A Classifying Space for Phases of Matrix Product States

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 $p:\mathcal{E}\to \mathcal{B}$ 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 ${\pi }_3(\mathcal{B})$.

63 pages, minor typos corrected, author order changed to align with conventions in condensed matter physics

References in corpus (14)

• The density-matrix renormalization group in the age of matrix product states
• Entanglement spectrum of a topological phase in one dimension
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Classifying quantum phases using Matrix Product States and PEPS
• Matrix product states represent ground states faithfully
• Entropy scaling and simulability by Matrix Product States
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Reflection positivity and invertible topological phases
• Quantum phase transitions in matrix product systems
• Geometry of Matrix Product States: metric, parallel transport and curvature
• Higher-dimensional generalizations of the Berry curvature
• Homotopical Foundations of Parametrized Quantum Spin Systems
• Topological Field Theory and Matrix Product States

Cited by in corpus (1)

• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation