Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2507.07287

Finitely Correlated States Driven by Topological Dynamics

Eric B. Roon, Jeffrey Schenker

Published Jul 9, 2025• 0 citations
arXivHTMLPDF

Abstract

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

DOI
10.48550/arxiv.2507.07287

Related papers

Parent Hamiltonians of Ergodic Matrix Product States

Owen Ekblad, Eloy Moreno-Nadales, Eric B. Roon +1 more

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.

math-phCross-listedarXiv:2607.01446•Jul 3, 2026•0 citations

Disordered Ground States of Ergodic Quantum Spin Systems

Eric B. Roon, Jeffrey H. Schenker

In this letter, we fill a hole in the existing literature about disordered quantum spin systems generated by a random local interaction $\{\mathfrak{h}(Z)\}_{Z\Subset \mathbb{Z}^ν}$ 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.

math-phCross-listedarXiv:2603.19475•Mar 23, 2026•0 citations

From Thermalization to Multifractality: Spin-Spin Correlation in Disordered $SU(2)$-Invariant 1D Heisenberg Spin Chains

Debasmita Giri, Julian Siegl, John Schliemann

We investigate spin correlations in one-dimensional $SU(2)$-invariant Heisenberg chains with exchange disorder for spin lengths $S=1/2$ and $S=1$. In the weak-disorder regime, the eigenmodes of the spin-spin correlation matrix are delocalized, consistent with ergodic behavior. Under strong disorder, the system enters a quasi-localized multifractal phase characterized by exponentially decaying, dimer-like spin correlations. Finite-size scaling of the inverse participation ratios of the correlation-matrix eigenmodes yields a correlation dimension, $D_2\approx 0.37-0.39$, confirming the stability of a multifractal regime that is distinct from both the ergodic limit ($D_2=1$) and the fully localized limit ($D_2=0$).

arXiv:2508.00573•Aug 1, 2025•0 citations

Probability transport on the Fock space of a disordered quantum spin chain

Isabel Creed, David E. Logan, Sthitadhi Roy

Within the broad theme of understanding the dynamics of disordered quantum many-body systems, one of the simplest questions one can ask is, given an initial state, how does it evolve in time on the associated Fock-space graph, in terms of the distribution of probabilities thereon? A detailed quantitative description of the temporal evolution of out-of-equilibrium disordered quantum states and probability transport on the Fock space is our central aim here. We investigate it in the context of a disordered quantum spin chain, which hosts a disorder-driven many-body localization transition. Real-time dynamics/probability transport is shown to exhibit a rich phenomenology, which is markedly different between the ergodic and many-body localized phases. The dynamics is, for example, found to be strongly inhomogeneous at intermediate times in both phases, but while it gives way to homogeneity at long times in the ergodic phase, the dynamics remain inhomogeneous and multifractal in nature for arbitrarily long times in the localized phase. Similarly, we show that an appropriately defined dynamical lengthscale on the Fock-space graph is directly related to the local spin autocorrelation, and as such sheds light on the (anomalous) decay of the autocorrelation in the ergodic phase, and lack of it in the localized phase.

arXiv:2212.14333•Physical review. B./Physical review. B•Mar 29, 2023•14 citations

Lower Bounds on Spectral Gaps of Parent Hamiltonians via Tensor Networks

Milán Ádám Rozmán, András Molnár, Norbert Schuch

Proving spectral gaps above the ground space is a central problem in quantum many-body physics. Yet, finding lower bounds on the gap is notoriously difficult. We revisit the martingale method, originally developed by Fannes, Nachtergaele, and Werner [Fannes '92, Nachtergaele '96] to prove the existence of spectral gaps of parent Hamiltonians of Matrix Product States (MPS), and provide improvements to the different steps of the method. Most importantly, we devise a new technique which allows to compute the key quantity in the martingale method -- the overlap of local ground spaces -- exactly and efficiently. This enables a clear improvement of the method, allowing it to outperform other existing techniques to lower bound gaps, which we demonstrate by benchmarking on several models. Remarkably, our -- numerically motivated -- approach at the same time also yields a significantly simplified proof of the fact that parent Hamiltonians of any (well-behaved) MPS are always gapped.

quant-pharXiv:2607.19078•Jul 22, 2026•0 citations

Enforced Gaplessness from States with Exponentially Decaying Correlations

Rahul Sahay, Curt W. von Keyserlingk, Ruben Verresen +1 more

It is well known that an exponentially localized Hamiltonian must be gapless if its ground state has algebraic correlations. We show that even certain exponentially decaying correlations can imply gaplessness. This is exemplified by the deformed toric code $\propto \exp(β\sum_{\ell} Z_{\ell}) |\mathsf{TC}\rangle$, where $|\mathsf{TC}\rangle$ is a fixed-point toric code wavefunction. Although it has a confined regime for $β> β_c$, recent work has drawn attention to its perimeter law loop correlations. Here, we show that these unusual loop correlations -- namely, perimeter law coexisting with a 1-form symmetry whose disorder operator has long-range order -- imply that any local parent Hamiltonian must either be gapless or have a degeneracy scaling with system size. Moreover, we construct a variational low-energy state for arbitrary local frustration-free Hamiltonians, upper bounding the finite-size gap by $O(1/L^3)$ on periodic boundary conditions. Strikingly, these variational states look like loop waves -- non-quasiparticle analogs of spin waves -- generated from the ground state by non-local loop operators. Our findings have implications for identifying the subset of Hilbert space to which gapped ground states belong, and the techniques have wide applicability. For instance, a corollary of our first result is that Glauber dynamics for the ordered phase of the two-dimensional classical Ising model on the torus must have a gapless Markov transition matrix, with our second result bounding its gap.

arXiv:2503.01977•Mar 3, 2025•0 citations
Let (Ω,¶)(Ω, ¶)(Ω,¶) be a standard probability space and let ϑ:Ω→Ω\vartheta:Ω\to Ωϑ:Ω→Ω be a measure preserving ergodic homeomorphism. Let A\mathcal{A}A be a C∗C^*C∗-algebra with a unit and let AZ\mathcal{A}_{\mathbb{Z}}AZ​ be the quasi-local algebra associated to the spin chain with one-site algebra A\mathcal{A}A. Equip AZ\mathcal{A}_{\mathbb{Z}}AZ​ with the group action of translation by kkk-units, τk∈Aut(AZ)τ_k\in Aut(\mathcal{A}_{\mathbb{Z}})τk​∈Aut(AZ​) for k∈Zk\in \mathbb{Z}k∈Z. We study the problem of finding a disordered matrix product state decomposition for disordered states ψ(ω)ψ(ω)ψ(ω) on AZ\mathcal{A}_{\mathbb{Z}}AZ​ with the covariance symmetry condition ψ(ω)∘τk=ψ(ϑkω)ψ(ω) \circ τ_k = ψ(\vartheta^k ω)ψ(ω)∘τk​=ψ(ϑkω). 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 νων_ωνω​ obtained by sampling the AKLT model [2] in parameter space. We go on to show that νων_ωνω​ has a nearest-neighbor parent Hamiltonian, its bulk spectral gap closes, but it has almost surely exponentially decaying correlations, and finally, that νων_ωνω​ is time-reversal invariant with a Tasaki index of −1-1−1 almost surely.
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 {Xj[k]}j=1D\{X_j^{[k]}\}_{j=1}^D{Xj[k]​}j=1D​ which are homogeneously distributed at every site kkk 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.
In this letter, we fill a hole in the existing literature about disordered quantum spin systems generated by a random local interaction {h(Z)}Z⋐Zν\{\mathfrak{h}(Z)\}_{Z\Subset \mathbb{Z}^ν}{h(Z)}Z⋐Zν​ 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 h\mathfrak{h}h. Along the way, we formalize the notion of a random state on a C∗C^*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.
From Thermalization to Multifractality: Spin-Spin Correlation in Disordered SU(2)SU(2)SU(2)-Invariant 1D Heisenberg Spin Chains
We investigate spin correlations in one-dimensional SU(2)SU(2)SU(2)-invariant Heisenberg chains with exchange disorder for spin lengths S=1/2S=1/2S=1/2 and S=1S=1S=1. In the weak-disorder regime, the eigenmodes of the spin-spin correlation matrix are delocalized, consistent with ergodic behavior. Under strong disorder, the system enters a quasi-localized multifractal phase characterized by exponentially decaying, dimer-like spin correlations. Finite-size scaling of the inverse participation ratios of the correlation-matrix eigenmodes yields a correlation dimension, D2≈0.37−0.39D_2\approx 0.37-0.39D2​≈0.37−0.39, confirming the stability of a multifractal regime that is distinct from both the ergodic limit (D2=1D_2=1D2​=1) and the fully localized limit (D2=0D_2=0D2​=0).
Within the broad theme of understanding the dynamics of disordered quantum many-body systems, one of the simplest questions one can ask is, given an initial state, how does it evolve in time on the associated Fock-space graph, in terms of the distribution of probabilities thereon? A detailed quantitative description of the temporal evolution of out-of-equilibrium disordered quantum states and probability transport on the Fock space is our central aim here. We investigate it in the context of a disordered quantum spin chain, which hosts a disorder-driven many-body localization transition. Real-time dynamics/probability transport is shown to exhibit a rich phenomenology, which is markedly different between the ergodic and many-body localized phases. The dynamics is, for example, found to be strongly inhomogeneous at intermediate times in both phases, but while it gives way to homogeneity at long times in the ergodic phase, the dynamics remain inhomogeneous and multifractal in nature for arbitrarily long times in the localized phase. Similarly, we show that an appropriately defined dynamical lengthscale on the Fock-space graph is directly related to the local spin autocorrelation, and as such sheds light on the (anomalous) decay of the autocorrelation in the ergodic phase, and lack of it in the localized phase.
Proving spectral gaps above the ground space is a central problem in quantum many-body physics. Yet, finding lower bounds on the gap is notoriously difficult. We revisit the martingale method, originally developed by Fannes, Nachtergaele, and Werner [Fannes '92, Nachtergaele '96] to prove the existence of spectral gaps of parent Hamiltonians of Matrix Product States (MPS), and provide improvements to the different steps of the method. Most importantly, we devise a new technique which allows to compute the key quantity in the martingale method -- the overlap of local ground spaces -- exactly and efficiently. This enables a clear improvement of the method, allowing it to outperform other existing techniques to lower bound gaps, which we demonstrate by benchmarking on several models. Remarkably, our -- numerically motivated -- approach at the same time also yields a significantly simplified proof of the fact that parent Hamiltonians of any (well-behaved) MPS are always gapped.
It is well known that an exponentially localized Hamiltonian must be gapless if its ground state has algebraic correlations. We show that even certain exponentially decaying correlations can imply gaplessness. This is exemplified by the deformed toric code ∝exp⁡(β∑ℓZℓ)∣TC⟩\propto \exp(β\sum_{\ell} Z_{\ell}) |\mathsf{TC}\rangle∝exp(β∑ℓ​Zℓ​)∣TC⟩, where ∣TC⟩|\mathsf{TC}\rangle∣TC⟩ is a fixed-point toric code wavefunction. Although it has a confined regime for β>βcβ> β_cβ>βc​, recent work has drawn attention to its perimeter law loop correlations. Here, we show that these unusual loop correlations -- namely, perimeter law coexisting with a 1-form symmetry whose disorder operator has long-range order -- imply that any local parent Hamiltonian must either be gapless or have a degeneracy scaling with system size. Moreover, we construct a variational low-energy state for arbitrary local frustration-free Hamiltonians, upper bounding the finite-size gap by O(1/L3)O(1/L^3)O(1/L3) on periodic boundary conditions. Strikingly, these variational states look like loop waves -- non-quasiparticle analogs of spin waves -- generated from the ground state by non-local loop operators. Our findings have implications for identifying the subset of Hilbert space to which gapped ground states belong, and the techniques have wide applicability. For instance, a corollary of our first result is that Glauber dynamics for the ordered phase of the two-dimensional classical Ising model on the torus must have a gapless Markov transition matrix, with our second result bounding its gap.