arXiv subjects

Eric B. Roon

Publications and source records attributed to Eric B. Roon.

5 recordsLinked to original sources

arXiv2026

Parent Hamiltonians of Ergodic Matrix Product States

Matrix product states (MPS) are quintessential examples of frustration-free gapped ground states of local interactions called parent Hamiltonians. In this work, we investigate parent Hamiltonians for a class of ergodic matrix product states (EMPS), which are MPS defined by site-dependent random tensors $\{X_j^{[k]}\}_{j=1}^D$ which are homogeneously distributed at every site $k$ in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.

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arXiv2026

Disordered Ground States of Ergodic Quantum Spin Systems

In this letter, we fill a hole in the existing literature about disordered quantum spin systems generated by a random local interaction $\{\mathfrak{h}(Z)\}_{Z\Subset \mathbb{Z}^\nu}$ satisfying a statistical version of translation invariance. We show such systems always have disordered ground states in the thermodynamic limit with the same symmetry. A key tool we use is a disordered version of the Lieb-Robinson bounds, which hold almost surely under mild conditions on $\mathfrak{h}$. Along the way, we formalize the notion of a random state on a $C^*$-algebra and prove a weak-$\ast$ version of the Riesz-Markov-Kakutani theorem, which seems not to have been recorded in the vector measures literature. As a consequence of the existence of the aforementioned disordered ground states, we show that the spectrum of the GNS Hamiltonain associated to the bulk dynamics is deterministic with respect to the disorder.

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arXiv2025

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 [31]. To reify our structure theory, we present a disordered state $\nu_\omega$ obtained by sampling the AKLT model [2] 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 invariant with a Tasaki index of $-1$ almost surely.

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arXiv2024

On Quasi-Locality and Decay of Correlations for Long-Range Models of Open Quantum Spin Systems

We consider models of open quantum spin systems with irreversible dynamics and show that general quasi-locality results for long-range models, e.g. as proven for the Heisenberg dynamics associated to quantum systems in [27], naturally extend to this setting. Given these bounds, we provide two applications. First, we use these results to obtain estimates on a strictly local approximation of these finite-volume, irreversible dynamics. Next, we show how these bounds can be used to estimate correlation decay in various states.

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arXiv2023

Ergodic Quantum Processes on Finite von Neumann Algebras

Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual and let $(\Omega, \mathbb{P})$ be a probability space. A bounded positive random linear operator on $L^1(M,\tau)$ is a map $\gamma : \Omega \times L^1(M,\tau) \to L^1(M,\tau)$ so that $\tau(\gamma_\omega(x)a)$ is measurable for all $x\in L^1(M,\tau)$ and $a\in M$, and $x\mapsto \gamma_\omega(x)$ is bounded, positive, and linear almost surely. Given an ergodic $T\in Aut(\Omega, \mathbb{P})$, we study quantum processes of the form $\gamma_{T^n \omega}\circ \gamma_{T^{n-1}\omega} \circ \cdots \circ \gamma_{T^m\omega}$ for $m,n\in \mathbb{Z}$. Using the Hennion metric introduced in [MS22], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when $\gamma_\omega$ is the predual of a normal positive linear map on $M$. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [MS22].

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