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Finitely Correlated States Driven by Topological Dynamics

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Pith's one-line read Disordered spin states with small correlations factor exactly through finite-dimensional random transfer maps, and the paper's AKLT example shows such states can be gapless, exponentially clustering, and topologically nontrivial.

desk verdict Novel and worth refereeing, but the central bundle construction currently rests on a false norm-identification lemma, so the converse classification is not yet proven as written. read the letter →

arxiv 2507.07287 v2 pith:AIF2FOLI submitted 2025-07-09 math-ph math.MP

Eric B. Roon , Jeffrey H. Schenker This is my paper · ORCID
classification math-phmath.MP MSC 46L6046L3082B2037A5582B44
keywords ergodicmatrixproductstatestranslationcovariantBanachbundlesdisorderedquantumspinchainsAKLTmodelparentHamiltonianspectralgapclosureTasakiindex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks which disordered quantum spin states — states whose expectation values depend on a disorder parameter — still admit a matrix-product description, meaning the state is computed by multiplying finitely many matrices along the chain. It proves an exact classification: a weakly* continuous, translation-covariant disordered state has finite-dimensional correlation span at every disorder value if and only if it is implemented by a finite-dimensional Banach bundle whose transfer maps, distinguished section, and boundary functional all depend continuously on the disorder (Theorem 3.34). This is the disordered, ergodic generalization of the finitely correlated state theory of [31], and the same construction shows that small-correlation states are weakly* dense among all translation-covariant states (Theorem 3.41). To show the structure carries real physics, the paper samples the one-parameter AKLT family independently at each site: the resulting state is almost surely pure with a nearest-neighbor, frustration-free, translation-covariant parent Hamiltonian, its bulk spectral gap is almost surely zero, and yet its correlations decay exponentially almost surely (Theorem 4.19). The state is also time-reversal invariant almost surely, with the $\mathbb{Z}_2$-valued Tasaki index equal to $-1$ (Theorem 4.24), so the example demonstrates that symmetry-protected index data can survive both disorder and a closed gap.

What carries the argument

The small-correlation bundle. Fix a cut separating the chain into a left half $\mathcal{A}^-$ and a right half $\mathcal{A}^+$; the paper declares two continuous right-half functions equivalent when the state $\psi_\omega$ cannot distinguish them against any left-half observable, and the quotient fibers $B^+_{\omega}$ record exactly the correlations across the cut. When these fibers are finite-dimensional, the Fell–Doran theorem (Theorem 2.11) gives the disjoint union a unique Banach bundle topology, and translation together with the Koopman map $\vartheta$ produce the transfer operators $E_{a,\omega}$ that shuttle vectors between fibers; the factorization (1.3) is then read off from iterating these maps. A minimality theorem (Theorem 3.32) identifies this bundle as the unique minimal implementation of the state. For the disordered AKLT example the machinery is the one-parameter family of isometries $V_\omega : \mathbb{C}^2 \to \mathbb{C}^3 \otimes \mathbb{C}^2$ with transfer maps $E_{a,\omega} = V_\omega^*(a \otimes \cdot)V_\omega$; the gaplessness proof uses rare regions where the sampled angle is close to zero to produce long stretches of slowly decaying correlations, contradicting the exponential clustering bound that a nonzero gap would impose, while the index computation analyzes the Affleck–Lieb twist operator through an explicit combinatorial description of the alternating spin configurations.

What would settle it

Simulate the IID AKLT transfer products and compare the decay of $|\nu_\omega(S^z_x S^z_{x+\ell})|$ with the claimed Lyapunov exponent $E\log\cos(2\omega)$: agreement confirms the exponential clustering, while systematically slower decay would falsify it. For the classification, construct a translation-covariant state whose correlation span is finite-dimensional for almost every $\omega$ but infinite-dimensional on a single orbit of $\vartheta$; Theorem 3.34's pointwise equivalence predicts that no finite-dimensional transfer apparatus can implement such a state, so producing one explicitly would falsify the theorem as stated.

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Extended reading notes

Core claim

The central claim is Theorem 3.34, stated for a weakly* continuous family of states $\psi_\omega$ on the quasi-local algebra $\mathcal{A}_{\mathbb{Z}}$ with the translation covariance $\psi_\omega \circ \tau_k = \psi_{\vartheta^k \omega}$. The following are equivalent: (a) for every $\omega$, the space $\{\psi_\omega(b \otimes (\cdot)) : b \in \mathcal{A}^-\}$ inside the dual of the right half-chain is finite-dimensional; (b) there is a finite-dimensional Banach bundle $E = \bigsqcup_\omega E_\omega$ over the disorder space together with a transfer apparatus — maps $E_{a,\omega} : E_\omega \to E_{\vartheta^{-1}\omega}$ linear in $a$, a continuous section $e(\omega)$, and linear functionals $\varrho_\omega$ — such that $\psi_\omega(a_m \otimes \cdots \otimes a_n) = \varrho_{\vartheta^{m-1}\omega} E_{a_m,\vartheta^m\omega} \cdots E_{a_n,\vartheta^n\omega}(e(\vartheta^n \omega))$. The proof takes, for each cut, the quotient of continuous right-half-chain functions by those annihilated against all left-half observables in the state, and the Fell–Doran theorem supplies the bundle topology making the transfer operators continuous. A further result, Theorem 3.41, states that the small-correlation states are weakly* dense among all translation-covariant states, uniformly in $\omega$. For the disordered AKLT model — a spin-1 valence-bond antiferromagnet whose one-parameter isometries $V_\omega$ are sampled independently at each site — Theorem 4.19 asserts that the bulk state is almost surely pure, is the ground state of a nearest-neighbor projection-valued parent Hamiltonian, is almost surely gapless, and yet has almost surely exponentially decaying correlations; Theorem 4.24 asserts that it is time-reversal invariant and that the Affleck–Lieb twist expectation converges almost surely to $-1$, giving a well-defined Tasaki index.

Load-bearing premise

The load-bearing premise is that the correlation space is finite-dimensional at every disorder value, because the Banach-bundle construction is proved only under that pointwise condition and has not been shown to work when small correlations hold merely almost surely.

Editorial extensions

If this is right

  • Every translation-covariant disordered state with small correlations can be evaluated exactly from finite matrix data — transfer maps, section, and boundary functional — making correlation functions, purity, and related observables of such disordered states as computable as for ordinary matrix product states.
  • Because small-correlation states are weakly* dense among translation-covariant states uniformly in $\omega$, any weak* continuous property that is uniform in the disorder can be established first on the finite-bundle class and then extended by approximation.
  • For the IID AKLT state, every translation-covariant finite-range Hamiltonian with this state as a ground state must have a vanishing bulk spectral gap, because the rare-region correlation lower bounds decay too slowly to be compatible with the uniform exponential clustering that a gap would force.
  • The example motivates a new class of 'quasi-gapped' ground states: almost surely exponentially clustering with random onset lengths, yet with no deterministic onset length and no bulk gap.
  • The almost sure Tasaki index of $-1$ shows that the $\mathbb{Z}_2$-valued symmetry-protected index survives disorder and gap closure, so index data remain meaningful for states that are not unique gapped ground states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The pointwise-versus-almost-sure gap in the theorem suggests a measurable-field version of the classification: a state with small correlations only almost surely should still admit a measurable transfer apparatus, with continuity of fiber norms the apparent obstruction in proof.
  • The rare-region mechanism makes a quantitative prediction: the statistics of slowly decaying stretches, and hence the rate of gap closure, should be governed by the tail probability $P(\omega_0 < \delta)$, and a distribution bounded away from the degenerate angle may keep the gap open — a question the paper itself leaves open.
  • If the Tasaki index is genuinely stable against disorder, other symmetry-protected markers such as string order parameters and reflection indices should also be almost surely constant across realizations, offering a disorder-based route to phase labels without gapped uniqueness.
  • The analogy to mobility-gapped fermion systems invites the test that low-lying excitations of the GNS Hamiltonian localize near the rare regions; a numerical study of the GNS spectrum on finite chains could indicate whether spatial localization accompanies the closed gap.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper develops a structure theory for translation-covariant disordered states on one-dimensional quantum spin chains. It defines a class of 'small correlations' states through finite-dimensional quotient spaces B^±_{x,ω}, constructs a Banach bundle from these fibers via the Fell–Doran theorem, and proves a factorization theorem (Theorem 3.34) in the spirit of the Fannes–Nachtergaele–Werner classification. The second half of the paper constructs an IID disordered deformation of the AKLT model and proves, almost surely, that the resulting state has a nearest-neighbor parent Hamiltonian, a deterministic but vanishing bulk spectral gap, exponentially decaying correlations, and a Tasaki index equal to −1. The paper also proves weak-* density of small-correlations states among all translation-covariant states.

Significance. If correct, the paper would provide a genuinely ergodic generalization of the FNW theory and a concrete disordered model exhibiting exponential clustering together with a closed bulk gap—an interesting phenomenon the authors call 'quasi-gapped.' The Banach-bundle transfer apparatus and the aromatic-averaging construction are original and potentially useful tools. The paper also ships explicit Wolfram Language computations in Appendix A, which is a helpful reproducible check of the AKLT ground-state intersections, and the IID AKLT example yields falsifiable predictions (gap closing, index −1) that go beyond the deterministic literature. However, the central bundle construction currently rests on a false norm-identification lemma, and several load-bearing results are deferred to unpublished companion works. The significance of the framework can only be assessed after these points are repaired.

major comments (5)
  1. Proposition 3.12 is false as stated. Consider the translation-invariant state ψ = (1/2)(δ_{0^Z} + δ_{1^Z}) on A_Z = (M_2)_Z. At the cut x = −1/2, B^+_{x,ω} is two-dimensional, so the hypothesis holds. For h = diag(1,0) at site 0, every d ∈ I_{x,ω} annihilates both |0^R⟩ and |1^R⟩, so (h+d)|0^R⟩ = |0^R⟩ and hence ||h+d|| ≥ 1; therefore ||[h]||_{B^+} = 1. On the other hand, n_ω(h) = sup_{||c||≤1} |ψ(c⊗h)| = 1/2, attained by taking c to be the projection onto the all-zero string on the left half-chain. Thus ||·||_{B^+} ≠ n_ω even in the finite-dimensional setting. The proof requires the reverse of the inequality in Remark 3.11—that norm-attaining functionals on B^+ are represented by classes in B^- of quotient norm at most 1—which fails: the evaluation-at-0 functional has dual norm 1 but is represented only by classes of quotient norm 2 (e.g. [2P_0^L]). Since Lemma 3.15 and Theorem 3.16 use Proposition 3.12 to obtain continuity of the quotient norm, the Fell–Doran construction of the small-correlations bundle and the (a)=>(b) direction of Theorem 3.34 are not justified as written.
  2. In the (b)=>(a) direction, after defining the spaces V_ω and W_ω, the proof states that by the Minimality Theorem 3.32 these spaces 'can be mapped isometrically into the small correlations bundle.' But Theorem 3.32(b)—the hypothesis under which such an isomorphism is obtained—is the dense-span condition, which is exactly what needs to be proved and has not been verified at that point. The subsequent boundedness estimates for Q^±_ω therefore rely on the desired conclusion. This is a genuine gap in the proof that finite-dimensionality of a transfer apparatus implies finite-dimensionality of the span in part (a).
  3. The informal Theorem A and the Abstract state the small-correlations condition only almost surely, whereas the formal Theorem 3.34(a) requires finite-dimensionality for every ω ∈ Ω, and its proof uses each fiber separately. No argument is given for passing from a full-measure subset to all of Ω, or for constructing a measurable Banach bundle on a full-measure subset. This matters for the AKLT application, where nondegeneracy conditions such as those in Lemma 4.7 and Theorem 4.11 hold only almost surely. The authors should either prove the almost-sure version of the structure theorem or explicitly restrict the formal statements to an invariant full-measure set with a measurable bundle.
  4. The quantitative Lyapunov bound in equation (4.37) has the wrong direction. From (4.38)–(4.39) the almost-sure Lyapunov exponent is Λ = E log cos(2Θ_0), which is negative. Since ∫_0^{π/4} |log cos(2x)| dx = (π/4) log 2 and f ≤ ||f||_∞, one obtains Λ = −∫ |log cos(2x)| f(x) dx ≥ −||f||_∞ (π/4) log 2, not ≤. The stated inequality is false; for example, if the density is concentrated near 0, Λ is close to 0, which is larger than the claimed negative upper bound. The almost-sure exponential decay and negativity of Λ remain valid, but the quantitative estimate must be corrected.
  5. Lemma 4.5, which supplies the Γ-map expansions and estimates used in the proof of purity (Theorem 4.11), the parent Hamiltonian analysis, and the index calculation, is deferred to the unpublished work [27] ('In Preparation'). Corollary 4.16 (deterministic spectrum) is likewise deferred to the preprint [72]. These are not standard external facts but load-bearing consequences specific to the present construction. The manuscript should include full proofs of these statements or replace the references with published, verifiable sources before the claims of Theorem C and Theorem D can be assessed.
minor comments (4)
  1. Equation (4.4) states that the transfer operators E_{a,•} are defined by (4.3) for a ∈ M_2, but the operators in (4.3) are defined for a ∈ M_3; this is a typo.
  2. The statement of Lemma 2.8 lists criteria (i) and (ii), but the proof refers to items (i) and (iii); the numbering should be made consistent.
  3. In inequality (3.29) the notation is inconsistent: the first line writes ϕ_{ϑ^{-n}ω} without the prime, while the second line writes ϕ'_{ϑ^nω}. The prime and the sign convention should be aligned with the definition of the aromatic average in (3.18).
  4. The Wolfram code line 'In [10}:= V = NullSpace[P]' appears to reference an undefined matrix P (probably L), and the displayed output is garbled; the appendix should be checked for consistency.

Circularity Check

2 steps flagged · score 4.0 of 10

Structure theorem is self-contained, but the AKLT example and deterministic-spectrum step delegate load-bearing proofs to same-author companion works.

  1. self citation load bearing [Section 4.1, proof of Lemma 4.5]
    "The proofs are straightforward extensions of Lemma 5.1 of [31], but we direct the reader to [27] for full details."

    Lemma 4.5 provides the estimates (4.12)-(4.14) that are used to prove purity (Theorem 4.11), the injectivity of the Gamma maps, and the Tasaki-index calculation via the expectation of the Affleck-Lieb twist in the state. Instead of a proof, the paper cites [27], an in-preparation work by Ekblad, Moreno-Nadales, Roon and Schenker, whose author list overlaps the present paper. The quoted step is load-bearing for the AKLT claims, and the cited work is not machine-checked, code-reproduced, or independently verified in the present manuscript. This is a self-citation used to carry a proof burden, although it is not a definitional equivalence: if [27] supplies the promised arguments, the example's derivation is independent of the theorem being proved.

  2. self citation load bearing [Section 4.2, proof of Lemma 4.14 and Corollary 4.16]
    "This essentially follows from the relation h_{j,j+1}(omega)=h_{j+k,j+k+1}(vartheta^k omega). For details, see our preprint [72]."

    The deterministic spectrum of the GNS Hamiltonian is the key input for Theorem 4.17 and for the contradiction in Theorem 4.19(d), where a nonzero gap would force uniform exponential clustering via [59]. The covariance relation and the deterministic spectrum are delegated to [72], a preprint by the same two authors. Corollary 4.16 is labeled an immediate consequence and again refers to [72] for details. This is a load-bearing reliance on a companion self-citation rather than a proof reproduced in the paper. It does not reduce the central classification theorem to its own input, but it makes a central part of the disordered-AKLT derivation depend on same-author work.

full rationale

The core structure theorem (Theorem 3.34) is not circular: the equivalence between finite-dimensional correlation spans and a transfer apparatus is proved by constructing the quotient bundle from the state itself and verifying the factorization, following FNW [31] rather than importing the conclusion. No parameter is fitted and no prediction is defined by the quantity it claims to predict; the IID AKLT state is explicitly built from transfer operators, so its small-correlations property and factorization hold by construction rather than by empirical fit. The main circularity-adjacent issue is the use of two same-author companion works at load-bearing points of the AKLT example: Lemma 4.5's key estimates are deferred to [27] (in preparation), and the covariance and deterministic-spectrum step is deferred to [72]. These raise the evidential burden and are flagged as load-bearing self-citations, but they are not equation-level reductions of a result to its own input. The skeptic's Proposition 3.12 counterexample, if correct, would be a mathematical gap in the Banach-bundle construction, not a circularity; the norm-identification issue is acknowledged in Remark 3.13 as resting on a specific representation of the dual functional. For that reason the score is 4 rather than 0 or 2: the central claim has independent content, but several load-bearing steps in the worked example are carried by same-author citations rather than by proofs in this paper.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

Everything in the paper is analytical and mathematical; no data are fitted. The main uncharged inputs are the compact-Hausdorff ergodic model, the all-omega finite-dimensionality condition, and the disorder distribution assumptions, plus reliance on several cited theorems including some from the authors' own companion works.

free parameters (2)
  • IID angle distribution for omega_0 = non-atomic, support [0, pi/4], P[omega_0 = 0] = 0, P[omega_0 < delta] > 0 for all delta > 0
    This distribution is an input chosen to make the rare-region gaplessness mechanism work; no independent evidence fixes it.
  • auxiliary sequence delta_n = delta_n decreases to 0
    Used in Theorem 4.19(c) to define a countable family of rare-region events whose intersection has probability one.
assumptions (6)
  • domain assumption The base probability space can be modeled as compact Hausdorff with an ergodic homeomorphism (Jewett-Krieger theorem).
    Assumption 1 and Lemma 2.3; needed to apply Banach bundle topology over the disorder space.
  • domain assumption Finite dimensionality of quotient fibers B^+_{x,omega} at every omega is sufficient for the Fell-Doran bundle construction.
    Theorem 3.16 and Remark 3.22; without it only a normed quotient bundle is available.
  • standard math Takeda's theorem guarantees a thermodynamic limit state from compatible local data.
    Theorem 2.23 is used to define nu_omega from finite-volume data.
  • standard math The exponential clustering theorem of Nachtergaele and Sims applies to pure gapped ground states.
    Lemma 4.18 uses this upper bound to prove gaplessness by contradiction.
  • domain assumption The random transfer maps phi_omega are strictly positive and satisfy the contraction estimates of [53, Theorem 2].
    Lemma 4.2 skips to [53, Assumption 1] and [27] for the full argument.
  • domain assumption The disorder distribution puts positive probability on arbitrarily small angles.
    Theorem 4.19(ii) requires P[omega_0 < delta] > 0 to create rare regions that force the gap to close.
invented entities (1)
  • quasi-gapped ground states
    purpose: A proposed class of disordered ground states with exponential clustering but vanishing bulk spectral gap.
    Section 1.3 introduces the concept conjecturally; no falsifiable observable prediction beyond clustering and gap behavior is defined.

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Cite this review

Pith. "Pith review of Finitely Correlated States Driven by Topological Dynamics." pith.science (2026). https://pith.science/paper/AIF2FOLI

@misc{pith2026250707287,
  author       = {Pith},
  title        = {Pith review of: Finitely Correlated States Driven by Topological Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AIF2FOLI}},
  note         = {Machine review of arXiv:2507.07287}
}
abstract

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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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Parent Hamiltonians of Ergodic Matrix Product States

    math-ph 2026-07 unverdicted novelty 7.0 of 10

    Ergodic MPS are the unique frustration-free ground states of parent Hamiltonians that may be infinite-range and need not be gapped.

Reference graph

Works this paper leans on

90 extracted references · 60 canonical work pages · cited by 1 Pith paper

  1. [27]

    Ekblad, E

    O. Ekblad, E. Moreno-Nadales, E. B. Roon, and J. H. Schenker. Parent hamiltonians for ergodic matrix product states.In Preparation

    work page

  2. [72]

    E. B. Roon and J. H. Schenker. Disordered ground states of ergodic quantum spin systems, 2026. arXiv:2603.19475

    arXiv 2026

  3. [1]

    Abanin, D. A. and De Roeck, W. and Huveneers, F. Theory of many-body localization in periodically driven systems.Ann. Physics, 372:1–11, 2016

    2016

  4. [2]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki. Valence bond ground states in isotropic quantum antiferromagnets.Communications in Mathematical Physics, 115(3):477–528, 1988

    1988

  5. [3]

    Aizenman and G

    M. Aizenman and G. M. Graf. Localization bounds for an electron gas.J. Phys. A, 31(32):6783–6806, 1998

    1998

  6. [4]

    Aizenman and S

    M. Aizenman and S. Warzel.Random operators, volume 168 ofGraduate Studies in Mathematics. Amer- ican Mathematical Society, Providence, RI, 2015. Disorder effects on quantum spectra and dynamics. 56

    2015

  7. [5]

    Beaudry, M

    A. Beaudry, M. Hermele, M. J. Pflaum, M. Qi, D. D. Spiegel, and D. T. Stephen. A classifying space for phases of matrix product states, 2025. arXiv:2501.14241

    arXiv 2025

  8. [6]

    Bhatia.Matrix analysis, volume 169 ofGraduate Texts in Mathematics

    R. Bhatia.Matrix analysis, volume 169 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1997

    1997

Show all 90 references
  1. [7]

    Bhatia.Positive definite matrices

    R. Bhatia.Positive definite matrices. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2007

    2007

  2. [8]

    Bols and W

    A. Bols and W. De Roeck. Asymptotic localization in the Bose-Hubbard model.J. Math. Phys., 59(2):021901, 28, 2018

    2018

  3. [9]

    A. Bols, W. De Roeck, M. De Wilde, and B. de O. Carvalho. Classification of locality preserving symmetries on spin chains, 2025. arXiv:2503.15088

    2025 arXiv

  4. [10]

    Bougerol and J

    P. Bougerol and J. Lacroix.Products of random matrices with applications to Schr¨ odinger operators, volume 8 ofProgress in Probability and Statistics. Birkh¨ auser Boston, Inc., Boston, MA, 1985

    1985

  5. [11]

    Bratteli and D

    O. Bratteli and D. W. Robinson.Operator algebras and quantum statistical mechanics. 1. Texts and Monographs in Physics. Springer-Verlag, New York, second edition, 1987.C ∗- andW ∗-algebras, sym- metry groups, decomposition of states

    1987

  6. [12]

    J. C. Bridgeman and C. T. Chubb. Hand-waving and interpretive dance: an introductory course on tensor networks.Journal of Physics A: Mathematical and Theoretical, 50(22):223001, may 2017

    2017

  7. [13]

    N. P. Brown and N. Ozawa.C ∗-algebras and finite-dimensional approximations, volume 88 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008

    2008

  8. [14]

    Carmona and J

    R. Carmona and J. Lacroix.Spectral theory of random Schr¨ odinger operators. Probability and its Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1990

    1990

  9. [15]

    L. Chen, R. J. Garcia, K. Bu, and A. Jaffe. Magic of random matrix product states.Phys. Rev. B, 109:174207, May 2024

    2024

  10. [16]

    Chirvasitu

    A. Chirvasitu. Small banach bundles and modules, 2024

    2024

  11. [17]

    J. I. Cirac, D. P´ erez-Garc ´ ıa, N. Schuch, and F. Verstraete. Matrix product states and projected entangled pair states: Concepts, symmetries, theorems.Rev. Mod. Phys., 93:045003, Dec 2021

    2021

  12. [18]

    Conway.A course in functional analysis, volume 96 ofGraduate Texts in Mathematics

    J. Conway.A course in functional analysis, volume 96 ofGraduate Texts in Mathematics. Springer- Verlag, New York, second edition, 1990

    1990

  13. [19]

    Conway.A course in operator theory, volume 21 ofGraduate Studies in Mathematics

    J. Conway.A course in operator theory, volume 21 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000

    2000

  14. [20]

    de Oliveira Carvalho, W

    B. de Oliveira Carvalho, W. De Roeck, and T. Jappens. Classification of symmetry protected states of quantum spin chains for continuous symmetry groups, 2024. arXiv:2409.01112

    2024 arXiv

  15. [21]

    De Roeck, F

    W. De Roeck, F. Huveneers, B. Meeus, and A. O. Pro´ sniak. Rigorous and simple results on very slow thermalization, or quasi-localization, of the disordered quantum chain.Phys. A, 631:Paper No. 129245, 20, 2023

    2023

  16. [22]

    Dixmier and A

    J. Dixmier and A. Douady. Champs continus d’espaces hilbertiens et deC ∗-alg` ebres.Bull. Soc. Math. France, 91:227–284, 1963

    1963

  17. [23]

    M. J. Dupr´ e and R. M. Gillette.Banach bundles, Banach modules and automorphisms ofC ∗-algebras, volume 92 ofResearch Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1983

    1983

  18. [24]

    Durrett.Probability—theory and examples, volume 49 ofCambridge Series in Statistical and Proba- bilistic Mathematics

    R. Durrett.Probability—theory and examples, volume 49 ofCambridge Series in Statistical and Proba- bilistic Mathematics. Cambridge University Press, Cambridge, fifth edition, 2019. 57

    2019

  19. [25]

    E. G. Effros and Z-J. Ruan.Operator spaces, volume 23 ofLondon Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York, 2000

    2000

  20. [26]

    Eisner, B

    T. Eisner, B. Farkas, M. Haase, and R. Nagel.Operator theoretic aspects of ergodic theory, volume 272 ofGraduate Texts in Mathematics. Springer, Cham, 2015

    2015

  21. [28]

    Fanizza, J

    M. Fanizza, J. Lumbreras, and A. Winter. Quantum theory in finite dimension cannot explain every general process with finite memory.Communications in Mathematical Physics, 405(2):50, 2024

    2024

  22. [29]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner. Abundance of translation invariant pure states on quantum spin chains.Lett. Math. Phys., 25(3):249–258, 1992

    1992

  23. [30]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner. Entropy estimates for finitely correlated states.Ann. Inst. H. Poincar´ e Phys. Th´ eor., 57(3):259–277, 1992

    1992

  24. [31]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner. Finitely correlated states on quantum spin chains. Communications in Mathematical Physics, 144(3):443–490, 1992

    1992

  25. [32]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner. Finitely correlated pure states.J. Funct. Anal., 120(2):511–534, 1994

    1994

  26. [33]

    J. M. G. Fell and R. S. Doran.Representations of ∗-algebras, locally compact groups, and Banach ∗- algebraic bundles. Vol. 1, volume 125 ofPure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1988. Basic representation theory of groups and algebras

    1988

  27. [34]

    Fern´ andez-Gonz´ alez, N

    C. Fern´ andez-Gonz´ alez, N. Schuch, M. M. Wolf, J. I. Cirac, and D. P´ erez-Garc ´ ıa. Frustration free gapless Hamiltonians for matrix product states.Comm. Math. Phys., 333(1):299–333, 2015

    2015

  28. [35]

    Gierz.Bundles of topological vector spaces and their duality, volume 57 ofQueen ’s Papers in Pure and Applied Mathematics

    G. Gierz.Bundles of topological vector spaces and their duality, volume 57 ofQueen ’s Papers in Pure and Applied Mathematics. Springer-Verlag, Berlin-New York, 1982. With an appendix by the author and Klaus Keimel, Lecture Notes in Mathematics, 955

    1982

  29. [36]

    A. E. Gutman. Banach bundles in the theory of lattice-normed spaces. I. Continuous Banach bundles. Siberian Adv. Math., 3(3):1–55, 1993. Siberian Advances in Mathematics

    1993

  30. [37]

    A. E. Gutman and A. V. Koptev. On the notion of the dual of a Banach bundle.Siberian Adv. Math., 9(1):46–98, 1999

    1999

  31. [38]

    M. Hamana. Injective envelopes of operator systems.Publ. Res. Inst. Math. Sci., 15(3):773–785, 1979

    1979

  32. [39]

    Hamza, R

    E. Hamza, R. Sims, and G. Stolz. Dynamical localization in disordered quantum spin systems.Comm. Math. Phys., 315(1):215–239, 2012

    2012

  33. [40]

    Hayden, S

    P. Hayden, S. Nezami, X-L. Qi, N. Thomas, M. Walter, and Z. Yang. Holographic duality from random tensor networks.Journal of High Energy Physics, 2016(11):9, 2016

    2016

  34. [41]

    Heikkinen

    N. Heikkinen. The canonical forms of matrix product states in infinite-dimensional hilbert spaces, 2025. arXiv:2502.12934

    2025 arXiv

  35. [42]

    Hu and I

    M. Hu and I. Nechita. Canonical partial ordering from min-cuts and quantum entanglement in random tensor networks, 2025

    2025

  36. [43]

    Husemoller.Fibre bundles, volume 20 ofGraduate Texts in Mathematics

    D. Husemoller.Fibre bundles, volume 20 ofGraduate Texts in Mathematics. Springer-Verlag, New York, third edition, 1994

    1994

  37. [44]

    Jauslin and M

    I. Jauslin and M. Lemm. Random translation-invariant Hamiltonians and their spectral gaps.Quantum, 6:790, September 2022. 58

    2022

  38. [45]

    R. V. Kadison and J. R. Ringrose.Fundamentals of the theory of operator algebras. Vol. I, volume 15 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. Elementary theory, Reprint of the 1983 original

    1997

  39. [46]

    Kapustin, N

    A. Kapustin, N. Sopenko, and B. Yang. A classification of invertible phases of bosonic quantum lattice systems in one dimension.J. Math. Phys., 62(8):Paper No. 081901, 16, 2021

    2021

  40. [47]

    A. S. Kavruk, V. I. Paulsen, I. G. Todorov, and M. Tomforde. Quotients, exactness, and nuclearity in the operator system category.Adv. Math., 235:321–360, 2013

    2013

  41. [48]

    Kirchberg and S

    E. Kirchberg and S. Wassermann.C ∗-algebras generated by operator systems.J. Funct. Anal., 155(2):324–351, 1998

    1998

  42. [49]

    Lancien and D

    C. Lancien and D. P´ erez-Garc ´ ıa. Correlation length in random MPS and PEPS.Ann. Henri Poincar´ e, 23(1):141–222, 2022

    2022

  43. [50]

    T. Matsui. The split property and the symmetry breaking of the quantum spin chain.Comm. Math. Phys., 218(2):393–416, 2001

    2001

  44. [51]

    T. Matsui. Boundedness of entanglement entropy and split property of quantum spin chains.Rev. Math. Phys., 25(9):1350017, 31, 2013

    2013

  45. [52]

    Movassagh and J

    R. Movassagh and J. Schenker. Theory of ergodic quantum processes.Phys. Rev. X, 11:041001, Oct 2021

    2021

  46. [53]

    Movassagh and J

    R. Movassagh and J. Schenker. An ergodic theorem for quantum processes with applications to matrix product states.Comm. Math. Phys., 395(3):1175–1196, 2022

    2022

  47. [54]

    J. R. Munkres.Topology. Prentice Hall, Inc., Upper Saddle River, NJ, second edition, 2000

    2000

  48. [55]

    G. J. Murphy.C ∗-algebras and operator theory. Academic Press, Inc., Boston, MA, 1990

    1990

  49. [56]

    Nachtergaele

    B. Nachtergaele. The spectral gap for some spin chains with discrete symmetry breaking.Comm. Math. Phys., 175(3):565–606, 1996

    1996

  50. [57]

    Nachtergaele, Y

    B. Nachtergaele, Y. Ogata, and R. Sims. Propagation of correlations in quantum lattice systems.J. Stat. Phys., 124(1):1–13, 2006

    2006

  51. [58]

    Nachtergaele and J

    B. Nachtergaele and J. Reschke. Slow propagation in some disordered quantum spin chains.J. Stat. Phys., 182(1):Paper No. 12, 28, 2021

    2021

  52. [59]

    Nachtergaele and R

    B. Nachtergaele and R. Sims. Lieb-Robinson bounds and the exponential clustering theorem.Comm. Math. Phys., 265(1):119–130, 2006

    2006

  53. [60]

    Nelson and E

    B. Nelson and E. B. Roon. Ergodic quantum processes on finite von Neumann algebras.J. Funct. Anal., 287(4):Paper No. 110485, 63, 2024

    2024

  54. [61]

    Y. Ogata. AZ 2-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains.Comm. Math. Phys., 374(2):705–734, 2020

    2020

  55. [62]

    Y. Ogata. AZ 2-index of symmetry protected topological phases with reflection symmetry for quantum spin chains.Comm. Math. Phys., 385(3):1245–1272, 2021

    2021

  56. [63]

    Y. Ogata. Classification of symmetry protected topological phases in quantum spin chains. InCurrent developments in mathematics 2020, pages 41–104. Int. Press, Boston, MA, 2022

    2020

  57. [64]

    Y. Ogata. Classification of gapped ground state phases in quantum spin systems. InICM—International Congress of Mathematicians. Vol. 5. Sections 9–11, pages 4142–4161. EMS Press, Berlin, [2023]©2023

    2023

  58. [65]

    Paulsen.Completely bounded maps and operator algebras, volume 78 ofCambridge Studies in Ad- vanced Mathematics

    V. Paulsen.Completely bounded maps and operator algebras, volume 78 ofCambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 2002. 59

    2002

  59. [66]

    Perez-Garcia, F

    D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix product state representations. Quantum Inf. Comput., 7(5-6):401–430, 2007

    2007

  60. [67]

    P´ erez-Garc ´ ıa, M

    D. P´ erez-Garc ´ ıa, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac. String order and symmetries in quantum spin lattices.Phys. Rev. Lett., 100:167202, Apr 2008

    2008

  61. [68]

    Petersen.Ergodic theory, volume 2 ofCambridge Studies in Advanced Mathematics

    K. Petersen.Ergodic theory, volume 2 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1989. Corrected reprint of the 1983 original

    1989

  62. [69]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa. Symmetry protection of topological phases in one-dimensional quantum spin systems.Phys. Rev. B, 85:075125, Feb 2012

    2012

  63. [70]

    Pollmann, A

    F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa. Entanglement spectrum of a topological phase in one dimension.Phys. Rev. B, 81:064439, Feb 2010

    2010

  64. [71]

    Resende and J

    P. Resende and J. P. Santos. Open quotients of trivial vector bundles.Topology Appl., 224:19–47, 2017

    2017

  65. [73]

    Rudin.Functional Analysis

    W. Rudin.Functional Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991

    1991

  66. [74]

    N. Sopenko. An index for two-dimensional SPT states.J. Math. Phys., 62(11):Paper No. 111901, 13, 2021

    2021

  67. [75]

    A. Souissi. Matrix product states as observations of entangled hidden markov models, 2025. arXiv: 2502.12641

    2025 arXiv

  68. [76]

    Souissi and A

    A. Souissi and A. Andolsi. A hidden quantum markov model framework for entanglement and topological order in the aklt chain, 2025. arXiv:2512.18642

    2025

  69. [77]

    Souissi and A

    A. Souissi and A. Andolsi. Aklt state is indeed the observation process of a causal hidden quantum markov model, 2026. arXiv:2605.24431

    2026 arXiv

  70. [78]

    D. D. Spiegel.A C*-Algebraic Approach to Parametrized Quantum Spin Systems and Their Phases in One Spatial Dimension. PhD thesis, University of Colorado at Boulder, 2023

    2023

  71. [79]

    Stollmann.Caught by Disorder: Bound states in random media

    P. Stollmann.Caught by Disorder: Bound states in random media. Progress in Mathematical Physics. Birkh¨ auser Boston, MA, 2001

    2001

  72. [80]

    G. Stolz. Aspects of the mathematical theory of disordered quantum spin chains. InAnalytic trends in mathematical physics, volume 741 ofContemp. Math., pages 163–197. Amer. Math. Soc., [Providence], RI, [2020]©2020

    2020

  73. [81]

    Z. Takeda. Inductive limit and infinite direct product of operator algebras.Tohoku Math. J. (2), 7:67–86, 1955

    1955

  74. [82]

    Takesaki.Theory of operator algebras

    M. Takesaki.Theory of operator algebras. I, volume 124 ofEncyclopaedia of Mathematical Sci- ences. Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non- commutative Geometry, 5

    1979

  75. [83]

    H. Tasaki. Topological phase transition and𭟋 2 index fors= 1 quantum spin chains.Phys. Rev. Lett., 121:140604, Oct 2018

    2018

  76. [84]

    Tasaki.Physics and mathematics of quantum many-body systems

    H. Tasaki.Physics and mathematics of quantum many-body systems. Graduate Texts in Physics. Springer, Cham, [2020]©2020

    2020

  77. [85]

    H. Tasaki. Rigorous index theory for one-dimensional interacting topological insulators.J. Math. Phys., 64(4):Paper No. 041903, 19, 2023. 60

    2023

  78. [86]

    The ground state of the s=1 antiferromagnetic heisenberg chain is topologically nontrivial if gapped, 2024

    Hal Tasaki. The ground state of the s=1 antiferromagnetic heisenberg chain is topologically nontrivial if gapped, 2024

    2024

  79. [87]

    van Luijk, R

    L. van Luijk, R. Schwonnek, A. Stottmeister, and R. F. Werner. The Schmidt rank for the commuting operator framework.Comm. Math. Phys., 405(7):Paper No. 152, 46, 2024

    2024

  80. [88]

    van Luijk, A

    L. van Luijk, A. Stottmeister, and H. Wilming. The large-scale structure of entanglement in quantum many-body systems, 2025. arXiv:2503.03833

    2025 arXiv

  81. [89]

    Verstraete and J

    F. Verstraete and J. I. Cirac. Matrix product states represent ground states faithfully.Phys. Rev. B, 73:094423, Mar 2006

    2006

  82. [90]

    G. Vidal. Efficient classical simulation of slightly entangled quantum computations.Phys. Rev. Lett., 91:147902, Oct 2003. 61

    2003

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