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Dimerization in $O(n)$-invariant quantum spin chains

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For n large enough, the one-dimensional O(n)-invariant quantum spin chain has two distinct ground states, each a one-site shift of the other, with exponentially decaying correlations.

desk verdict Theorem 2.1 as stated can't be right for small v, but the core result is new and the fix looks local. read the letter →

arxiv 2506.06103 v1 pith:SFDKWVRL submitted 2025-06-06 math-ph math.MPmath.PR

J. E. Björnberg , K. Ryan This is my paper · ORCID
classification math-phmath.MPmath.PR MSC 82B2082B2660K3582B10
keywords dimerizationO(n)quantumspinchainsrandomloopmodelgroundstatesexponentialdecayofcorrelationsrepairmapPoissonprocesstranslationsymmetrybreaking
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

The paper proves that the one-dimensional O(n)-invariant quantum spin chain with Hamiltonian $H=-\sum_{xy}[uT_{xy}+(1-u)Q_{xy}]$, for fixed $u\in[0,1)$ and $n$ sufficiently large, has two distinct infinite-volume ground states. The states are translations of each other by one lattice site, each is invariant under translations by two sites, and correlations between local observables decay exponentially in both spatial and time separation. This establishes dimerization throughout $[0,1)$ for large $n$, going beyond the exactly solvable point $u=0$ and its neighbourhood where dimerization was previously known. The proof works through a probabilistic loop representation of the quantum system, adapting a repair-map method developed for loop O(n) models on the hexagonal lattice.

What carries the argument

The load-bearing object is the repair map. A trivial loop is a loop that visits exactly two double-bars spanning the same edge; it is called small if its vertical height is below $1/(\kappa n)$. The repair map takes a configuration whose 'outside' $O$ contains no small loops and remaps it to one dominated by small primal loops, by shifting dual clusters left one step and turning every crossing in $O$ into a double-bar. This increases the loop count by at least $|\omega_{\mathrm{ex}}|/4$, while the preimage count is at most $4^{|\bar{\omega}_{\mathrm{out}}|}$. The argument uses a block discretisation (blocks of height $h/n$ spanning one primal and one dual column) to control the entropy of possible outsides, with the number of connected block-outsides of $m$ blocks bounded by $16^m$. This machinery proves Proposition 3.1, from which the exponential perimeter bound (Theorem 2.1) and the convergence of infinite-volume measures (Theorem 2.2) are derived.

What would settle it

Enumerate, by transfer-matrix or exhaustive methods, the number of connected subgraphs of size $m$ in the block adjacency graph that contain a fixed block and surround a fixed point; if the growth exceeds $m\cdot16^m$ for any $m$, the repair-map entropy bound fails and the proof of Theorem 2.1, hence of Theorem 1.1, collapses.

Watch

Extended reading notes

Core claim

The paper establishes that dimerization occurs in the one-dimensional O(n)-invariant quantum spin chain for every fixed $u\in[0,1)$ once $n$ is large enough. Concretely, Theorem 1.1 asserts the existence of two distinct infinite-volume ground states $\langle\cdot\rangle_1$ and $\langle\cdot\rangle_2$, obtained as limits from even and odd finite volumes (with either Gibbs or seeded boundary conditions), which are $2\mathbb{Z}$-invariant, related by a one-site translation, and satisfy exponential decay of truncated correlations in both spatial and time separations. In the loop representation, Theorem 2.2 states that the primal and dual finite-volume loop measures converge to two distinct infinite-volume Gibbs measures $P^1_{n,u}$ and $P^2_{n,u}$, while Theorem 2.1 gives the uniform exponential perimeter decay of the defect component around any point, which is the estimate that transfers these results back to the quantum system.

Load-bearing premise

Everything depends on the assertion, stated without a full proof, that the number of connected block-outsides of $m$ blocks surrounding a point grows at most like $17^m$; this count underpins the exponential perimeter decay of the defect component in Theorem 2.1.

Editorial extensions

If this is right

  • For every fixed $u\in[0,1)$, once $n$ exceeds the threshold $n_0(u)$, the quantum chain has two extremal ground states rather than a unique one, so one-step translation symmetry is spontaneously broken.
  • Both limiting states satisfy exponential space-time clustering, so local perturbations in one ground state have no long-range influence.
  • The same loop-model estimates produce, for large $n$, two distinct periodic Gibbs measures in the discrete mirror model, a non-quantum analogue of dimerization.
  • The range of $u$ covered is the whole segment from $u=0$ up to (but not including) $u=1$, where the ferromagnetic boundary lies; previous rigorous results were confined to $u=0$ and a neighbourhood of it.

Reading between the lines

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

  • Inference: The paper's threshold $n_0(u)$ diverges as $u\to1$; a sharper conjecture implicit in the physics literature is that dimerization holds already for $n\ge3$ on the same interval, but the present method is too weak to settle that.
  • Inference: The repair-map construction should adapt to other planar loop models with large loop weight, where the same block-outside entropy control could prove analogous ordering transitions.
  • Inference: A direct enumeration of the counting bound (3.49) would either certify or falsify the weakest step of the proof, and could be carried out by transfer-matrix or generating-function methods.
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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

2 major / 3 minor

Summary. This paper proves dimerization in O(n)-invariant quantum spin chains on Z with Hamiltonian H_Λ = -Σ_xy [u T_xy + (1-u) Q_xy] for fixed u ∈ [0,1) and n > n_0(u). Two distinct infinite-volume ground states are constructed as limits of periodic and seeded finite-volume states with even/odd L; the states are 2Z-periodic shifts of one another and have exponential space-time clustering. The proof works through a continuous loop model: Theorem 2.1 gives exponential perimeter decay for defect components, Theorem 2.2 gives convergence to two distinct invariant Gibbs measures, and Lemma 2.5 together with the loop representation transfers these facts to the quantum correlation functions. The method adapts the repair-map strategy of Duminil-Copin, Peled, Samotij and Spinka from the hexagonal-lattice loop O(n) model to a Poisson-process-based loop model.

Significance. The result is significant: it establishes a long-expected phase in a family of O(n)-invariant quantum spin chains for large n, going beyond the previously known u=0 case and the perturbative cluster-expansion result. The proof is probabilistic and self-contained; the parameters u, n, κ are explicit, no constants are fitted, and the loop-model theorems hold for non-integer n and also yield a mirror-model analogue. If the gap in Theorem 2.1 pointed out below is repaired, the paper will be a valuable contribution to the rigorous theory of dimerization and of loop O(n) models.

major comments (2)
  1. [Theorem 2.1 / §3.2, Eqs. (3.50)–(3.51)] Theorem 2.1 as stated is false. For every configuration, perim(C_κ(x_0)) is at least 4: the boundary γ is a simple closed curve composed of horizontal segments of length 2 and vertical segments in dual columns, so its Euclidean length is at least 4+2h > 4 if it encloses x_0, and if no small-loop circuit encloses x_0 then the component is the whole primal domain, whose perimeter is again at least 4. Consequently P[perim(C)>v] = 1 for all v < 4, contradicting (2.3) for any C>0. The proof obtains (3.50) only for v>n by substituting w=v/n, and the step 'By adjusting the constants in the exponents... for v>1' is not justified: for v ∈ (1,n] the hypothesis w>1 of (3.47)–(3.48) fails, and no adjustment of constants can make e^{-Cv} dominate a probability equal to 1 for small v while retaining decay. The same defect propagates to Corollary 3.8 (for small v) and to Lemma 2.5 (for small d(A,B^c)). This appears locally fixable by restating the perimeter bounds for v ≥ v_0 with an absolute v_0 ≥ 4 and by using a prefactor or a large-distance condition in Lemma 2.5, but the current statements are incorrect.
  2. [§3.2, Eq. (3.49)] The proof of Theorem 2.1 reduces the defect-component estimate to the repair-map lemmas by replacing the block-counting bound (3.17) with (3.49) for block-outsides that surround x_0 rather than contain it. The derivation of (3.49) is compressed into a single sentence ('by counting according to which is the rightmost block along the x-axis'), and the following sentence asserts that the analogues of Lemmas 3.2 and 3.3 are 'proved exactly as before'. Since this counting feeds directly into (3.47)–(3.48) and hence into the total-variation bound of Lemma 2.5, the authors should supply a complete argument showing that a bound of the form m 16^m, or a comparable one, holds uniformly for outsides surrounding x_0, and should verify that the large-deviation and repair-map steps are unchanged when the block-outside is not anchored at x_0.
minor comments (3)
  1. [Abstract] The abstract contains the typo 'eachother'; it should read 'each other'.
  2. [Corollary 3.8] The statement defines n_0 = n_0(u, ε) although ε has not been introduced; the intended dependence is presumably n_0(u, κ), matching the statement of Theorem 2.1.
  3. [Theorem 2.1 and Corollary 3.8] In Theorem 2.1 the constant C is allowed to depend on n, while n_0 depends on u and κ; the corresponding dependence in Corollary 3.8 should be stated with the same conventions to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the loop-model derivation is self-contained, with background self-citations that are not load-bearing.

full rationale

The paper's central claim (Theorem 1.1) is derived from the loop-model Theorems 2.1 and 2.2, which are proved from the model definition via the repair-map argument (Section 3). No parameter is fitted to data, no observable is defined in terms of the quantity it predicts, and no load-bearing step invokes the authors' prior work as the source of the main result. The citations to the authors' own work ([8], [9]) appear only as background on existing dimerization results and planned work in other dimensions; they are not used to justify the new dimerization statement. The DPSS method [15] is an external technique that is adapted, not imported as an unverified ansatz. The proof of Theorem 2.1 does contain a possibly unjustified extension from v>n to v>1 in (3.50)-(3.51), but that is a correctness gap in the proof as written, not a circularity: the argument does not assume the desired exponential decay or define the model in terms of it. Since no reduction of a claimed result to its own inputs by construction, fit, or self-citation chain is present, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the loop representation, which is standard, and on a set of standard probabilistic tools (Mecke, large deviations, stochastic domination, subgraph counting). The proof introduces auxiliary objects such as the repair map, small loops, gardens, and block-outsides, but these are proof devices, not new physical entities. No free parameters are fitted to data.

assumptions (6)
  • domain assumption Loop representation of the quantum Gibbs state (Eq. (2.1)) and seeded states (Lemma 2.4)
    Bridges the quantum Hamiltonian to the Poissonian loop model; standard result (Tóth, Aizenman-Nachtergaele, Ueltschi) but load-bearing for transferring loop-model convergence to quantum ground states.
  • domain assumption Poisson process with nonnegative intensities u and 1-u for crosses and double bars (Section 2.1)
    Requires u in [0,1), which is exactly the segment of the phase diagram studied; outside this range the loop measure is signed.
  • standard math Mecke's formula for Poisson processes (Lemma 2.6)
    Used throughout Section 3 to condition on exact locations of links, e.g., in Lemmas 3.2-3.4 and Lemma 2.5.
  • standard math Large deviation estimates for Poisson and binomial variables (Lemma 3.7)
    Provides the exponential bounds used in Lemmas 3.2 and 3.3 to control bad events.
  • standard math Stochastic domination by a Poisson process of intensity n (Lemma 3.5)
    Via Georgii-Küneth (1997); reduces link-count bounds to Poisson tails in several places.
  • standard math Counting of connected subgraphs of bounded-degree graphs (Eq. (3.17), from Bollobás)
    Controls the number of possible block-outsides in the repair-map entropy estimates.

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Cite this review

Pith. "Pith review of Dimerization in $O(n)$-invariant quantum spin chains." pith.science (2026). https://pith.science/paper/SFDKWVRL

@misc{pith2026250606103,
  author       = {Pith},
  title        = {Pith review of: Dimerization in $O(n)$-invariant quantum spin chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SFDKWVRL}},
  note         = {Machine review of arXiv:2506.06103}
}
abstract

We establish dimerization in $O(n)$-invariant quantum spin chains with big enough $n$, in a large part of the phase diagram where this result is expected. This includes identifying two distinct ground states which are translations of one unit of eachother, and which both have exponentially decaying correlations. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop $O(n)$ models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.

Figures

Figures reproduced from arXiv: 2506.06103 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Mirror configuration in Z 2 with loops. The boundary condition favours loops surrounding black faces. The parameters of the model are numbers pv, ph, p∅ ∈ [0, 1] satisfying pv + ph + p∅ = 1, as well as n > 0. Here n plays the same role as the spin-parameter in the quantum system, but is not restricted to be an integer. A mirror configuration is chosen at random, with probability (1.10) P mir Λ,n(ξ) ∝ p #{x∈Λ∶ξx=v} v… view at source ↗
Figure 3
Figure 3. Rescaled mirror configuration with vertical distances scaled by ε. In the limit ε → 0 we obtain the continuous loop-model (2.1) which is a probabilistic representation of the quantum spin system (1.6); horizontal mirrors become double bars and missing mirrors become crosses . (Only part of one loop is drawn in this picture.) 1.4. Organization of the paper. In Section 2 we define the probabilistic representation of t… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Two pictures of configurations ω in dimension d = 1. On the left, in ΛL × [0, β] (where L = 4) with periodic boundary condition, and on the right in a primal domain DΓ with primal boundary condition. The links ( and ) create loops, which in the left picture wrap around…
Figure 5
Figure 5. Figure 5: Illustration of the connected component P(ω) of small loops (drawn turquoise) adjacent to the boundary Γ of a primal domain DΓ, as well as C(x0) = Dγ with its boundary γ drawn dashed. Long loops are drawn off-blue. The rightmost loop in C(x0) is a tall, trivial loop. S…
Figure 6
Figure 6. Figure 6: Illustration of the event U. In red, γ is the innermost circuit of small loops surrounding Λ(A) × {0}, and the links (all ) included in γ form the configuration ξ. The inside and outside of γ define two primal domains DA ξ and DB ξ . Since DB is at distance at least R/…
Figure 7
Figure 7. Figure 7: A configuration ω in a primal domain DΓ. Long loops are drawn off-blue, small primal loops green, and small dual loops orange. Primal clusters are shaded green while dual clusters are shaded orange. The small dual cluster in the lower left coincides with the support o …
Figure 8
Figure 8. Figure 8: The repaired version ω¯ of the configuration ω in [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: The primal domain B ⊆ D1 ∩ D2 (the domains D1 ∩ D2 not depicted) as well as the component K ′ B , with ω1 drawn green and ω2 drawn orange. Long loops not part of K ′ B , as well as most small loops, are not depicted. The exception is on the right part of the picture, w…
Figure 10
Figure 10. Figure 10: A domain D and a link-configuration τ outside D, including some links crossing ∂vD. The configuration τ defines a partial pairing of ∂τD, the latter illustrated using red dots. Two points of ∂τD, on the top boundary of D (highlighted), are unpaired. that for such k, t…
Figure 11
Figure 11. Figure 11: Illustration of part of a sample of PL,∞. The rectangle RL ⊇ A is exponentially likely (in L) to be surrounded by a circuit of small primal loops, and since PL,∞ is a Gibbs measure, the conditional distribution inside that circuit is P 1 Dξ,n,u. The marginal distribut…

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Cited by 1 Pith paper

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

  1. Exponential decay in $O(n)$-invariant quantum spin systems

    math-ph 2025-06 conditional novelty 7.0 of 10

    For large n, spin-spin correlations in O(n)-invariant reflection-positive quantum spin models decay exponentially in space-time distance for every d ≥ 1.

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Works this paper leans on

43 extracted references · 35 canonical work pages · cited by 1 Pith paper

  1. [1]

    Affleck, T

    I. Affleck, T. Kennedy, E. Lieb, H. Tasaki,Rigorous Results on Valence-Bond Ground States in Antiferro- magnets, Phys. Rev. Lett. 59 (1987), 799-802

    work page 1987

  2. [2]

    Affleck, T

    I. Affleck, T. Kennedy, E. Lieb, H. Tasaki,Valence Bond Ground States in Isotropic Quantum Antiferromag- nets, Comm. Math. Phys. 115 (1987), 477-528

    work page 1987

  3. [3]

    Affleck and E

    I. Affleck and E. Lieb,A Proof of Part of Haldane’s Conjecture on Spin Chains, Lett. Math. Phys. 12 (1986), 57

    work page 1986

  4. [4]

    Aizenman, H

    M. Aizenman, H. Duminil-Copin and S. Warzel,Dimerization and Néel order in different quantum spin chains through a shared loop representation, Ann. Henri Poincaré 21 (2020), 2737–2774

    work page 2020

  5. [5]

    Aizenman and B

    M. Aizenman and B. Nachtergaele,Geometric aspects of quantum spin states, Comm. Math. Phys. 164 (1994), 17–63

    1994

  6. [6]

    Alcaraz, S

    F. Alcaraz, S. Salinas and W. Wreszinski,Anisotropic Ferromagnetic Quantum Domains, Phys. Rev. Lett. 75 (1995), 930-933

    work page 1995

  7. [7]

    Araki and T

    H. Araki and T. Matsui,Ground states of the XY-model, Comm. Math. Phys. 101 (1985), 213–245

    work page 1985

  8. [8]

    J. E. Björnberg, P. Mühlbacher, B. Nachtergaele and D. Ueltschi,Dimerization in quantum spin chains with O(n)symmetry, Comm. Math. Phys. (2021)

    work page 2021

Show all 43 references
  1. [9]

    J. E. Björnberg and K. Ryan,Exponential decay inO(n)-invariant quantum spin systems, in preparation (2025)

    2025

  2. [10]

    Bollobás,The art of mathematics

    B. Bollobás,The art of mathematics. Coffee time in Memphis.Cambridge University Press, New York, 2006, MR 2285090 (2007h:00005)

    2006

  3. [11]

    D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes, Volume I: Elementary Theory and Methods, Probability and Its Applications, Springer, 2008

    2008

  4. [12]

    D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure, Probability and Its Applications, Springer, 2008

    2008

  5. [13]

    Duminil-Copin, K

    H. Duminil-Copin, K. Kozlowski, D. Krachun, I. Manolescu and M. Oulamara,Rotational invariance in critical planar lattice models, arxiv:2012.11672

    2012 arXiv

  6. [14]

    Duminil-Copin, J-H

    H. Duminil-Copin, J-H. Li and I. Manolescu,Universality for the random-cluster model on isoradial graphs, Elec. Jour. Prob. 0 (2016), 1-70. 32 J. E. BJÖRNBERG AND K. RYAN

    2016

  7. [15]

    Duminil-Copin, R

    H. Duminil-Copin, R. Peled, W. Samotij, Y. Spinka,Exponential Decay of Loop Lengths in the Loop O(n) Model with Large n, Comm. Math. Phys. 349, pages 777–817, (2017)

    2017

  8. [16]

    Duminil-Copin, V

    H. Duminil-Copin, V. Sidoravicius and V. Tassion,Continuity of the Phase Transition for Planar Random- Cluster and Potts Models with1≤q≤4, Comm. Math. Phys. 349 (2017), 47–107

    2017

  9. [17]

    Fáth and J

    G. Fáth and J. Sólyom,Isotropic spin-1 chain with twisted boundary condition, Phys. Rev. B. 47 (1993), 872-881

    1993

  10. [18]

    Fröhlich, C

    J. Fröhlich, C. Pfister,On the Absence of Spontaneous Symmetry Breaking and of Crystalline Ordering in Two-Dimensional Systems, Comm. Math. Phys. 81 (1981), 277–298

    1981

  11. [19]

    Georgii and T

    H.-O. Georgii and T. Küneth.Stochastic comparison of point random fields.Journal of Applied Probability, 34(4): 868-881, 1997

    1997

  12. [20]

    Gottstein and R

    C.-T. Gottstein and R. Werner,Ground states of the infinite q-deformed Heisenberg ferromagnet, arXiv:cond- mat/9501123

  13. [21]

    C.ItoiandM-H.Kato,Extended massless phase and the Haldane phase in a spin-1 isotropic antiferromagnetic chain, Phys. Rev. B. 55 (1997), 8295–8303

    1997

  14. [22]

    Koma and B

    T. Koma and B. Nachtergaele,The Complete set of ground states of the ferromagnetic XXZ chains, Adv. Theor. Math. Phys. 2 (1998), 533–558

    1998

  15. [23]

    T. Koma, H. Tasaki,Decay of superconducting and magnetic correlations in one- and two-dimensional Hub- bard models, Phys. Rev. Lett. 68 (1992)

    1992

  16. [24]

    Korepin, N

    V. Korepin, N. Bogoliubov and A. Izergin,Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, 1993

    1993

  17. [25]

    Last and M

    G. Last and M. Penrose.Lectures on the Poisson process. Cambridge University Press, 2017

    2017

  18. [26]

    Läuchli, G

    A. Läuchli, G. Schmid and S. Trebst,Spin nematics correlations in bilinear-biquadraticS=1spin chains, Phys. Rev. B. 74 (2006)

    2006

  19. [27]

    E. Lieb, T. Schultz, D. Mattis,Two Soluble Models of an Antiferromagnetic Chain, Ann. Phys. 16 (1961), 407-466

    1961

  20. [28]

    McBryan, T

    O. McBryan, T. Spencer,On the decay of correlations in SO(n)-symmetric ferromagnets, Comm. Math. Phys. 53 (1977), 299–302

    1977

  21. [29]

    Matsui,On the absence of non-periodic ground states for the antiferromagnetic XXZ model, Comm

    T. Matsui,On the absence of non-periodic ground states for the antiferromagnetic XXZ model, Comm. Math. Phys. 253 (2005), 585–609

    2005

  22. [30]

    Matsui,On Ground States of the One-Dimensional Ferromagnetic XXZ Model, Lett

    T. Matsui,On Ground States of the One-Dimensional Ferromagnetic XXZ Model, Lett. Math. Phys. 37 (1996), 397–403

    1996

  23. [31]

    Mermin, H

    N. Mermin, H. Wagner,Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett. 17 (1966)

    1966

  24. [32]

    Nachtergaele, R

    B. Nachtergaele, R. Sims and A. Young,Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems, Lett. Math. Phys. 114 (2024)

    2024

  25. [33]

    Nachtergaele and D

    B. Nachtergaele and D. Ueltschi.A direct proof of dimerization in a family of SU (n)-invariant quantum spin chains. Letters in Mathematical Physics 107 (2017): 1629-1647

    2017

  26. [34]

    Peled and Y

    R. Peled and Y. Spinka.Lectures on the spin and loopO(n)models.In Sojourns in Probability Theory and Statistical Physics-I: Spin Glasses and Statistical Mechanics, A Festschrift for Charles M. Newman (pp. 246-320). Springer Singapore, 2019

    2019

  27. [35]

    Ragone,SO(n) AKLT Chains as Symmetry Protected Topological Quantum Ground States, arXiv:2403.09951

    M. Ragone,SO(n) AKLT Chains as Symmetry Protected Topological Quantum Ground States, arXiv:2403.09951

    arXiv

  28. [36]

    Ray and Y

    G. Ray and Y. Spinka.A short proof of the discontinuity of phase transition in the planar random-cluster model withq>4.Communications in Mathematical Physics 378.3 (2020): 1977-1988

    2020

  29. [37]

    Reshetikhin,A method of functional equations in the theory of exactly solvable quantum systems, Lett

    N. Reshetikhin,A method of functional equations in the theory of exactly solvable quantum systems, Lett. Math. Phys. 7 (1983), 205-213

    1983

  30. [38]

    Sutherland,Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems, IOP Publishing, 2004

    B. Sutherland,Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems, IOP Publishing, 2004

    2004

  31. [39]

    Takahashi,Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press, 1999

    M. Takahashi,Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press, 1999

    1999

  32. [40]

    Tasaki,Physics and mathematics of quantum many-body systems, Springer 2020

    H. Tasaki,Physics and mathematics of quantum many-body systems, Springer 2020

    2020

  33. [41]

    Tóth,Improved lower bound on the thermodynamic pressure of the spin1/2Heisenberg ferromagnet, Lett

    B. Tóth,Improved lower bound on the thermodynamic pressure of the spin1/2Heisenberg ferromagnet, Lett. Math. Phys. 28 (1993), 75–84

    1993

  34. [42]

    H-H. Tu, G-M. Zhang and T. Xiang,Class of exactly solvableSO(n)symmetric spin chains with matrix product ground states, Phys. Rev. B. 78 (2008)

    2008

  35. [43]

    Ueltschi,Random loop representations for quantum spin systems, J

    D. Ueltschi,Random loop representations for quantum spin systems, J. Math. Phys. 54 (2013), 083301

    2013

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