Finitely Correlated States Driven by Topological Dynamics

Let $(\Omega, \P)$ be a standard probability space and let $\vartheta:\Omega \to \Omega$ be a measure preserving ergodic homeomorphism. Let $\mathcal{A}$ be a $C^*$-algebra with a unit and let $\mathcal{A}_{\mathbb{Z}}$ be the quasi-local algebra associated to the spin chain with one-site algebra $\mathcal{A}$. Equip $\mathcal{A}_{\mathbb{Z}}$ with the group action of translation by $k$-units, $\tau_k\in Aut(\mathcal{A}_{\mathbb{Z}})$ for $k\in \mathbb{Z}$. We study the problem of finding a disordered matrix product state decomposition for disordered states $\psi(\omega)$ on $\mathcal{A}_{\mathbb{Z}}$ with the covariance symmetry condition $\psi(\omega) \circ \tau_k = \psi(\vartheta^k \omega)$. This can be seen as an ergodic generalization of the results of Fannes, Nachtergaele, and Werner \cite{FannesNachtergaeleWerner}. To reify our structure theory, we present a disordered state $\nu_\omega$ obtained by sampling the AKLT model \cite{AKLT} in parameter space. We go on to show that $\nu_\omega$ has a nearest-neighbor parent Hamiltonian, its bulk spectral gap closes, but it has almost surely exponentially decaying correlations, and finally, that $\nu_\omega$ is time-reversal symmetry protected with a Tasaki index of $-1$.