Baxter (J. Phys. A 50, 014001 (2017)) derived exactly the bulk, surface and corner free energies fb,fs,fs′,fcf_b, f_s, f_s', f_c of the anisotropic zero-field square-lattice Ising model on a rectangle with open boundaries below TcT_c, confirming the Vernier-Jacobsen conjecture for the corner term. The spinor-based derivation is rigorous in substance, but the paper also gives a far shorter inversion-relation derivation that rests on unproved analyticity assumptions: the reduced free energies are assumed to admit Laurent expansions in the elliptic variable ww convergent in an annulus containing the inversion points w2=1w^2=1 and w2=qw^2=q. These assumptions are not evident because the transfer matrices of the unrotated lattice with open boundaries do not commute as the spectral parameter varies. The open problem is twofold: (1) give an a priori rigorous justification of the required analyticity properties, or otherwise establish rigorously the inversion and rotation relations for fb,fs,fs′f_b, f_s, f_s' and the anisotropy-independence of fcf_c, e.g. via reflection Yang-Baxter commutativity; (2) prove the summand factorization S1S4=S2S3S_1S_4=S_2S_3 linking the four free-energy series and determine the additive logarithmic terms, so that the corner free energy follows from the bulk and surface ones without the lengthy parity-split determinant calculation. The follow-up literature (Hucht's rectangle papers, Baxter's triangular-lattice series, scaling and conformal-field-theory analyses, and current work still using conjectural inversion relations) confirms and extends the values but leaves the rigor question open.

Background

The zero-field Ising model on a square lattice assigns spins σi=±1\sigma_i=\pm1 to the sites of an M×NM\times N rectangle, with dimensionless couplings H=βJH=\beta J on vertical edges and H′=βJ′H'=\beta J' on horizontal edges, and partition function Z=∑σexp⁡(H∑σiσj+H′∑σiσj)Z=\sum_\sigma \exp(H\sum\sigma_i\sigma_j + H'\sum\sigma_i\sigma_j) over the (M−1)N(M-1)N vertical and M(N−1)M(N-1) horizontal edges. 'Open' (free) boundary conditions mean the lattice simply ends at its four sides with no interaction across and no spins fixed on the boundary; Baxter calls these 'closed' boundary conditions (the boundary is a closed curve with no fixed spins), which is the same geometry other authors call open boundaries in both directions. In the thermodynamic limit the reduced free energy decomposes as −βF=MN βfb+M βfs+N βfs′+βfc+o(1)-\beta F = MN\,\beta f_b + M\,\beta f_s + N\,\beta f_s' + \beta f_c + o(1): a bulk free energy per site fbf_b, surface free energies fsf_s and fs′f_s' per unit length for the two boundary orientations, and a corner free energy fcf_c, a pure constant contributed by the four corners, which exists only when both directions have free boundaries.

The bulk free energy is Onsager's 1944 result. Kaufman's spinor method reformulated the calculation: the row-to-row transfer matrices (the 2N2^N-dimensional matrices V1V_1 carrying the vertical bonds and V2V_2 the horizontal ones, whose products propagate rows) are represented by Clifford-algebra (spinor) matrices, reducing ZZ to finite determinants. McCoy and Wu obtained fsf_s and fs′f_s' in 1967 by Toeplitz determinant methods (determinants of matrices whose entries depend only on the difference of the indices). The corner term resisted exact treatment longer: Cardy and Peschel (1988) predicted its logarithmic critical behaviour from conformal field theory, and Vernier and Jacobsen (2012), combining finite-lattice series with exact transfer-matrix enumeration, observed that all free energies take infinite-product forms with exponents linear in the product index and conjectured the exact fcf_c. Baxter (2017) then derived all four free energies exactly for the anisotropic model in the ordered phase below the critical temperature TcT_c (criticality is sinh⁡2H sinh⁡2H′=1\sinh 2H\,\sinh 2H'=1), confirming the conjectures; for instance

−βfc=log⁡2+14log⁡k′+4∑m oddqm/2(1+q2m)m(1+qm)2(1−qm).-\beta f_c = \log 2 + \tfrac{1}{4}\log k' + 4\sum_{m\ \mathrm{odd}} \frac{q^{m/2}(1+q^{2m})}{m(1+q^m)^2(1-q^m)}.

The working is parametrized by elliptic functions: k=(sinh⁡2H sinh⁡2H′)−1k=(\sinh 2H\,\sinh 2H')^{-1} is the elliptic modulus, k′=(1−k2)1/2k'=(1-k^2)^{1/2} its complement, KK and K′K' the complete elliptic integrals, q=e−πK′/Kq=e^{-\pi K'/K} the nome, and vv a spectral parameter whose physical regime is 0<Im v<K′0<\mathrm{Im}\,v<K'; setting w=eiπv/2Kw=e^{i\pi v/2K} maps this regime to the annulus q1/2<∣w∣<1q^{1/2}<|w|<1. Interchanging HH and H′H' corresponds to v→iK′−vv\to iK'-v, i.e. rotating the lattice through 90∘90^\circ. A striking output is that fcf_c depends on kk (equivalently qq) but not on vv (equivalently ww): like the spontaneous magnetization k′1/4k'^{1/4} of Yang, the corner free energy is independent of the anisotropy.

Alongside the rigorous spinor derivation, Baxter presented a much shorter route: the inversion relation method, introduced by Stroganov and developed by Baxter and Pearce. The transfer matrix T=V11/2V2V11/2T=V_1^{1/2}V_2V_1^{1/2} satisfies the exact matrix identity T(v) T(2iK′−v)=(2isinh⁡2H)N1T(v)\,T(2iK'-v)=(2i\sinh 2H)^N\mathbf{1}. Plausibly, the largest (Perron) eigenvalue at 2iK′−v2iK'-v is the analytic continuation of the largest eigenvalue at vv (all eigenvalues degenerate at the midpoint v=iK′v=iK') and the Perron eigenvector is unchanged; from the eigenvalue decomposition of ZZ this yields inversion relations for the four free energies, which, combined with the rotation relations from v→iK′−vv\to iK'-v, 'almost define' the four functions: if the reduced combinations −βfb−H−H′-\beta f_b-H-H', −βfs+H′-\beta f_s+H', −βfs′+H-\beta f_s'+H, −βfc-\beta f_c admit Laurent expansions in ww (expansions in integer powers, positive and negative) convergent in the annulus q1/2<∣w∣<1q^{1/2}<|w|<1, with singularities only at the known poles and zeros at the inversion points w2=1w^2=1 and w2=qw^2=q, then equating coefficients reproduces the closed forms of fb,fs,fs′f_b,f_s,f_s' and forces every non-constant coefficient of fcf_c to vanish, proving its anisotropy-independence.

The gap: for the square lattice rotated through 45∘45^\circ with cylindrical boundary conditions the row-to-row transfer matrices at different vv commute, as a consequence of the Yang-Baxter relation (the algebraic consistency identity underlying exact solvability), so their eigenvalues are meromorphic in vv even at finite width and the inversion argument can be made rigorous. For the geometry actually used, the unrotated lattice with open boundaries, the transfer matrices do not commute, the eigenvectors depend on vv through the zeros z1,…,zNz_1,\dots,z_N of a polynomial P(z)P(z), and Baxter states plainly that the section 'is not rigorous' and that 'we do not have any a priori reason for believing the surface and corner free energies to be meromorphic functions of vv', while noting that reflection Yang-Baxter relations, boundary analogues of the Yang-Baxter equation that restore commutativity for open systems and were used by O'Brien and Pearce to obtain surface free energies of the ABF restricted solid-on-solid models (the Andrews-Baxter-Forrester height models, in which each site carries an integer height restricted to a finite set of consecutive values), might provide a rigorous route.

A second, structural gap concerns the corner term specifically. The integrand of Baxter's free-energy integral factorizes into a part linear in MM and a part linear in NN, which implies that the summands S1,S2,S3,S4S_1,S_2,S_3,S_4 of the four final series satisfy S1S4=S2S3S_1S_4=S_2S_3: the corner summand is algebraically determined by the bulk and surface summands up to simple additive terms, logarithms of simple functions of the Boltzmann weights such as log⁡2\log 2 and 14log⁡k′\tfrac{1}{4}\log k'. If this factorization could be justified and the additive terms explained, the corner free energy would follow from the bulk and surface free energies and the lengthy calculation, in particular the parity-split (NN even/odd) determinant evaluations of the appendices, could be eliminated. Baxter's summary poses exactly this: 'If this could be justified, and the additional simple algebraic terms ... explained, then we could eliminate the lengthy calculation herein of the corner free energy.'

Problem Statement

For the anisotropic zero-field square-lattice Ising model on a rectangle with open boundaries in both directions, in the ordered ferromagnetic regime below TcT_c treated in the source, replace the heuristic inversion-relation derivation by a rigorous one and justify the algebraic structure that links the four free energies, so that the corner free energy is obtained without the lengthy parity-split determinant calculation. Two results are required.

(1) Analyticity foundation of the inversion-relation method. Prove, a priori (without assuming the closed forms of the free energies), that the reduced free energies −βfb−H−H′-\beta f_b-H-H', −βfs+H′-\beta f_s+H', −βfs′+H-\beta f_s'+H and −βfc-\beta f_c admit Laurent expansions in the elliptic variable ww that converge in the annulus q1/2<∣w∣<1q^{1/2}<|w|<1, whose boundary carries the inversion points w2=1w^2=1 and w2=qw^2=q and where the only permitted singularities are the known poles and zeros at those points; or otherwise establish rigorously, for the transfer matrices of the unrotated lattice with open boundaries (which do not commute as vv varies), the inversion relations

−βfb(v)−βfb(2iK′−v)=iπ/2+log⁡[2sinh⁡2H],−βfs(v)−βfs(2iK′−v)=0,-\beta f_b(v)-\beta f_b(2iK'-v)=i\pi/2+\log[2\sinh 2H], \quad -\beta f_s(v)-\beta f_s(2iK'-v)=0,

−βfs′(v)+βfs′(2iK′−v)=iπ/2+log⁡tanh⁡H,−βfc(v)+βfc(2iK′−v)=0,-\beta f_s'(v)+\beta f_s'(2iK'-v)=i\pi/2+\log\tanh H, \quad -\beta f_c(v)+\beta f_c(2iK'-v)=0,

together with the rotation relations obtained from v→iK′−vv\to iK'-v (e.g. fs(iK′−v)=fs′(v)f_s(iK'-v)=f_s'(v), fc(iK′−v)=fc(v)f_c(iK'-v)=f_c(v)), for instance via reflection Yang-Baxter commutativity of suitable double-row transfer matrices (the open-boundary transfer matrices built from a row operator together with its mirror reflection). As consequences, the coefficient-matching derivation of fbf_b, fsf_s and fs′f_s', and the vanishing of all non-constant Laurent coefficients of fcf_c (the anisotropy-independence of the corner free energy), must become theorems rather than plausibility arguments.

(2) Structure of the corner term. Prove the summand factorization S1S4=S2S3S_1S_4=S_2S_3 linking the summands of the four free-energy series (ideally at finite system size, before the thermodynamic limit), and determine the remaining additive algebraic terms, logarithms of simple functions of the Boltzmann weights (in the source's convention, log⁡2\log 2 and 14log⁡k′\tfrac{1}{4}\log k', noting the later literature's reattribution of the constant log⁡2\log 2 to the symmetry-breaking residual part), so that the corner free energy is rigorously obtained, up to its explicitly identified constant term, from the bulk and surface free energies alone.

Extensions to the disordered phase, to other boundary conditions, or to other lattices are natural generalizations but are not required for completion: for the same geometry they have been substantially addressed in the later literature by other methods, as recorded in the References.

References

  1. R. J. Baxter, The bulk, surface and corner free energies of the square lattice Ising model, J. Phys. A: Math. Theor. 50 (2017) 014001, DOI 10.1088/1751-8113/50/1/014001, arXiv:1606.02029, https://arxiv.org/abs/1606.02029
  2. Eric Vernier and Jesper Lykke Jacobsen, Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices, J. Phys. A: Math. Theor. 45 (2012) 045003, DOI 10.1088/1751-8113/45/4/045003, arXiv:1110.2158, https://arxiv.org/abs/1110.2158
  3. R. J. Baxter, Surface and corner free energies of the self-dual Potts model, arXiv:1606.01616 (2016), https://arxiv.org/abs/1606.01616
  4. Rodney J. Baxter, The bulk, surface and corner free energies of the anisotropic triangular Ising model, Proc. R. Soc. A 476 (2020) 20190713, DOI 10.1098/rspa.2019.0713 (preprint arXiv:1909.00518, titled 'Bulk, surface and corner free energies of the anisotropic triangular Ising model: series expansions and critical behaviour'), https://arxiv.org/abs/1909.00518
  5. Alfred Hucht, The square lattice Ising model on the rectangle I: finite systems, J. Phys. A: Math. Theor. 50 (2017) 065201, DOI 10.1088/1751-8121/aa5535, arXiv:1609.01963, https://arxiv.org/abs/1609.01963
  6. Alfred Hucht, The square lattice Ising model on the rectangle II: finite-size scaling limit, J. Phys. A: Math. Theor. 50 (2017) 265205, DOI 10.1088/1751-8121/aa6b7a, arXiv:1701.08722, https://arxiv.org/abs/1701.08722
  7. Alfred Hucht, The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants, J. Phys. A: Math. Theor. 54 (2021) 375201, DOI 10.1088/1751-8121/ac0983, arXiv:2103.10776, https://arxiv.org/abs/2103.10776
  8. Hendrik Hobrecht and Alfred Hucht, Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields, SciPost Phys. 8 (2020) 032, DOI 10.21468/SciPostPhys.8.3.032, arXiv:1805.00369, https://arxiv.org/abs/1805.00369
  9. Doron Gepner, On the free energy of solvable lattice models, Nucl. Phys. B 971 (2021) 115532, DOI 10.1016/j.nuclphysb.2021.115532, arXiv:2011.14108, https://arxiv.org/abs/2011.14108
  10. Florian Kischel and Stefan Wessel, Quantifying nonuniversal corner free-energy contributions in weakly-anisotropic two-dimensional critical systems, arXiv:2403.10422 (2024), https://arxiv.org/abs/2403.10422
  11. Lars Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65 (1944) 117-149, DOI 10.1103/PhysRev.65.117, https://doi.org/10.1103/PhysRev.65.117
  12. Bruria Kaufman, Crystal statistics. II. Partition function evaluated by spinor analysis, Phys. Rev. 76 (1949) 1232-1243, DOI 10.1103/PhysRev.76.1232, https://doi.org/10.1103/PhysRev.76.1232
  13. Barry M. McCoy and Tai Tsun Wu, Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. IV, Phys. Rev. 162 (1967) 436-475, DOI 10.1103/PhysRev.162.436, https://doi.org/10.1103/PhysRev.162.436
  14. John L. Cardy and Ingo Peschel, Finite-size dependence of the free energy in two-dimensional critical systems, Nucl. Phys. B 300 (1988) 377-392, DOI 10.1016/0550-3213(88)90604-9, https://doi.org/10.1016/0550-3213(88)90604-9
  15. R. J. Baxter, The inversion relation method for some two-dimensional exactly solved models in lattice statistics, J. Stat. Phys. 28 (1982) 1-41, DOI 10.1007/BF01011621, https://doi.org/10.1007/BF01011621
  16. Paul A. Pearce, Transfer-matrix inversion identities for exactly solvable lattice-spin models, Phys. Rev. Lett. 58 (1987) 1502-1504, DOI 10.1103/PhysRevLett.58.1502, https://doi.org/10.1103/PhysRevLett.58.1502
  17. David L. O'Brien and Paul A. Pearce, Surface free energies, interfacial tensions and correlation lengths of the ABF models, J. Phys. A: Math. Gen. 30 (1997) 2353-2366, DOI 10.1088/0305-4470/30/7/017, https://doi.org/10.1088/0305-4470/30/7/017
  18. Xintian Wu and Shenghong Sun, The critical two dimensional Ising model on right-triangle-shaped square lattices, Physica A 558 (2020) 124961, DOI 10.1016/j.physa.2020.124961, https://doi.org/10.1016/j.physa.2020.124961
  19. C. N. Yang, The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev. 85 (1952) 808-816, DOI 10.1103/PhysRev.85.808, https://doi.org/10.1103/PhysRev.85.808
  20. Yu. G. Stroganov, A new calculation method for partition functions in some lattice models, Phys. Lett. A 74 (1979) 116-118, DOI 10.1016/0375-9601(79)90601-7, https://doi.org/10.1016/0375-9601(79)90601-7