Mathematics Colloquia and Seminars

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In this talk, I will discuss the construction of a topological space B consisting of translation invariant injective matrix product states (MPS) of all physical and bond dimensions. Having such a space is a useful tool in the discussion of parametrized phases of MPS; it simultaneously defines a parametrized family of MPS as a map into the classifying space B and defines a parametrized phase as a homotopy class of such maps. The classifying space 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, which allows us to 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), partially confirming a conjecture by Kitaev.