pith. sign in

arxiv: 2604.16992 · v1 · submitted 2026-04-18 · ❄️ cond-mat.stat-mech · math-ph· math.MP

Solution of the Ising model with Brascamp-Kunz boundary conditions by the transfer matrix method

De-Zhang Li , Xin Wang This is my paper

Pith reviewed 2026-05-10 06:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords Ising modelBrascamp-Kunz boundary conditionstransfer matrix methodpartition functionFisher zeroscritical pointfermionic representationtoroidal boundary conditions
0
0 comments X

The pith

The Ising model with Brascamp-Kunz boundary conditions is exactly solved by mapping to toroidal boundaries and using the transfer matrix in fermionic representation.

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

This paper derives an exact solution for the square-lattice Ising model under Brascamp-Kunz boundary conditions by means of the transfer matrix formalism. Special interactions are imposed on the boundaries and a limit is taken to convert the system into an equivalent one with toroidal boundaries. The Schultz-Mattis-Lieb method is then applied to the mapped system, yielding the partition function in fermionic form. From this expression the Fisher zeros are obtained analytically and the physical critical point is located. The derivation complements the existing Pfaffian solution and extends the set of transfer-matrix treatments for Ising models with different boundary conditions.

Core claim

By setting special interactions on the boundaries and taking certain limits, the Brascamp-Kunz system is transformed into a system under toroidal boundary conditions. The Schultz-Mattis-Lieb method is applied to the mapping system and the partition function is exactly solved in the fermionic representation. The Fisher zeros are analytically calculated and the physical critical point is identified.

What carries the argument

The boundary mapping via special interactions and limits, followed by the fermionic transfer-matrix formalism of the Schultz-Mattis-Lieb method.

If this is right

  • The partition function is obtained exactly in fermionic variables.
  • Fisher zeros are calculated analytically from the closed-form expression.
  • The physical critical point is identified directly from the location of the zeros.
  • Explicit differences between the transfer-matrix treatments of Brascamp-Kunz and toroidal systems are clarified.
  • The solution joins the collection of transfer-matrix derivations for the Ising model under varied boundary conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same boundary-mapping technique could be tested on other non-periodic boundary conditions to obtain exact solutions.
  • The fermionic representation may permit direct evaluation of additional quantities such as magnetization or spin correlations.
  • The work illustrates how controlled boundary adjustments can enable exact solvability while leaving bulk thermodynamics unchanged.
  • Similar mappings might link the Brascamp-Kunz case to other exactly solvable lattice models.

Load-bearing premise

Setting special interactions on the boundaries and taking a certain limit transforms the Brascamp-Kunz system into an equivalent toroidal-boundary system without changing the essential thermodynamics or introducing artifacts.

What would settle it

An independent calculation of the Fisher zeros or the critical temperature on the original Brascamp-Kunz system, for instance by Monte Carlo simulation, that yields values different from those obtained after the boundary mapping and limit.

Figures

Figures reproduced from arXiv: 2604.16992 by De-Zhang Li, Xin Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. The Ising model under the B-K BCs on the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The new system under the toroidal BCs, with the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
read the original abstract

The square lattice Ising model under the Brascamp-Kunz boundary conditions is a well-known exactly solvable lattice model. The exact solution of this system has been derived within the framework of Pfaffian-type method. In this paper we provide a derivation for the solution by the Schultz-Mattis-Lieb method in the transfer matrix formalism. We set special interactions on the boundaries and take certain limit of these interactions, so that the system under the Brascamp-Kunz boundary conditions is transformed into another system under the toroidal boundary conditions. The Schultz-Mattis-Lieb method is applied to the mapping system and the partition function is exactly solved in the fermionic representation. The Fisher zeros are analytically calculated and the physical critical point is identified. We also discuss the difference between the transfer matrix approaches to the Brascamp-Kunz and to the toroidal boundary conditions. Our work introduces a member to the family of transfer-matrix-based studies for Ising model under various boundary conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to derive an exact solution for the square-lattice Ising model with Brascamp-Kunz boundary conditions via the transfer-matrix formalism. Special boundary interactions are introduced and a limit is taken to map the system onto an equivalent toroidal-boundary system; the Schultz-Mattis-Lieb method is then applied to obtain the partition function in the fermionic representation. Analytic expressions for the Fisher zeros are derived and the physical critical point is identified. Differences between the transfer-matrix treatments of the two boundary conditions are discussed.

Significance. If the mapping is rigorously justified, the work supplies a transfer-matrix route to a previously Pfaffian-solved model, enlarges the set of boundary conditions for which closed-form solutions exist, and yields explicit Fisher-zero loci that can be compared across boundary conditions. The analytic treatment of the zeros and the explicit discussion of methodological differences constitute concrete strengths.

major comments (2)
  1. [Abstract and mapping procedure] Abstract and the mapping procedure (presumably §2–3): the claim that the Brascamp-Kunz system is transformed into the toroidal system by special boundary couplings and a subsequent limit is load-bearing for every subsequent result. No explicit demonstration is given that the limit commutes with the thermodynamic limit, that the partition function remains identical (rather than merely asymptotically equivalent), or that the loci of the Fisher zeros are unchanged. Because Brascamp-Kunz conditions rely on a distinct Pfaffian structure, any mismatch would invalidate the fermionic diagonalization and the analytic zero expressions derived from it.
  2. [Fisher-zero calculation] Fisher-zero calculation (presumably §4): the analytic expressions for the zeros are obtained after the mapping. Without a direct verification that the zero loci of the original Brascamp-Kunz partition function coincide with those of the mapped toroidal system, the claimed analytic results cannot be attributed to the Brascamp-Kunz model.
minor comments (2)
  1. Notation for the boundary couplings and the precise form of the limit should be introduced with an equation rather than described only in words.
  2. The discussion of differences between the two transfer-matrix approaches would benefit from a side-by-side comparison of the resulting fermionic spectra or characteristic equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major points raised below, providing our response and indicating the revisions we will implement.

read point-by-point responses

  1. Referee: [Abstract and mapping procedure] Abstract and the mapping procedure (presumably §2–3): the claim that the Brascamp-Kunz system is transformed into the toroidal system by special boundary couplings and a subsequent limit is load-bearing for every subsequent result. No explicit demonstration is given that the limit commutes with the thermodynamic limit, that the partition function remains identical (rather than merely asymptotically equivalent), or that the loci of the Fisher zeros are unchanged. Because Brascamp-Kunz conditions rely on a distinct Pfaffian structure, any mismatch would invalidate the fermionic diagonalization and the analytic zero expressions derived from it.

    Authors: We agree that a rigorous justification of the mapping is essential. In the manuscript we introduce auxiliary boundary interactions and take their strength to a limiting value to enforce the Brascamp-Kunz conditions, thereby reproducing the toroidal system. For any finite lattice this limit makes the Boltzmann weights (and hence the partition function) exactly identical to those of the toroidal system; the thermodynamic limit is subsequently taken on the equivalent finite-size system. We will revise Sections 2 and 3 to include an explicit demonstration of this finite-size equivalence, together with a discussion confirming that the order of limits does not affect the result. Because the finite-size partition functions coincide exactly, the loci of their zeros are identical before the thermodynamic limit is taken. revision: yes

  2. Referee: [Fisher-zero calculation] Fisher-zero calculation (presumably §4): the analytic expressions for the zeros are obtained after the mapping. Without a direct verification that the zero loci of the original Brascamp-Kunz partition function coincide with those of the mapped toroidal system, the claimed analytic results cannot be attributed to the Brascamp-Kunz model.

    Authors: The mapping establishes exact equality of the finite-size partition functions, so the zeros coincide by construction. The analytic expressions obtained via the Schultz-Mattis-Lieb diagonalization of the toroidal system therefore apply directly to the Brascamp-Kunz model. In the revised manuscript we will add a short paragraph in Section 4 that recalls this equality and, for additional transparency, includes a brief numerical check of the zero loci for small lattices before and after the limit is taken. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation maps to known toroidal system via explicit limit and applies standard fermionic method

full rationale

The paper's central chain sets special boundary interactions and takes a limit to map Brascamp-Kunz conditions onto a toroidal system, then applies the established Schultz-Mattis-Lieb transfer-matrix technique to obtain the fermionic representation, partition function, and Fisher zeros. This mapping is presented as a concrete transformation rather than a definitional equivalence or fitted parameter; the subsequent analytic steps follow from the known toroidal solution without reducing the output to the input by construction. No self-citations bear the load of the result, no ansatz is smuggled, and no predictions are statistically forced from subsets of data. The derivation remains self-contained against external benchmarks such as the pre-existing toroidal Ising solution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard nearest-neighbor Ising Hamiltonian and the validity of the boundary-interaction limit mapping to toroidal conditions; no new free parameters or invented entities are introduced beyond the established fermionic representation of the transfer matrix.

axioms (1)
  • domain assumption The square-lattice Ising model with nearest-neighbor interactions is exactly solvable under toroidal boundary conditions via the Schultz-Mattis-Lieb method.
    Invoked when the Brascamp-Kunz system is mapped to the toroidal case.

pith-pipeline@v0.9.0 · 5472 in / 1220 out tokens · 45948 ms · 2026-05-10T06:39:13.180330+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    2N−1P j=1 (C† j −Cj )(C† j+1 +Cj+1 )−U(C † 2N −C2N )(C† 1 +C1) # , V ′ 1 =e K3

    Jordan-Wigner transformation Define the spin raising and lowering operators σ+ i = 1 2(σx i +iσ y i ), σ − i = 1 2(σx i −iσ y i ).(15) It is straightforward to find σx i =σ + i +σ − i , σ z i = 2σ+ i σ− i −1(16) (here1is a2 2N ×2 2N identity matrix). Equation (12) is then written as V1 =e K1[(σ+ 1 +σ− 1 )(σ+ 2 +σ− 2 )+···+(σ+ 2N +σ− 2N )(σ+ 1 +σ− 1 )], V ...

    work page

  2. [2]

    It is expected to perform a direct product decompo- sition to these matrices so that the diagonalization will be simplified

    Direct product decomposition Via the Jordan-Wigner transformation we obtain the expressions ofV ± 1 ,V ′ 1 ± andV 2 in terms of fermion opera- tors. It is expected to perform a direct product decompo- sition to these matrices so that the diagonalization will be simplified. Following the SML method we take the linear canonical transformation to a new set o...

    work page

  3. [3]

    (50) V ′′± 1,l andV ′± 1,l are defined similarly. Obviously the oper- atorUhas the form U + = NY ⊗ l=1 e iπ η† 2l−1η2l−1+η† −(2l−1)η−(2l−1) ,for the even part; U − =   N−1Y ⊗ l=1 eiπ(η† 2lη2l+η† −2lη−2l)   ⊗e iπη † 0η0 ⊗e iπη † πηπ , for the odd part.(51) Then we can return to Eq. (34). The even and odd parts ofV M V ′′V ′ can be given in terms of dir...

    work page

  4. [4]

    Ising,Beitrag zur Theorie des Ferromagnetismus, Z

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

    work page 1925

  5. [5]

    S. G. Brush,History of the Lenz-Ising Model, Rev. Mod. Phys.39, 883–893 (1967)

    work page 1967

  6. [6]

    Niss,History of the Lenz-Ising Model 1920–1950: From Ferromagnetic to Cooperative Phenomena, Arch

    M. Niss,History of the Lenz-Ising Model 1920–1950: From Ferromagnetic to Cooperative Phenomena, Arch. Hist. Exact Sci.59, 267–318 (2005)

    work page 1920

  7. [7]

    Peierls,On Ising’s model of ferromagnetism, Proc

    R. Peierls,On Ising’s model of ferromagnetism, Proc. Camb. Phil. Soc.32, 477–481 (1936)

    work page 1936

  8. [8]

    H. A. Kramers and G. H. Wannier,Statistics of the Two- Dimensional Ferromagnet. Part I, Phys. Rev.60, 252– 262 (1941)

    work page 1941

  9. [9]

    H. A. Kramers and G. H. Wannier,Statistics of the Two- Dimensional Ferromagnet. Part II, Phys. Rev.60, 263– 276 (1941)

    work page 1941

  10. [10]

    Onsager,Crystal Statistics

    L. Onsager,Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev.65, 117–149 (1944)

    work page 1944

  11. [11]

    Husimi and I

    K. Husimi and I. Syôzi,The Statistics of Honeycomb and Triangular Lattice. I, Prog. Theor. Phys.5, 177–186 (1950)

    work page 1950

  12. [12]

    II, Prog

    I.Syôzi,The Statistics of Honeycomb and Triangular Lat- tice. II, Prog. Theor. Phys.5, 341–351 (1950)

    work page 1950

  13. [13]

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

    work page 1950

  14. [14]

    Syôzi,Statistics of Kagomé Lattice, Prog

    I. Syôzi,Statistics of Kagomé Lattice, Prog. Theor. Phys. 6, 306–308 (1951)

    work page 1951

  15. [15]

    Kanô and S

    K. Kanô and S. Naya,Antiferromagnetism. The Kagomé Ising Net, Prog. Theor. Phys.10, 158–172 (1953)

    work page 1953

  16. [16]

    H. J. Giacomini,Exact results for a checkerboard Ising model with crossing and four-spin interactions, J. Phys. A: Math. Gen.18, L1087–L1093 (1985)

    work page 1985

  17. [17]

    T. D. Lee and C. N. Yang,Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model, Phys. Rev.87, 410–419 (1952)

    work page 1952

  18. [18]

    G.Baxter,Weight Factors for the Two-Dimensional Ising Model, J. Math. Phys.6, 1015–1021 (1965)

    work page 1965

  19. [19]

    B. M. McCoy and T. T. Wu,Theory of Toeplitz Determi- nants and the Spin Correlations of the Two-Dimensional Ising Model. II, Phys. Rev.155, 438–452 (1967)

    work page 1967

  20. [20]

    F. Y. Wu,Two-dimensional Ising model with crossing and four-spin interactions and a magnetic field i(π/2)kT, J. Stat. Phys.44, 455–463 (1986)

    work page 1986

  21. [21]

    K. Y. Lin and F. Y. Wu,Ising Model In The Magnetic Field iπkT/2, Int. J. Mod. Phys. B02, 471–481 (1988)

    work page 1988

  22. [22]

    D.-Z. Li, X. Wang, and X.-B. Yang,Free-Fermion Models and Two-Dimensional Ising Models Under Zero Field and Imaginary Fieldi(π/2)k BT, Entropy27, 799 (2025)

    work page 2025

  23. [23]

    R. J. Baxter,Exactly solved models in statistical mechan- ics(Academic Press, London, 1982)

    work page 1982

  24. [24]

    McCoy and T

    B. McCoy and T. Wu,The Two-Dimensional Ising Model: Second Edition(Dover Publications, New York, 2014)

    work page 2014

  25. [25]

    Kaufman,Crystal Statistics

    B. Kaufman,Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis, Phys. Rev.76, 1232–1243 (1949)

    work page 1949

  26. [26]

    Nambu,A Note on the Eigenvalue Problem in Crystal Statistics, Prog

    Y. Nambu,A Note on the Eigenvalue Problem in Crystal Statistics, Prog. Theor. Phys.5, 1–13 (1950)

    work page 1950

  27. [27]

    T. D. Schultz, D. C. Mattis, and E. H. Lieb,Two- Dimensional Ising Model as a Soluble Problem of Many Fermions, Rev. Mod. Phys.36, 856–871 (1964)

    work page 1964

  28. [28]

    C. J. Thompson,Algebraic Derivation of the Partition Function of a Two-Dimensional Ising Model, J. Math. Phys.6, 1392–1395 (1965)

    work page 1965

  29. [29]

    Kastening,Simplifying Kaufman’s solution of the two- dimensional Ising model,Phys.Rev.E64,066106(2001)

    B. Kastening,Simplifying Kaufman’s solution of the two- dimensional Ising model,Phys.Rev.E64,066106(2001)

    work page 2001

  30. [30]

    Kac and J

    M. Kac and J. C. Ward,A Combinatorial Solution of the Two-Dimensional Ising Model, Phys. Rev.88, 1332–1337 (1952)

    work page 1952

  31. [31]

    R. B. Potts and J. C. Ward,The Combinatrial Method and the Two-Dimensional Ising Model, Prog. Theor. Phys.13, 38–46 (1955)

    work page 1955

  32. [32]

    Sherman,Combinatorial Aspects of the Ising Model for Ferromagnetism

    S. Sherman,Combinatorial Aspects of the Ising Model for Ferromagnetism. I. A Conjecture of Feynman on Paths and Graphs, J. Math. Phys.1, 202–217 (1960)

    work page 1960

  33. [33]

    Sherman,Combinatorial aspects of the Ising model 12 for ferromagnetism

    S. Sherman,Combinatorial aspects of the Ising model 12 for ferromagnetism. II. An analogue to the Witt identity, Bull. Amer. Math. Soc.68, 225–229 (1962)

    work page 1962

  34. [34]

    P. N. Burgoyne,Remarks on the Combinatorial Approach to the Ising Problem, J. Math. Phys.4, 1320–1326 (1963)

    work page 1963

  35. [35]

    N.V.Vdovichenko,A calculation of the partition function for a plane dipole lattice, Sov. Phys. JETP20, 477–479 (1965)

    work page 1965

  36. [36]

    M. L. Glasser,Exact Partition Function for the Two- Dimensional Ising Model, Am. J. Phys.38, 1033–1036 (1970)

    work page 1970

  37. [37]

    da Costa and A

    G. da Costa and A. L. Maciel,Combinatorial formulation of Ising model revisited, Rev. Bras. Ensino Fís.25, 49–61 (2003)

    work page 2003

  38. [38]

    C. A. Hurst and H. S. Green,New Solution of the Ising Problem for a Rectangular Lattice, J. Chem. Phys.33, 1059–1062 (1960)

    work page 1960

  39. [39]

    C. A. Hurst,Applicability of the Pfaffian Method to Com- binatorial Problems on a Lattice, J. Math. Phys.5, 90– 100 (1964)

    work page 1964

  40. [40]

    C. A. Hurst,New Approach to the Ising Problem, J. Math. Phys.7, 305–310 (1966)

    work page 1966

  41. [41]

    M. E. Fisher,On the Dimer Solution of Planar Ising Models, J. Math. Phys.7, 1776–1781 (1966)

    work page 1966

  42. [42]

    R. W. Gibberd and C. A. Hurst,New Approach to the Ising ModeI. II, J. Math. Phys.8, 1427–1435 (1967)

    work page 1967

  43. [43]

    E. W. Montroll, Lattice Statistics, inApplied Combina- torial Mathematics(Wiley, New York, 1964)

    work page 1964

  44. [44]

    Samuel,The use of anticommuting variable integrals in statistical mechanics

    S. Samuel,The use of anticommuting variable integrals in statistical mechanics. I. The computation of partition functions, J. Math. Phys.21, 2806–2814 (1980)

    work page 1980

  45. [45]

    V. N. Plechko,Simple solution of two-dimensional ising model on a torus in terms of Grassmann integrals, Theor. Math. Phys.64, 748–756 (1985)

    work page 1985

  46. [46]

    H.J.BrascampandH.Kunz,Zeroes of the partition func- tion for the Ising model in the complex temperature plane, J. Math. Phys.15, 65–66 (1974)

    work page 1974

  47. [47]

    M. E. Fisher, The nature of critical points, inLectures in Theoretical Physics: Volume VII C - Statistical Physics, Weak Interactions, Field Theory(University of Colorado Press, Boulder, 1965)

    work page 1965

  48. [48]

    W. T. Lu and F. Y. Wu,Density of the Fisher Zeroes for the Ising Model, J. Stat. Phys.102, 953–970 (2001)

    work page 2001

  49. [49]

    B. M. McCoy and T. T. Wu,Theory of Toeplitz Determi- nants and the Spin Correlations of the Two-Dimensional Ising Model. IV, Phys. Rev.162, 436–475 (1967)

    work page 1967

  50. [50]

    Kastening,Simplified transfer matrix approach in the two-dimensional Ising model with various boundary con- ditions, Phys

    B. Kastening,Simplified transfer matrix approach in the two-dimensional Ising model with various boundary con- ditions, Phys. Rev. E66, 057103 (2002)

    work page 2002

  51. [51]

    Lyberg,The Ising lattice with Brascamp-Kunz bound- ary conditions and an external magnetic field, arXiv preprint arXiv:0805.2497 10.48550/arXiv.0805.2497 (2008)

    I. Lyberg,The Ising lattice with Brascamp-Kunz bound- ary conditions and an external magnetic field, arXiv preprint arXiv:0805.2497 10.48550/arXiv.0805.2497 (2008)

    work page doi:10.48550/arxiv.0805.2497 2008

  52. [52]

    Lyberg,Free energy of the anisotropic Ising lattice with Brascamp-Kunz boundary conditions, Phys

    I. Lyberg,Free energy of the anisotropic Ising lattice with Brascamp-Kunz boundary conditions, Phys. Rev. E87, 062141 (2013)

    work page 2013

  53. [53]

    Li and X

    D.-Z. Li and X. Wang,Free-fermion approach to the par- tition function zeros: Special boundary conditions and product form of solution, Phys. Rev. Research7, 043258 (2025)

    work page 2025

  54. [54]

    D. B. Abraham,On the Transfer Matrix for the Two- Dimensional Ising Model, Stud. Appl. Math.50, 71–88 (1971)

    work page 1971

  55. [55]

    N. S. Izmailian and Y.-N. Yeh,Ising model with mixed boundary conditions: Universal amplitude ratios, Nucl. Phys. B814, 573–581 (2009)

    work page 2009

  56. [56]

    R.J.Baxter,The bulk, surface and corner free energies of the square lattice Ising model, J. Phys. A: Math. Theor. 50, 014001 (2017)

    work page 2017

  57. [57]

    Poghosyan, N

    A. Poghosyan, N. Izmailian, and R. Kenna,Exact solu- tion of the critical Ising model with special toroidal bound- ary conditions, Phys. Rev. E96, 062127 (2017)

    work page 2017

  58. [58]

    Jordan and E

    P. Jordan and E. Wigner,Über das Paulische Äquivalen- zverbot, Z. Phys.47, 631–651 (1928)

    work page 1928

  59. [59]

    Beichert,Phases at complex temperature : spiral cor- relation functions and regions of Fisher zeros for Ising models, Ph.D

    F. Beichert,Phases at complex temperature : spiral cor- relation functions and regions of Fisher zeros for Ising models, Ph.D. thesis (University of St Andrews, 2013)

    work page 2013