A Classifying Space for Phases of Matrix Product States
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914255200059392 |
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| author | Spiegel, Daniel D. Qi, Marvin Stephen, David T. Hermele, Michael Pflaum, Markus J. Beaudry, Agnes |
| author_facet | Spiegel, Daniel D. Qi, Marvin Stephen, David T. Hermele, Michael Pflaum, Markus J. Beaudry, Agnes |
| contents | 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 $π_3(\mathcal{B})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_14241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Classifying Space for Phases of Matrix Product States Spiegel, Daniel D. Qi, Marvin Stephen, David T. Hermele, Michael Pflaum, Markus J. Beaudry, Agnes Mathematical Physics Strongly Correlated Electrons Algebraic Topology Quantum Physics 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 $π_3(\mathcal{B})$. |
| title | A Classifying Space for Phases of Matrix Product States |
| topic | Mathematical Physics Strongly Correlated Electrons Algebraic Topology Quantum Physics |
| url | https://arxiv.org/abs/2501.14241 |