A Classifying Space for Phases of Matrix Product States

• URL: http://arxiv.org/abs/2501.14241v1
• Date: Fri, 24 Jan 2025 04:58:32 GMT
• Title: A Classifying Space for Phases of Matrix Product States
• Authors: Agnes Beaudry, Michael Hermele, Markus J. Pflaum, Marvin Qi, Daniel D. Spiegel, David T. Stephen,
• Abstract summary: A topological space $mathcalB$ is defined as the quotient of a contractible space $mathcalE$ of MPS tensors. We prove that the projection map $p:mathcalE rightarrow mathcalB$ is a quasifibration.
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• 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 $\pi_3(\mathcal{B})$.

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