Pith. sign in

REVIEW 5 major objections 5 minor 42 references

Computational complexity of spin-glass three-dimensional (3D) Ising model

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that any exact algorithm for a spin-glass 3D Ising model on an $l\times m\times n$ lattice needs at least $O(2^{mn})$ time, making the problem subexponential but superpolynomial in the site count.

desk verdict An unsupported lower-bound claim for the spin-glass 3D Ising model, whose proof is circular and not ready for serious review. read the letter →

arxiv 2506.02067 v1 pith:5V3FQNIM submitted 2025-06-02 physics.gen-ph

Zhidong Zhang This is my paper
classification physics.gen-ph MSC 82B2082D3068Q2568Q17
keywords 3DIsingmodelspinglasscomputationalcomplexityabsoluteminimumcorelong-rangeentanglementfrustrationsubexponentialsuperpolynomial
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 sets out to prove a lower bound on how hard it is to compute exact properties of a spin-glass 3D Ising model, the disordered nearest-neighbour magnet whose bond signs are random and can be frustrated. Its target result is Theorem 4: for a lattice of $N=lmn$ sites, no algorithm can solve the model in less than $O(2^{mn})$ time, where $m,n$ are the dimensions of its largest two-dimensional cross-section. The significance is that this puts the problem in a narrow complexity class—subexponential and superpolynomial in $N$—and attributes the difficulty to the area of a plane rather than to the volume. The proof route is to claim that the irreducible core of the model is a spin-glass 2D Ising plane coupled to its nearest neighbouring plane; anything smaller destroys basic information. If right, this transfers a concrete exponential barrier to the exact study of three-dimensional spin glasses.

What carries the argument

The load-bearing object is the absolute minimum core (AMC): a spin-glass 2D Ising plane of $m\times n$ spins interacting with its nearest neighbouring plane. The author treats this two-plane object as the smallest unit that still preserves the defining features of the 3D model—nonplanar graphs, long-range entanglement, non-locality, frustration, and random bond disorder. Inside the argument, the AMC does two jobs: it fixes the combinatorial floor $2^{mn}$ (all configurations of one plane must be counted), and it turns the 3D lattice into $l$ independent copies of that floor, so any exact algorithm must cope with $l$ AMC-sized subproblems. The relation $2^{mn}=2^{N^{2/3}}$ for roughly cubic lattices, together with a limit argument comparing it with $2^N$ and $N^p$, converts the AMC floor into the claimed subexponential/superpolynomial classification.

What would settle it

Settle the claim by finding one exact algorithm for the spin-glass 3D Ising model with random nearest-neighbour bonds and $l\ge3$ whose worst-case running time is $o(2^{mn})$, for instance a contraction or branch-and-bound method that solves every $l\times m\times n$ instance in $O(2^{c\,mn})$ elementary operations with $c<1$; on a lattice family with $m=n=30$ and $l=3$, the claimed floor is $2^{900}$ steps, so any demonstrated algorithm below that bound would contradict Theorem 4.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that spin-glass 3D Ising models contain an absolute minimum core (AMC) whose computational cost is $O(2^{mn})$, and that no exact algorithm can go below that core. The AMC consists of a two-dimensional spin-glass Ising plane of $mn$ sites interacting with the nearest plane next to it; the paper argues this two-plane unit carries the non-locality and long-range entanglement of the 3D lattice, so cutting it loses indispensable information. Since the full lattice is made of $l$ stacked AMC units, the lower bound for the 3D problem is at least $l$ times the AMC cost, which in the symmetric limit $n=m=l=N^{1/3}$ is $O(2^{N^{2/3}})$. The paper then compares this with $O(2^N)$ and with polynomial $O(N^p)$ to conclude that the complexity is superpolynomial yet subexponential.

Load-bearing premise

The load-bearing premise is that 3D Ising systems are so entangled across each plane that an exact computation must consider all $2^{mn}$ spin configurations of at least one whole plane, and that any algorithm working on a smaller sub-block would destroy information essential to the model; the paper assumes this follows from non-locality inherited from the ferromagnetic 3D case rather than deriving it.

Editorial extensions

If this is right

  • If the bound holds, exact ground-state and partition-function computations for spin-glass 3D Ising models inherit a worst-case floor of $O(2^{mn})$; adding layers multiplies the cost but does not change the exponential set by the largest plane.
  • The floor is far below brute force: for roughly cubic lattices it is $O(2^{N^{2/3}})$, which is superpolynomial but subexponential in the site count $N$.
  • If correct, the result excludes polynomial-time exact algorithms for the model and any reduction that compresses the problem below a full two-plane core.
  • The claimed floor is algorithm-independent: transfer-matrix, branch-and-bound, or algebraic methods all face at least the AMC cost.

Reading between the lines

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

  • One testable consequence the author leaves implicit is that the same $O(2^{mn})$ barrier should apply to any three-dimensional disordered spin model whose transfer matrix crosses a full plane; a 3D random-field Ising model would be a clean discriminating test because it keeps dimensionality while removing frustration.
  • The bound is worst-case and exact; it says nothing about average-case difficulty or about approximate and heuristic solvers, so practical computation on typical instances could remain easy even if Theorem 4 is true.
  • If the paper's companion mapping to Boolean satisfiability is faithful, this lower bound would transfer as a $2^{N^{2/3}}$-type barrier to those satisfiability encodings, but the transfer is only as strong as the mapping itself.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The manuscript claims to prove a lower bound of O(2^mn) on the computational complexity of exact algorithms for the spin-glass 3D Ising model on an l×m×n lattice, defining an 'absolute minimum core' (AMC) model consisting of a spin-glass 2D Ising plane interacting with its nearest neighboring plane. Four theorems are presented: Theorem 1 asserts the AMC model is the smallest sub-model preserving all intrinsic characters; Theorem 2 asserts the full model's complexity is at least l times the AMC complexity; Theorem 3 asserts the AMC complexity cannot be reduced below O(2^mn), described as subexponential and superpolynomial; Theorem 4 concludes the spin-glass 3D Ising model has complexity at least O(2^mn). The proof of Theorem 3 consists of an assertion about combinatorial complexity followed by asymptotic comparisons of 2^mn with 2^N and N^p, and Theorems 1–2 rest on the existence and uniqueness of the AMC model as a primitive. No formal model of computation, input encoding, decision problem, or reduction from a known hard problem is given.

Significance. If the claimed lower bound were rigorously established, it would be a striking result: an unconditional exponential lower bound for a natural statistical-mechanics problem, with implications for complexity theory and for the boundary between polynomial and superpolynomial algorithms. The manuscript also usefully highlights genuine physical features—frustration, randomness, and long-range entanglement—that plausibly obstruct the analytic methods used for ferromagnetic 2D and 3D Ising models. However, the manuscript does not provide a proof in any standard sense. The central object, the AMC model, is defined by the property that it 'must' preserve all intrinsic characters, and this property is then used as the basis for the lower bound, making the argument circular. The paper also ships no machine-checked proofs, no reproducible code, and no falsifiable algorithmic prediction that could be tested independently. The significance is therefore potential rather than demonstrated.

major comments (5)
  1. [§2, Theorem 3] The proof of Theorem 3 does not establish a lower bound. It states that 'one has to take into account all the combinatorial complexity of this AMC model' and then compares the asymptotic order of O(2^mn) with O(2^N) and O(N^p). No argument is given to rule out algorithms that compute the AMC model without enumerating all 2^mn spin configurations, no reduction from a known hard problem is supplied, and no model of computation or input size measure is defined. A lower bound of the form 'cannot be reduced below O(2^mn) by any algorithms' requires exactly such missing ingredients.
  2. [§2, Theorem 1] The AMC model is introduced in Theorem 1 as the 'absolute minimum core' that must retain all intrinsic characters of the spin-glass 3D Ising model, with the assertion that any smaller model would 'cut the basic element.' This is an assumption about what any exact algorithm must compute, not a derived fact. Theorem 3 then relies on this assumption to conclude that the AMC complexity cannot be reduced. The lower bound therefore follows from the definition of the AMC model rather than from a proof about the spin-glass 3D Ising model itself, which is circular.
  3. [§2, Theorem 2] The claim that 'any algorithms to solve exactly the spin-glass 3D Ising model must deal with at least one AMC model' is asserted without proof. Moreover, the argument that the probability of all states factorizes as a direct product over sub-models does not imply a computational lower bound for the full model: independent subsystems can often be solved separately, and the claimed 'l terms of the AMC model' does not establish that the whole system is harder than one such term. The connection between the physical decomposition into planes and the computational complexity of the full lattice is not made rigorous.
  4. [§2, Theorems 3–4] The subexponential/superpolynomial classification of O(2^mn) is not valid as stated. The derivation that mn = N^{2/3} requires the special choice n = m = l = N^{1/3}; for other aspect ratios, mn can be arbitrarily close to N, in which case O(2^mn) is exponential in N, not subexponential. The theorems state a lower bound 'for any' lattice sizes, so a single asymptotic comparison under a symmetric-lattice assumption does not support the claimed classification for general l, m, n.
  5. [§1 and §2] The manuscript never formally defines 'computational complexity,' 'exact solution,' or the algorithmic problem being studied. It is unclear whether the object is computing the partition function, finding a ground state, or determining some other property, and no input encoding (e.g., binary encoding of J_ij values) is specified. Without such definitions, statements like 'cannot be reduced below O(2^mn) by any algorithms' are not well-formed and cannot be evaluated.
minor comments (5)
  1. [Abstract and §2, Theorem 3] The mathematical display comparing O(2^mn) with O(N^p) contains unreadable and garbled notation, and the claimed use of L'Hôpital's rule is not shown step by step; the limit computation should be rewritten carefully or omitted.
  2. [§2, Theorem 3] The statement 'The results obtained in this work provide a proof of Theorems on NP-complete problems [34,35]' is vague and unsupported; the paper does not connect its lower bound to any specific NP-complete problem or formal complexity class.
  3. [§1] The Hamiltonian and partition-function displays are incomplete: the Hamiltonian sum has no explicit indices or domain, and the partition function display contains undefined symbols such as K_c, K_c', and K_c'' even though the variables K_ij are introduced later.
  4. [Throughout] Typos and formatting errors appear throughout, including 'O(mn2)' in the abstract, 'configurations' misspelled as 'configurations' in the passage after the partition function, and missing signs in the Hamiltonian equation.
  5. [§1] The references [13]–[15] and [33] are cited as evidence for non-locality and long-range entanglement, but these are controversial and are not independently established in this manuscript; the argument should not depend on the contested interpretation of those works without further justification.

Circularity Check

3 steps flagged · score 9.0 of 10

The claimed lower bound O(2mn) is not derived; it is injected by defining the 'absolute minimum core' as the object whose entanglements must be kept and then citing the author's own prior work for those entanglements.

  1. self definitional [Section 2, Theorem 1 proof (definition of AMC)]
    "Therefore, it is clear that there exists an absolute minimum core (AMC) model in the spin-glass 3D Ising model, in which the entanglements between the spins should not be broken. The AMC model of the spin-glass 3D Ising model consists of a spin-glass 2D Ising model interacting with its nearest neighboring plane."

    The AMC is stipulated, not proved, as the smallest model that preserves all 'intrinsic characters' (non-locality, long-range entanglement, frustration). Every later lower bound refers back to this stipulated object. No argument from the spin-glass Hamiltonian shows that any exact algorithm must solve this particular two-plane submodel; the paper's own phrase 'the entanglements between the spins should not be broken' is the conclusion restated as a definition.

  2. self citation load bearing [Introduction, after the transfer-matrix discussion; used in Theorem 1 proof]
    "The only thing in our previous work, which we use in this work, is a fact that there exists the long-range entanglement, due to the internal factors in the transfer matrices (as clearly seen in Eq. (15) in [14], Eq. (A10) in [15], Eq. (3) in [33] and Eq. (73) in [9]), which is the character of the 3D Ising model."

    The entire non-locality premise on which AMC and O(2mn) rest is imported from the author's own prior papers [14,15] (with [9] and [33] also tied to that program). Those papers do not prove any computational-complexity lower bound; they propose a Clifford-algebra representation. The present paper treats that proposed 'fact' as an established external result, making the lower bound depend on a self-citation rather than on a derivation from the model.

1 more flagged steps
  1. other [Section 2, Theorem 3 proof]
    "The computational complexity of the AMC model of a spin-glass 3D Ising model cannot be reduced to be less than O(2mn) by any algorithms, because of the non-locality in the system. Indeed, one has to take into account all the combinatorial complexity of this AMC model."

    The proof of the central lower bound is the sentence 'one has to take into account all the combinatorial complexity of this AMC model' - that is the theorem statement itself. The only justification offered is 'non-locality', which was already assumed in Theorem 1 from self-cited work. Thus Theorem 3, and consequently Theorem 4 ('immediate consequence of Theorems 1-3'), reduce by construction to the definition of AMC plus the imported non-locality premise.

full rationale

The paper's central claim is Theorem 4: no algorithm can reduce the computational complexity below O(2mn). The derivation chain is Theorem 1 -> Theorem 2 -> Theorem 3 -> Theorem 4. Theorem 1 introduces the AMC as a stipulated object: it is 'clear' that an absolute minimum core exists in which entanglements should not be broken, and this core is a 2D spin-glass plane interacting with its nearest neighboring plane. The only evidence offered for the non-locality that forces this core is a 'fact' from the author's own earlier papers [14,15] (with [9,33]) regarding internal factors in transfer matrices. Those papers do not prove any lower bound; they propose a Clifford-algebra representation. Theorem 3 then asserts that the complexity of the AMC model cannot be less than O(2mn) 'because of the non-locality' and because 'one has to take into account all the combinatorial complexity of this AMC model' - a restatement rather than a proof. Theorem 4 is merely an 'immediate consequence'. Thus the lower bound is equivalent to the definition of AMC plus an imported self-cited premise; it is not derived from any independent property of spin-glass 3D Ising Hamiltonians or from any known NP-hardness result. The citation is not machine-checked, code-reproduced, or externally validated, so it is load-bearing self-citation rather than independent support. Score 9.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper's central claim rests on two postulates about non-locality and on an invented minimal core model. There are no fitted numerical parameters. The axioms and invented entities are not independently verified, so the argument's burden is carried by the author's own prior framework.

assumptions (4)
  • domain assumption Non-local internal factors and long-range entanglement in the 3D Ising model exist as described in the author's earlier papers (Eq. (15) in [14], Eq. (A10) in [15]).
    This property is cited from prior work by the same author and is not re-derived or independently verified in the present paper. It is the basis for claiming the AMC model cannot be split.
  • ad hoc to paper An "absolute minimum core" (AMC) model exists and is the smallest sub-model that retains all intrinsic characters of the spin-glass 3D Ising model.
    The AMC is introduced in Theorem 1 by verbal definition; no formal characterization is given. The lower bound depends entirely on this entity.
  • ad hoc to paper Any exact algorithm must solve at least one AMC model, and any smaller sub-model would cut the basic element and lose information.
    Stated immediately after Theorem 2 and used in Theorem 3; it is asserted without proof and is load-bearing for the O(2^mn) bound.
  • domain assumption Randomness in spin-glass interactions invalidates periodic boundary conditions and the largest-eigenvalue method, so no analytic reduction to a smaller problem is possible.
    Argued in Section 1 based on general properties of spin glasses; plausible but not proven, and it is used to exclude alternative algorithmic strategies.
invented entities (2)
  • Absolute minimum core (AMC) model
    purpose: A spin-glass 2D Ising plane interacting with its neighbor plane, defined as the smallest irreducible subproblem whose complexity determines the lower bound.
    The AMC is introduced verbally in Theorem 1, carries the entire proof, and has no falsifiable prediction or external validation.
  • Non-local internal factors in transfer matrices
    purpose: Invoked to justify why the AMC model cannot be decomposed into independent smaller pieces.
    These factors are inherited from the author's prior ferromagnetic 3D Ising work and are not independently evidenced in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Computational complexity of spin-glass three-dimensional (3D) Ising model." pith.science (2026). https://pith.science/paper/5V3FQNIM

@misc{pith2026250602067,
  author       = {Pith},
  title        = {Pith review of: Computational complexity of spin-glass three-dimensional (3D) Ising model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5V3FQNIM}},
  note         = {Machine review of arXiv:2506.02067}
}
read the original abstract

In this work, the computational complexity of a spin-glass three-dimensional (3D) Ising model (for the lattice size N = lmn, where l, m, n are the numbers of lattice points along three crystallographic directions) is studied. We prove that an absolute minimum core (AMC) model consisting of a spin-glass 2D Ising model interacting with its nearest neighboring plane, has its computational complexity O(2^mn). Any algorithms to make the model smaller (or simpler) than the AMC model will cut the basic element of the spin-glass 3D Ising model and lost many important information of the original model. Therefore, the computational complexity of the spin-glass 3D Ising model cannot be reduced to be less than O(2^mn) by any algorithms, which is in subexponential time, superpolynomial.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [27]

    Ghirlanda and F

    S. Ghirlanda and F. Guerra, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A 31, (1998) 9149–9155

    work page 1998

  2. [1]

    The study on the phase transition between the order and the disorder states is a very active topic in condensed matter physics and mathematical physics

    Introduction In nature, order and disorder are two different states in a material . The study on the phase transition between the order and the disorder states is a very active topic in condensed matter physics and mathematical physics. In a ferromagnet, the magnetic order (ferromagnetic state) is emerged at the critical point, which is transformed from t...

    work page

  3. [2]

    inside and outside

    Topological effect: The existence of topological effect in the ferromagnetic 3D Ising model has been pointed out in [11], which should be true for the spin -glass 3D Ising model . The combinatorial method of counting the closed graph for the 2D Ising model cannot be generated in any obvious way to the 3D problem (see page 366 in [11]). For the 3D Ising mo...

    work page

  4. [3]

    The randomness of the distribution of the interactions causes the computational complexity becomes muc h more complicated than that of the ferromagnetic counterpart

    Randomness: One of the most important characters of the spin -glass systems is the randomly distributed interactions between spins , and the presence of frustration [1-6,22-24]. The randomness of the distribution of the interactions causes the computational complexity becomes muc h more complicated than that of the ferromagnetic counterpart. This is becau...

    work page

  5. [4]

    Geometrical frustration is an important feature in magnetism, where it stems from the topological arrangement of spins

    Frustration: Normally, frustration consists of geometrical frustration and compositional frustration. Geometrical frustration is an important feature in magnetism, where it stems from the topological arrangement of spins. A spin at a particular lattice (such as triangular, honeycomb, Kagome, etc. ) can be frustrated because its two possible orientations, ...

    work page 1950

  6. [5]

    Non-ergodic behavior: A so -called non-ergodic behavior happens in spin glasses below the freezing temperature Tf , since the system cannot escape from the ultradeep minima of the hierarchically -disordered energy landscape [6,32]. The hierarchical disorder of the energy landscape may be verbally characterized by a scenario that there are (random) valleys...

    work page

  7. [6]

    Stein and C

    D.L. Stein and C. M. Newman, Spin Glasses and Complexity , Princeton University Press, Princeton & Oxford (2010)

    work page 2010

  8. [7]

    Proof: In the ferromagnetic 3D Ising model, it was revealed in Eq

    Computational complexity of spin-glass 3D Ising model In this section, we prove four theorems for the computational complexity of the spin-glass 3D Ising model: Theorem 1 The core model of the spin -glass 3D Ising model consists is much more complex than an absolute minimum core (AMC) model consisting of a spin-glass 2D Ising model interacting with its ne...

    work page

Show all 42 references
  1. [8]

    This causes the simplicity of the calculations, since the dominant contribution to the partition function is the large st eigenvalue of the models [8,12-15]

    for the ferromagnetic 2D Ising model, and Zhang [13,14] for the ferromagnetic3D Ising model, in which we wrap our crystal on cylinders. This causes the simplicity of the calculations, since the dominant contribution to the partition function is the large st eigenvalue of the m...

  2. [9]

    Binder and A

    K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, (1986) 801-976

    1986

  3. [10]

    S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F: Metal Phys.. 5, (1975) 965-974

    1975

  4. [11]

    Kirkpatrick and D

    S. Kirkpatrick and D. Shcrrington, Infinite-ranged models of spin -glasses, Phys. Rev. B 17, (1978) 4384-4403

    1978

  5. [12]

    Parisi, Infinite number of order parameters for spin-glasses, Phys

    G. Parisi, Infinite number of order parameters for spin-glasses, Phys. Rev. Lett.43, (1979) 1754-1756

    1979

  6. [13]

    Sherrington and S

    D. Sherrington and S. Kirkpatrick, Solvable Model of a Spin -Glass, Phys. Rev. Lett. 35, (1975) 1792-1796

    1975

  7. [14]

    E. Ising. Beitrag zur Theorie des Ferromagnetismus, Z Phys, 31, (1925) 253-258

    1925

  8. [15]

    Kaufman, Crystal Statistics II: Partition function evaluated by spinor analysis, Phys

    B. Kaufman, Crystal Statistics II: Partition function evaluated by spinor analysis, Phys. Rev. 76, (1949) 1232-1243

    1949

  9. [16]

    S. L. Lou, S.H. Wu, Three-dimensional Ising model and transfer matrices,. Chin J Phys, 38, (2000) 841-854

    2000

  10. [17]

    March, G.G.N

    N.H. March, G.G.N. Angilella, Exactly solvable models in many -body theory World Scientific, Singapore, Chapter 9, (2016) 147-191

    2016

  11. [18]

    Newell and E.W

    G.F. Newell and E.W. Montroll, On the theory of the Ising model with ferromagnetism, Rev. Mod. Phys. 25, (1953) 353-389

    1953

  12. [19]

    Onsager, Crystal Statistics I: A two-dimensional model with an order-disorder transition, Phys

    L. Onsager, Crystal Statistics I: A two-dimensional model with an order-disorder transition, Phys. Rev. 65, (1944) 117-149

    1944

  13. [20]

    Z.D.Zhang, Conjectures on the exact solution of three - dimensional (3D) simple orthorhombic Ising lattices, Phil. Mag. 87, (2007) 5309-5419

    2007

  14. [21]

    Zhang, Mathematical structure of the three - dimensional (3D) Ising model, Chinese Phys

    Z.D. Zhang, Mathematical structure of the three - dimensional (3D) Ising model, Chinese Phys. B 22, (2013) 030513-1-15

    2013

  15. [22]

    Zhang, O

    Z.D. Zhang, O. Suzuki and N.H. March, Clifford algebra approach of 3D Ising model, Advances in Applied Clifford Algebras. 29, (2019) 12

    2019

  16. [23]

    Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction: Oxford: Oxford University Press

    H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction: Oxford: Oxford University Press. (2001) p. 243

    2001

  17. [24]

    Guerra, F

    F. Guerra, F. Lucio Toninelli , The thermodynamic limit in mean field spin glass , Commun. Math. Phys. 230, (2002) 71–79

    2002

  18. [25]

    Mezard, G

    M. Mezard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, Nature of the spin-glass phase, Phys. Rev. Lett. 52, (1984) 1156-1159

    1984

  19. [26]

    Ramrnal, G

    R. Ramrnal, G. Toulouse and M. A. Virasoro, Ultrametricity for physicists, Rev. Mod. Phys. 58, (1986) 765-788

    1986

  20. [28]

    Talagrand, The Parisi formula, Annals of Mathematics, 163, (2006) 221–263

    M. Talagrand, The Parisi formula, Annals of Mathematics, 163, (2006) 221–263

    2006

  21. [29]

    D. S. Fisher, D. A. Huse, Equilibrium behavior of the spin -glass ordered phase, Phys. Rev.B 38, (1988) 386-411

    1988

  22. [30]

    Stauffer, K

    D. Stauffer, K. Binder, On the nature of the ordering in Ising spin glasses, Z. Physik B 30, (1978) 313 -324

    1978

  23. [31]

    Suzuki, Phenomenological theory of spin -glasses and some rigorous results, Prog

    M. Suzuki, Phenomenological theory of spin -glasses and some rigorous results, Prog. Theor.Phys., 58, (1977) 1151-1165

    1977

  24. [32]

    J. A. Blackman, Two-dimensional frustrated Ising network as an eigenvalue problem, Phys. Rev. B 26, (1982) 4987-4996

    1982

  25. [33]

    Kirkpatrick, Frustration and ground -state degeneracy in spin glasses, Phys

    S. Kirkpatrick, Frustration and ground -state degeneracy in spin glasses, Phys. Rev. B 16, (1977) 4630-4641

    1977

  26. [34]

    L. Saul, M. Kardar, The 2D -+-J Ising spin glass: exact partition functions in polynomial time, Nuclear Physics B 432, (1994) 641-667

    1994

  27. [35]

    General formalism and two-dimensional model, J

    J Villain, Two-level systems in a spin -glass model : I. General formalism and two-dimensional model, J. Phys. C 10, (1977) 4793-4803

    1977

  28. [36]

    Wannier, Antiferromagnetism

    G.H. Wannier, Antiferromagnetism. The triangular Ising net . Phys. Rev. 79, (1950) 357–364

    1950

  29. [37]

    Toulouse, The Frustration Model, In Pekalski, Andrzej; Przystawa, Jerzy

    G. Toulouse, The Frustration Model, In Pekalski, Andrzej; Przystawa, Jerzy. Modern Trends in the Theory of Condensed Matter . Lecture Notes in Physics 115. Springer Berlin / Heidelberg. (1980) 195–203

    1980

  30. [38]

    Vannimenus, G

    J. Vannimenus, G. Toulouse, Theory of the frustration effect. II. Ising spins on a square lattice. J. Phys. C 10, (1977) L537-L542

    1977

  31. [39]

    N. D. Mackenzie and A. P. Young, Lack of ergodicity in the infinite -range Ising spin-glass, Phys. Rev. Lett. 49, (1982) 301-304

    1982

  32. [40]

    Perk, Comment on 'Conjectures on exact solution of three -dimensional (3D) simple orthorhombic Ising lattices', Phil

    J.H.H. Perk, Comment on 'Conjectures on exact solution of three -dimensional (3D) simple orthorhombic Ising lattices', Phil. Mag. 89, (2009) 761–764

    2009

  33. [41]

    Zhang, Mapping between spin -glass three -dimensional (3D) Ising model and Boolean satisfiability problems, Mathematics, 11, (2023) 237

    Z.D. Zhang, Mapping between spin -glass three -dimensional (3D) Ising model and Boolean satisfiability problems, Mathematics, 11, (2023) 237

    2023

  34. [42]

    Zhang, Lower bound of computational complexity of Knapsack problems , AIMS Mathematics, 10, (2025) 11918–11938

    Z.D. Zhang, Lower bound of computational complexity of Knapsack problems , AIMS Mathematics, 10, (2025) 11918–11938

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.