A Classifying Space for Phases of Matrix Product States
Daniel D. Spiegel, Marvin Qi, David T. Stephen, Michael Hermele, Markus J. Pflaum, Agnes Beaudry
TL;DR
This work constructs a universal classifying space $\mathcal{B}$ for phases of translation invariant injective MPS across all physical and bond dimensions, proving its weak homotopy type is $K(\mathbb{Z},2)\times K(\mathbb{Z},3)$ and that phases of families over $X$ correspond to $[X,\mathcal{B}]\cong H^2(X;\mathbb{Z})\times H^3(X;\mathbb{Z})$, encoded respectively by a Chern-number per unit cell and a KS-number. The construction uses a contractible gauge-tensor space $\mathcal{E}$ and forms $\mathcal{B}=\mathcal{E}/\sim$; the projection $p:\mathcal{E}\to\mathcal{B}$ is shown to be a quasifibration, enabling a long exact sequence of homotopy groups that yields the weak homotopy type of $\mathcal{B}$. Explicit generators for $\pi_2(\mathcal{B})$ and $\pi_3(\mathcal{B})$ are given by a basic $S^2$-family $\psi_2$ and the Chern-number pump $\psi_3$, connecting to Berry-phase physics and the KS invariant. This framework aligns with Kitaev’s classification program for gapped invertible lattice systems and clarifies how translation invariance influences the topological data of 1D MPS phases, suggesting routes to extend to non-translation-invariant settings.
Abstract
We construct a topological space $\mathcal{B}$ consisting of translation invariant injective matrix product states (MPS) of all physical and bond dimensions and show that it has the weak homotopy type $K(\mathbb{Z}, 2) \times K(\mathbb{Z}, 3)$. The implication is that the phase of a family of such states parametrized by a space $X$ is completely determined by two invariants: a class in $H^2(X; \mathbb{Z})$ corresponding to the Chern number per unit cell and a class in $H^3(X; \mathbb{Z})$, the so-called Kapustin-Spodyneiko (KS) number. The space $\mathcal{B}$ is defined as the quotient of a contractible space $\mathcal{E}$ of MPS tensors by an equivalence relation describing gauge transformations of the tensors. We prove that the projection map $p:\mathcal{E} \rightarrow \mathcal{B}$ is a quasifibration, and this allows us to determine the weak homotopy type of $\mathcal{B}$. As an example, we review the Chern number pump-a family of MPS parametrized by $S^3$-and prove that it generates $π_3(\mathcal{B})$.