact it allows us to define a parametrized phase as a homotopy class of maps into B. The space B is constructed as the quotient of a contractible space E of MPS tensors modulo gauge transformations. The projection map from E to B is a quasifibration, from which we can compute the homotopy groups of the classifying space B by a long exact sequence. In particular, B has the weak homotopy type K(Z, 2) x K(Z, 3), shedding light on Kitaev’s conjecture in the context of MPS.

and Sunghyuk Park also speaking tomorrow 10/7 (15 min talk)

For the list of upcoming speakers, see: https://people.math.harvard.edu/~vkrylov/seminar1.html