Finitely Correlated States Driven by Topological Dynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915380704837632 |
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| author | Roon, Eric B. Schenker, Jeffrey H. |
| author_facet | Roon, Eric B. Schenker, Jeffrey H. |
| contents | Let $(Ω, ¶)$ be a standard probability space and let $\vartheta:Ω\to Ω$ 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, $τ_k\in Aut(\mathcal{A}_{\mathbb{Z}})$ for $k\in \mathbb{Z}$. We study the problem of finding a disordered matrix product state decomposition for disordered states $ψ(ω)$ on $\mathcal{A}_{\mathbb{Z}}$ with the covariance symmetry condition $ψ(ω) \circ τ_k = ψ(\vartheta^k ω)$. 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 $ν_ω$ obtained by sampling the AKLT model \cite{AKLT} in parameter space. We go on to show that $ν_ω$ has a nearest-neighbor parent Hamiltonian, its bulk spectral gap closes, but it has almost surely exponentially decaying correlations, and finally, that $ν_ω$ is time-reversal symmetry protected with a Tasaki index of $-1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_07287 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finitely Correlated States Driven by Topological Dynamics Roon, Eric B. Schenker, Jeffrey H. Mathematical Physics Let $(Ω, ¶)$ be a standard probability space and let $\vartheta:Ω\to Ω$ 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, $τ_k\in Aut(\mathcal{A}_{\mathbb{Z}})$ for $k\in \mathbb{Z}$. We study the problem of finding a disordered matrix product state decomposition for disordered states $ψ(ω)$ on $\mathcal{A}_{\mathbb{Z}}$ with the covariance symmetry condition $ψ(ω) \circ τ_k = ψ(\vartheta^k ω)$. 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 $ν_ω$ obtained by sampling the AKLT model \cite{AKLT} in parameter space. We go on to show that $ν_ω$ has a nearest-neighbor parent Hamiltonian, its bulk spectral gap closes, but it has almost surely exponentially decaying correlations, and finally, that $ν_ω$ is time-reversal symmetry protected with a Tasaki index of $-1$. |
| title | Finitely Correlated States Driven by Topological Dynamics |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2507.07287 |