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The Frustration of being Odd: How Boundary Conditions can destroy Local Order

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read On an odd ring of frustrated antiferromagnetic spins, the local order parameter vanishes in the thermodynamic limit.

desk verdict The exact finite-size magnetization calculation is solid and the 1/N decay is real, but the paper's headline claim that boundary conditions destroy local order depends on a nonstandard order-parameter definition that the standard staggered-field limit probably refutes. read the letter →

arxiv 1908.10876 v3 pith:SWQGQCEM submitted 2019-08-28 cond-mat.stat-mech cond-mat.str-elhep-thquant-ph

Vanja Marić , Salvatore Marco Giampaolo , Domagoj Kuić , Fabio Franchini This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.str-elhep-thquant-ph
keywords quantumspinchainsXYZmodelgeometricfrustrationodd-lengthringsboundaryconditionsspontaneoussymmetrybreakingorderparameterToeplitzdeterminants
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 studies a spin-$1/2$ XYZ chain closed into a ring with an odd number of sites, so the antiferromagnetic bonds cannot all be satisfied simultaneously. It claims that in the phase with dominant antiferromagnetic coupling along $x$ (the xAFM phase), the spontaneous $x$-magnetization, the standard local order parameter, decays to zero as $1/N$ as the ring grows, and it is not staggered. The same techniques reproduce the expected finite magnetization in the ferromagnetic phase, so the vanishing is specific to the frustrated odd ring. If true, this is a direct counterexample to the usual assumption that boundary conditions cannot affect bulk local order in the thermodynamic limit.

What carries the argument

The load-bearing device is the parity-twist identity of Eq. (9): because the zero-field Hamiltonian commutes with all three parity operators $\Pi^\alpha$, and these anticommute when $N$ is odd, the states $|g_\alpha\rangle \propto (1+\Pi^\alpha)|g_z\rangle$ are exact degenerate ground states at every finite $N$. The identity rewrites the local one-point magnetization as the string expectation value $\langle g_z|\tilde{\Pi}^x_j|g_z\rangle$, which is a determinant of a Toeplitz matrix. In the frustrated phase the symbol of that matrix carries a delta-function singularity coming from the single delocalized excitation, and its asymptotic analysis produces the $1/N$ decay instead of exponential saturation.

What would settle it

Compute the single-site $x$-magnetization in the xAFM phase on an odd ring at $\phi=-0.25$ for increasing $N$: if it follows $m_x \sim \frac{1}{N}(1-\tan^2\phi)^{1/4}$ down to $N\sim10^4$, the paper's central claim is supported; if it saturates toward $(1-\tan^2\phi)^{1/4}$, the standard limit-ordering prescription wins and the claimed boundary effect is not a bulk phenomenon.

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Extended reading notes

Core claim

On a zero-field odd ring, the degenerate ground states with definite parity along $x$ are constructed as $|g_x\rangle = (1+\Pi^x)|g_z\rangle/\sqrt{2}$; in the xAFM phase their $x$-magnetization is $m_x \simeq \frac{1}{N}(1-\tan^2\phi)^{1/4}$ (Eq. A.42), vanishing algebraically while remaining uniform rather than staggered. The paper calls the resulting finite-size state a mesoscopic ferromagnetic phase (MFM). In the yFM phase the same construction gives $m_y=(1-\cot^2\phi)^{1/4}$, which shows the procedure is not biased toward zero. Hence the authors conclude that frustrated periodic boundary conditions destroy the local order parameter in the infinite-size limit, contradicting the standard expectation that boundary terms are sub-extensive.

Load-bearing premise

The load-bearing premise is that the spontaneous magnetization should be defined by taking the thermodynamic limit of the one-point expectation value in the finite-size parity-symmetry-broken states $|g_\alpha\rangle$, rather than by taking $N\to\infty$ first in the two-point correlator and then breaking the symmetry; the paper itself states that this latter prescription gives $m_x=(1-\tan^2\phi)^{1/4}$.

Editorial extensions

If this is right

  • Boundary conditions acquire a thermodynamic-limit effect on a local observable: the same XYZ Hamiltonian on an even ring or open chain has a finite staggered $x$-magnetization, while the odd ring has none.
  • Finite odd rings in the xAFM phase should display a measurable uniform $x$-magnetization that shrinks as $1/N$; this mesoscopic ferromagnetic phase is the experimental signature of the effect.
  • The delocalized-excitation mechanism ties the vanishing order parameter to an algebraically closing excitation gap, so the thermodynamic limit is approached slowly rather than exponentially in the frustrated phase.
  • The numerical results for $\delta\neq 0$ show the algebraic decay persists away from the free-fermion line, so the effect is not special to the exactly solvable XY point.

Reading between the lines

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

  • Editorial inference: the limit-ordering question is not merely technical; under the conventional prescription of taking $N\to\infty$ first in the two-point function, the same model has $m_x=(1-\tan^2\phi)^{1/4}$ and the paradox disappears. The paper's case therefore stands or falls on whether the finite-size symmetry-broken state is the physically relevant one.
  • Editorial inference: the non-staggered local moment coexists with a staggered two-point correlator at fixed $r$ in Eq. (10), so on an odd ring the one-point and two-point functions encode incompatible-looking orders; comparing them directly in a numerical or experimental setting would isolate which limiting prescription is realized.
  • Editorial inference: the same parity-twist construction applies to any zero-field chain with three noncommuting parities, so other frustrated geometries, such as odd ladders or rings with a single defect, are natural places to look for the same boundary-induced destruction of local order.
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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

3 major / 5 minor

Summary. The paper studies an odd-length spin-1/2 XYZ ring with periodic boundary conditions and no external field, choosing parameters so that one coupling (x) is antiferromagnetic and the other two are ferromagnetic. In the xAFM phase (φ in (-π/4,0]), the authors construct finite-size parity-projected states |g_α> via Eq. (9) and compute the one-point magnetizations. For the XY case δ=0 they use a Jordan-Wigner mapping and express the magnetizations as Toeplitz determinants, obtaining asymptotic formulas m_x ~ (1/N)(1-tan^2 φ)^{1/4}, m_y ~ (2/N)(1-tan φ)^{1/4}(1+tan φ)^{-3/4}, and m_z=2/N; for δ≠0 they provide exact diagonalization data up to N=23. They conclude that in the frustrated AFM phase all spontaneous magnetizations decay algebraically to zero and are not staggered, a behavior they call 'ferromagnetic mesoscopic magnetization' (MFM), and they interpret this as evidence that frustrated boundary conditions destroy local order in the thermodynamic limit. In the yFM phase they recover the standard finite magnetizations. The supplementary material contains the determinant representations, the asymptotic results quoted from the companion paper [54], and a perturbative kink calculation near the Ising point.

Significance. If the central claim were established, this would be a striking counterexample to the standard assumption that boundary conditions cannot affect local bulk order parameters. The paper has real strengths: the finite-N determinant representation for the XY case is exact and parameter-free, the results are cross-checked numerically, and the perturbative analysis in Appendix A.6 provides an independent consistency check. However, the physical conclusion depends on a nonstandard definition of the order parameter: the one-point function is evaluated in translationally invariant parity-projected states and then the thermodynamic limit is taken, rather than using the standard staggered-field symmetry-breaking protocol. The paper itself acknowledges that the standard N→∞-first prescription gives a finite staggered magnetization from Eq. (10). Therefore the significance is conditional on resolving this methodological issue.

major comments (3)
  1. [Section 4, Eq. (10) and following paragraph] The extraction of m_x from the antipodal value of Cxx(r) is internally inconsistent. Eq. (10) gives Cxx(r) ~ (-1)^r sqrt(1-tan^2 φ) (1-2r/N) for fixed r, and at r ≈ N/2 this is of order 1/N. If cluster decomposition were used, one would expect m_x^2 ~ Cxx(r≈N/2), yielding m_x ~ N^{-1/2}, not the quoted (1/N)(1-tan^2 φ)^{1/4}. The paper instead identifies the correlation value itself with m_x, and later notes that cluster decomposition is 'spoiled'. But then the quantity computed in Eq. (A.42) is not connected to the standard order parameter extracted from two-point correlations, and the antipodal argument does not support the claimed 1/N decay. This point needs to be resolved before the central claim can be assessed.
  2. [Section 4, Eq. (10); Section 1] The central claim that frustrated boundary conditions destroy local order rests on defining the order parameter as the thermodynamic limit of the one-point function in the finite-size parity-projected states |g_x> of Eq. (9). This is not the standard SSB definition for an antiferromagnet, which is m_s = lim_{h→0+} lim_{N→∞} (1/N) Σ_j (-1)^j ⟨σ^x_j⟩_h with a staggered field selecting one Néel state. On an odd ring a perfectly staggered field cannot be periodic, but a field with a single sign defect becomes a staggered field in the bulk; the paper does not compute this limit. Moreover, Eq. (10) shows that for fixed r the correlation has the staggered envelope (-1)^r and tends to a finite value if N→∞ first. Thus the evidence presented supports a finite staggered order under the standard protocol, and the paper's conclusion depends on a methodological choice that is asserted rather than derived.
  3. [Appendix A.6, Eqs. (A.56), (A.59), (A.60)] The perturbative calculation illustrates the same definitional issue. In the uniform kink superposition |s_{q=0}>, the local magnetization is 1/N (Eq. A.59), but a single-kink state |l> has ⟨σ^x_j|l> = (-1)^{l+j} or (-1)^{l+j+1} (Eq. A.56), which is staggered away from the defect and would give a finite bulk staggered magnetization in the N→∞ limit. The vanishing 1/N therefore arises from choosing the translationally invariant superposition of kinks, not from the impossibility of staggered order on an odd ring. The authors should justify why |s_{q=0}>, rather than a state with a localized symmetry-breaking defect, is the correct finite-size representative of the AFM phase.
minor comments (5)
  1. [Abstract] 'Central tenant' should be 'central tenet'.
  2. [Section 2] The sentence containing '([H, Πα])' is malformed; it should read [H, Πα]=0.
  3. [Appendix A.5, Eqs. (A.42)-(A.43)] The main asymptotic formulas are imported from the companion paper [54] without derivation or a statement of their regime of validity. Since these formulas carry the quantitative claim, the authors should either include a proof sketch or clearly state the theorem and its conditions in the main text.
  4. [Fig. 2 caption] 'gathered settings δ=0' should be 'gathered setting δ=0'; also the legend markers are described as dots, while filled squares, circles, and diamonds are used.
  5. [Appendix A.2, Eq. (A.10)] There is a typographical issue in the presentation of Cxx(r) = (-i)^r Δ(ρ_xx); please check the formatting of all determinant formulas for missing parentheses.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central finite-size magnetization is derived from the Hamiltonian by exact fermionization and Toeplitz determinants, with an independent perturbative check.

full rationale

The paper's central result, Eq. (A.42) with mx ~ (1/N)(1 - tan^2 phi)^(1/4), is derived within the paper from the Hamiltonian by mapping to Majorana fermions, using Wick's theorem, and evaluating Toeplitz determinants. The state |g_x> in Eq. (9) is explicitly constructed from the Hamiltonian's zero-field degenerate ground states, and the one-point function is converted into the determinant of a finite matrix. This is not a fitted parameter, and the result is checked independently by both numerical diagonalization and by the perturbative kink calculation in Appendix A.6, where the same 1/N magnetization emerges from the exact classical ground-state subspace for phi -> 0. The citations to the authors' previous work are not circularly load-bearing: Ref. [54] is a parameter-free mathematical result about Toeplitz determinants with delta-function singularities, used as a lemma; Ref. [15] is cited for the physical motivation of evaluating correlations at antipodal points, but the paper does not rely on that citation for the proof of Eq. (A.42), which is worked out in the appendix. The paper also explicitly acknowledges the alternative standard prescription: 'using the standard prescription of taking N->infinity first, one would get mx = (1 - tan^2 phi)^(1/4)' and then explains the antipodal-point procedure. This is a transparent choice of limiting prescription and definition of the order parameter, not a hidden identification of output with input. Whether that choice is the physically correct order-parameter definition is a scientific judgment about the model, not a circularity in the derivation itself. Thus no step in the claimed derivation chain reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard integrable-model techniques plus two nonstandard assumptions: the validity of the companion-paper determinant asymptotics, and the choice of how the thermodynamic limit is taken for the order parameter. No parameters are fitted to data; phi and delta are Hamiltonian parameters.

assumptions (5)
  • standard math Jordan-Wigner transformation and Wick's theorem apply to the XY/XYZ chain with odd periodic boundary conditions.
    Used throughout Section 3 and Appendices A.1-A.3 to reduce spin correlators to Toeplitz determinants.
  • domain assumption Asymptotic evaluation of Toeplitz determinants with delta-function singularities given in the companion paper [54] is correct.
    The central formulas (A.42,A.43) for the frustrated-phase magnetizations are quoted from [54], not proven here.
  • domain assumption The finite-size parity-projected states |g_alpha> are the appropriate symmetry-broken ground states whose N-to-infinity one-point functions define the order parameter.
    The result depends on this choice; the standard prescription, taking N to infinity at fixed r in Eq. (10), would give a finite mx (Section 4).
  • standard math Szego limit theorem and Wiener-Hopf method for Toeplitz determinants.
    Used in Appendix A.4 to evaluate the ferromagnetic-phase determinants.
  • domain assumption The algebraic decay observed for N up to 23 in the XYZ case continues to the thermodynamic limit.
    The XYZ generalization is supported only by numerical diagonalization on small systems (Fig. 2, lower panel).

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Pith. "Pith review of The Frustration of being Odd: How Boundary Conditions can destroy Local Order." pith.science (2026). https://pith.science/paper/SWQGQCEM

@misc{pith2026190810876,
  author       = {Pith},
  title        = {Pith review of: The Frustration of being Odd: How Boundary Conditions can destroy Local Order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWQGQCEM}},
  note         = {Machine review of arXiv:1908.10876}
}
read the original abstract

A central tenant in the classification of phases is that boundary conditions cannot affect the bulk properties of a system. In this work, we show striking, yet puzzling, evidence of a clear violation of this assumption. We use the prototypical example of an XYZ chain with no external field in a ring geometry with an odd number of sites and both ferromagnetic and antiferromagnetic interactions. In such a setting, even at finite sizes, we are able to calculate directly the spontaneous magnetizations that are traditionally used as order parameters to characterize the system's phases. When ferromagnetic interactions dominate, we recover magnetizations that in the thermodynamic limit lose any knowledge about the boundary conditions and are in complete agreement with standard expectations. On the contrary, when the system is governed by antiferromagnetic interactions, the magnetizations decay algebraically to zero with the system size and are not staggered, despite the AFM coupling. We term this behavior {\it ferromagnetic mesoscopic magnetization}. Hence, in the antiferromagnetic regime, our results show an unexpected dependence of a local, one--spin expectation values on the boundary conditions, which is in contrast with predictions from the general theory.

Figures

Figures reproduced from arXiv: 1908.10876 by the authors.

Figure 1
Figure 1. Magnetizations along the three axis (in absolute value) as a function of the chain length for the yFM phase at φ = −1.32. The upper panel dots represent the data gathered settings δ = 0 and using the “trick” discussed in the text to evaluate the magnetizations as determinants of N−1 2 × N−1 2 matrices, while the dots in the lower one are obtained taking δ = .3 and using exact numerical diagonalization. Regardless th… view at source ↗
Figure 2
Figure 2. Magnetizations along the three axis (in absolute value) as a function of the chain length for the xAFM/MFM phase at φ = −0.25. The upper panel dots represent the data gathered settings δ = 0 using the “trick” discussed in the text to evaluate the magnetizations as determinants of N−1 2 × N−1 2 matrices, while the dots in the lower one are obtained taking δ = .3 and using exact numerical diagonalization. Regardless o… view at source ↗
Figure 3
Figure 3. Plot of the magnetizations as a function of φ for δ = .3 for several system sizes. The yFM phase (φ < −π/4) shows a fast approach to saturation, while for the frustrated case the decay toward zero is algebraically slow. weakest spontaneous magnetization in that direction, with my being the strongest one (once more, these magnetization refers to different states |gαi). Finally, we remark that FBC also seems to somewh… view at source ↗

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Works this paper leans on

66 extracted references · 65 canonical work pages

  1. [54]

    Sachdev, Quantum Phase Transitions, Cambridge University Press (2011)

    S. Sachdev, Quantum Phase Transitions, Cambridge University Press (2011)

    work page 2011

  2. [1]

    sufficiently large system

    Introduction Landau theory is one of the most impactful constructions of the last century. It allows distinguishing between different phases through different local order parameters, quantities which are finite or vanish depending on the phase of a system [1, 3, 2, 4]. Although the new century has taught us that this classification is not complete, because ce...

    work page

  3. [2]

    Crucially, we apply periodic boundary conditions σα j+N =σα j

    The spin chains and their properties We consider an anisotropic spin– 1 2 chain with Hamiltonian H = N∑ j=1 cosδ ( cosφσx jσx j+1 + sinφσy jσy j+1 ) − sinδσz jσz j+1, (1) where σα j , with α = x,y,z , are Pauli operators and N is the number of lattice sites, which we henceforth set to be odd N = 2M + 1. Crucially, we apply periodic boundary conditions σα ...

    work page

  4. [3]

    and separates two phases characterized by a two–fold degenerate ground state. In particular, forφ∈ [−π/2,−π/4) the phase favors a ferromagnetic alignment along the y direction (yFM), while for φ∈ (−π/4, 0] the dominant interaction is AFM along the x direction (xAFM) and thus topologically frustrated. With no external field, all three parity operators along...

    work page

  5. [4]

    (1) and focus on the ferromagnetic region φ∈ [−π/2,−π/4)

    The ferromagnetic case Let us now turn back to the system in eq. (1) and focus on the ferromagnetic region φ∈ [−π/2,−π/4). The (quasi–)long–range order represented by the order parameter can be extracted in two ways: either from the two–point function or by selecting a suitable superposition of states at finite sizes and then following their magnetization ...

    work page

  6. [5]

    mesoscopic magnetization

    The frustrated case We now turn to the case with ( φ∈ (−π/4, 0]), where the boundary conditions induce topological frustration. The effect of frustration has been recently studied in detail in Refs. [14, 15, 53]. For δ = 0, the model can be solved through the same steps used in the traditional cases and exactly mapped into a system of free fermions. In the...

    work page

  7. [6]

    We have done so, by realizing that, with no external field, we can exploit particle/hole duality to construct an exact ground state at finite sizes that break the Z2 symmetry

    Conclusions We have presented a comparative study of the ferromagnetic and AFM frustrated case for a XYZ chain, showing that, contrary to expectations, the boundary conditions are able to destroy local order. We have done so, by realizing that, with no external field, we can exploit particle/hole duality to construct an exact ground state at finite sizes th...

    work page 2016

  8. [7]

    To overcome this problem, we proceed as in Ref

    (A.28) The magnetization in the x direction, instead, is more complicated, because the generating function of the corresponding Toeplitz matrix has a non–zero winding number. To overcome this problem, we proceed as in Ref. [57] and notice that the determinant in eq. (A.26) can be seen as the minor of ∆ r+1(ρy) in eq. (A.27) obtained removing the first row ...

    work page

Show all 66 references
  1. [8]

    Landau, E.M

    L.D. Landau, E.M. Lifshitz, & L.P. Pitaevskij, Statistical Physics, Pergamon Press, Oxford (1978)

    1978

  2. [9]

    Chandler, Introduction to Modern Statistical Mechanics , Oxford University Press; 1 edition (1987)

    D. Chandler, Introduction to Modern Statistical Mechanics , Oxford University Press; 1 edition (1987)

    1987

  3. [10]

    Anderson, Basic Notions Of Condensed Matter Physics , Addison-Wesley (1997)

    P.W. Anderson, Basic Notions Of Condensed Matter Physics , Addison-Wesley (1997)

    1997

  4. [11]

    Coleman, Introduction to Many-Body Physics , Cambridge University Press (2016)

    P. Coleman, Introduction to Many-Body Physics , Cambridge University Press (2016)

    2016

  5. [12]

    Stone (Ed.), Quantum Hall Effect , World Scientific (1992)

    M. Stone (Ed.), Quantum Hall Effect , World Scientific (1992)

    1992

  6. [13]

    Wen, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons , Oxford University Press (2004)

    X.-G. Wen, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons , Oxford University Press (2004)

    2004

  7. [14]

    Nayak, S.H

    C. Nayak, S.H. Simon, A. Stern, M. Freedman, & S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). Non-abelian anyons and topological quantum computation

    2008

  8. [15]

    Hasan & C.L

    M.Z. Hasan & C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010) Colloquium: topological insulators

    2010

  9. [16]

    Fradkin, Field theories of condensed matter physics , Cambridge University Press (2013)

    E. Fradkin, Field theories of condensed matter physics , Cambridge University Press (2013)

    2013

  10. [17]

    B. A. Bernevig & T.L. Hughes, Topological Insulators And Topological Superconductors, Princeton University Press (2013). The Frustration of being Odd: How Boundary Conditions can destroy Local Order 23

    2013

  11. [18]

    S. M. Giampaolo & B. C. Hiesmayr, Phys. Rev. A 92, 012306 (2015). Topological and nematic ordered phases in many–body cluster–Ising models

    2015

  12. [19]

    Witten, Rev

    E. Witten, Rev. Mod. Phys. ‘bf 88, 35001 (2016). Free fermions And Topological Phases

    2016

  13. [20]

    B. Zeng, X. Chen, D.-L. Zhou, & X.-G. Wen, Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems , Springer (2019)

    2019

  14. [21]

    J.-J. Dong, P. Li, & Q.-H. Chen, J. Stat. Mech. P113102 (2016) The A-Cycle Problem for Transverse Ising Ring

    2016

  15. [22]

    S. M. Giampaolo, F. B. Ramos, & F. Franchini, J. Phys. Commun. 3, 081001 (2019) The Frustration in being Odd: Area Law Violation in Local Systems

    2019

  16. [23]

    Affleck, Acta Phys

    I. Affleck, Acta Phys. Polon B 26 , 1869 (1995) Conformal Field Theory Approach to the Kondo Effect

    1995

  17. [24]

    Durganandini, Phys

    P. Durganandini, Phys. Rev. B 53 , R8832(R) (1996). Kondo effect in a Luttinger liquid: A boundary-conformal-field-theory approach

    1996

  18. [25]

    Cardy, Nucl

    J. Cardy, Nucl. Phys. B 240 , 4 (1984) Conformal invariance and surface critical behavior

    1984

  19. [26]

    Cardy, arXiv:hep-th/0411189 (2004)

    J. Cardy, arXiv:hep-th/0411189 (2004). Boundary Conformal Field Theory

    2004 arXiv

  20. [27]

    Di Francesco, P

    P. Di Francesco, P. Mathieu, & D. Senechal, Conformal Field Theory , Springer (1999)

    1999

  21. [28]

    Korepin, N.M

    V.E. Korepin, N.M. Bogoliubov, & A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1997)

    1997

  22. [29]

    Korepin & P

    V.E. Korepin & P. Zinn-Justin, J. Phys. A 33 , 7053 (2000) Thermodynamic limit of the six-vertex model with domain wall boundary conditions

    2000

  23. [30]

    Zinn-Justin, (2002), arXiv:cond-mat/0205192 (2002)

    P. Zinn-Justin, (2002), arXiv:cond-mat/0205192 (2002). The influence of boundary conditions in the six-vertex model

    2002 arXiv

  24. [31]

    Colomo & A

    F. Colomo & A. G. Pronko, J. Stat. Phys. 138, 662 (2010) The arctic curve of the domain-wall six-vertex model

    2010

  25. [32]

    Bleher & K

    P. Bleher & K. Liechty, Random Matrices and the Six-Vertex Model, CRM monographs series, vol. 32, American Mathematical Society, Providence (2013)

    2013

  26. [33]

    Colomo & A

    F. Colomo & A. Sportiello, J. Stat. Phys. 164, 1488 (2016) Arctic curves of the six-vertex model on generic domains: the Tangent Method

    2016

  27. [34]

    Allegra, J

    N. Allegra, J. Dubail, J.-M. St´ ephan, & J. Viti, J. Stat. Mech. 2016, 053108 (2016). Inhomogeneous field theory inside the arctic circle

    2016

  28. [35]

    Reshetikhin & A

    N. Reshetikhin & A. Sridhar, Commun. Math. Phys. 356, 535 (2017) Integrability of limit shapes of the six-vertex model

    2017

  29. [36]

    Di Francesco & E

    P. Di Francesco & E. Guitter, J. Phys. A: Math. Theor. 51, 355201 (2018) Arctic curves for paths with arbitrary starting points: a tangent method approach

    2018

  30. [37]

    Colomo, A.G

    F. Colomo, A.G. Pronko, & A. Sportiello, J. Stat. Phys. 174, 1 (2018) Arctic Curve of the Free-Fermion Six-Vertex Model in an L-Shaped Domain

    2018

  31. [38]

    Toulouse, Commun

    G. Toulouse, Commun. Phys. 2, 115 (1977) Theory of the frustration effect in spin glasses: I

    1977

  32. [39]

    Vannimenus & G

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

    1977

  33. [40]

    M.M. Wolf, F. Verstraete & J.I. Cirac, Int. Journal of Quantum Information 1, 465 (2003) Entanglement and Frustration in Ordered Systems

    2003

  34. [41]

    S. M. Giampaolo, G. Gualdi, A. Monras, & F. Illuminati, Phys. Rev. Lett. 107, 260602 (2011) Characterizing and quantifying frustration in quantum many-body systems

    2011

  35. [42]

    Marzolino, S

    U. Marzolino, S. M. Giampaolo, & F. Illuminati, Phys. Rev. A 88, 020301(R) (2013) Frustration, entanglement, and correlations in quantum many body systems

    2013

  36. [43]

    J. F. Sadoc & R. Mosseri, Geometrical frustration. Cambridge University Press (2007)

    2007

  37. [44]

    Lacroix, P

    C. Lacroix, P. Mendels, & F. Mila (eds), Introduction to Frustrated Magnetism: Materials, The Frustration of being Odd: How Boundary Conditions can destroy Local Order 24 Experiments, Theory. Springer Series in Solid-State Sciences, Vol. 164 (2011)

    2011

  38. [45]

    H. T. Diep, Frustrated Spin Systems, World Scientific (2013)

    2013

  39. [46]

    Wannier, Phys

    G.H. Wannier, Phys. Rev. 79, 357 (1950) Antiferromagnetism. The Triangular Ising Net

    1950

  40. [47]

    Burkhardt & I

    T.W. Burkhardt & I. Guim, J. Phys. A: Math. Gen 18, L33 (1985) Finite-size scaling of the quantum Ising chain with periodic, free, and antiperiodic boundary conditions

    1985

  41. [48]

    Cabrera & R

    G.G. Cabrera & R. Jullien, Phys. Rev. B 35 , 7062 (1987) Role of boundary conditions in the finite-size Ising model

    1987

  42. [49]

    Campostrini, A

    M. Campostrini, A. Pelissetto, & E. Vicari, Phys. Rev. E 91 , 042123 (2015) Quantum transitions driven by one-bond defects in quantum Ising rings

    2015

  43. [50]

    Ercolessi, S

    E. Ercolessi, S. Evangelisti, F. Franchini, & F. Ravanini, Phys. Rev. B 88, 104418 (2013) Modular invariance in the gapped XYZ spin- 1 2 chain

    2013

  44. [51]

    Franchini, An introduction to integrable techniques for one-dimensional quantum systems , Lecture Notes in Physics 940, Springer (2017)

    F. Franchini, An introduction to integrable techniques for one-dimensional quantum systems , Lecture Notes in Physics 940, Springer (2017)

    2017

  45. [52]

    Mari´ c, S

    V. Mari´ c, S. M. Giampaolo, D. Kui´ c, & F. Franchini, In preparation The Frustration in being Odd: Exact finite size degeneracies

  46. [53]

    Mari´ c, S

    V. Mari´ c, S. M. Giampaolo, D. Kui´ c, & F. Franchini, In preparation The Frustration in being Odd: Can Boundary Conditions induce a Quantum Phase Transition?

  47. [55]

    Damski & M

    B. Damski & M. M. Rams, J. Phys. A 47 , 025303 (2014) Exact results for fidelity susceptibility of the quantum Ising model: The interplay between parity, system size, and magnetic field

    2014

  48. [56]

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

    1961

  49. [57]

    Jordan & E

    P. Jordan & E. Wigner, Z. Phys. 47, 631 (1928) ¨Uber das Paulische ¨Aquivalenzverbot

    1928

  50. [58]

    McCoy, Phys

    B.M. McCoy, Phys. Rev. 173, 531 (1968) Spin Correlation Functions of the X-Y Model

    1968

  51. [59]

    Barouch & B.M

    E. Barouch & B.M. McCoy, Phys. Rev. A 3 , 786 (1971) Statistical Mechanics of the XY Model. II. Spin-Correlation Functions

    1971

  52. [60]

    Dong & P

    J.-J. Dong & P. Li, Mod. Phys. Lett. B 31 , 1750061 (2017) The a-cycle problem in XY model with ring frustration

    2017

  53. [61]

    Mari´ c, & F

    V. Mari´ c, & F. Franchini, Asymptotic behavior of Toeplitz determinants with delta function singularities , arXiv:2006.01922 (2020)

    2020 arXiv

  54. [62]

    Mari´ c, S

    V. Mari´ c, S. M. Giampaolo, & F. Franchini, In preparation The Frustration in being Odd: Resilience against perturbations

  55. [63]

    Hirschman, Jr., Amer

    I.I. Hirschman, Jr., Amer. J. Math. 88, 577 (1966). The Strong Szeg¨ o Limit Theorem for Toeplitz Determinants

    1966

  56. [64]

    Wu, Phys

    T.T. Wu, Phys. Rev. 149, 380 (1966). Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. I

    1966

  57. [65]

    Campostrini, A

    M. Campostrini, A. Pelissetto, & E. Vicari, J. Stat. Mech. P11015 (2015) Quantum Ising chains with boundary fields

    2015

  58. [66]

    McCoy & T.T

    B. McCoy & T.T. Wu, The Two-Dimensional Ising Model , Harvard University Press (1973)

    1973

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Reviewed August 14, 2026 · model on record in the stance chip above.