The six-vertex model with domain-wall boundary conditions and one reflecting end (U-turn boundary) on an N×2NN\times 2N lattice is an integrable boundary class whose partition function is the Tsuchiya determinant and whose configurations at the combinatorial point enumerate vertically symmetric alternating sign matrices. Ribeiro and Korepin (2015) derived its bulk free energy per site only in the disordered regime −1<Δ<1-1<\Delta<1, by showing that the homogeneous Tsuchiya determinant satisfies the bidimensional Toda equation and reducing the large-NN problem to a Liouville equation whose solution is fixed by exactly solvable special points. For the ferroelectric regime Δ>1\Delta>1 they could only propose a leading-state expression, e−2F(λ,μ)=sinh⁡(λ−∣μ∣+∣γ∣)sinh⁡(λ+∣μ∣−γ) sinh⁡(λ+∣μ∣+γ)e^{-2F(\lambda,\mu)}=\sinh(\lambda-|\mu|+|\gamma|)\sqrt{\sinh(\lambda+|\mu|-\gamma)\,\sinh(\lambda+|\mu|+\gamma)}, and stated they were 'unable to fix a solution of Liouville equation which agrees with our formula'; the antiferroelectric regime Δ<−1\Delta<-1 is untouched, and the analog of the arctic-circle phase-separation phenomenon is undetermined outside the disordered phase. The problem is to complete the phase-dependent thermodynamic limit of this model: derive the large-NN free energy in the ferroelectric regime (establishing or correcting the conjectural formula, e.g. by supplying suitable homogeneous-limit boundary data for the Toda-to-Liouville reduction or by an alternative asymptotic analysis of the determinant), determine the antiferroelectric free energy, and characterize spatial phase separation (the arctic curve) in both regimes. The ferroelectric and antiferroelectric regimes of the reflecting-end six-vertex model remain open, and the rigorous asymptotic program that solved all phases for domain-wall and half-turn boundaries has not been extended to the reflecting end.

Background

The six-vertex model places arrows on the edges of a square lattice subject to the ice rule — two arrows point into and two out of every vertex — leaving six local vertex configurations with Boltzmann weights aa, bb, cc. Its phase structure is governed by the anisotropy parameter Δ=(a2+b2−c2)/(2ab)\Delta=(a^2+b^2-c^2)/(2ab): Δ>1\Delta>1 is the ferroelectric regime (hyperbolic parametrization a=sinh⁡(λ−γ)a=\sinh(\lambda-\gamma), b=sinh⁡(λ+γ)b=\sinh(\lambda+\gamma), c=sinh⁡(2∣γ∣)c=\sinh(2|\gamma|), so Δ=cosh⁡2γ\Delta=\cosh 2\gamma), −1<Δ<1-1<\Delta<1 the disordered regime (trigonometric weights), and Δ<−1\Delta<-1 the antiferroelectric regime.

Boundary conditions drastically change the thermodynamics. With domain-wall boundary conditions (DWBC — the external arrows on all four sides are fixed so that they point in on two opposite sides and out on the other two), the partition function is the Izergin–Korepin determinant; Korepin and Zinn-Justin showed that the thermodynamic-limit free energy differs from the periodic-boundary value, and Bleher, Fokin and Liechty later derived it rigorously in all three regimes by a Riemann–Hilbert steepest-descent analysis (a rigorous asymptotic technique from the theory of orthogonal polynomials) applied to the determinant.

The model considered here replaces one horizontal side of the domain-wall square by a reflecting end, also called a U-turn boundary: horizontal lines are pairwise connected by U-turns that reflect arrows back into the lattice, the U-turns carrying weights given by a diagonal reflection (KK-)matrix in Sklyanin's reflection algebra. On the resulting N×2NN\times 2N lattice the partition function is the Tsuchiya determinant, and at the combinatorial point a=b=ca=b=c it enumerates vertically symmetric alternating sign matrices (ASMs — 0,±10,\pm1 matrices whose rows and columns sum to 1 and alternate in sign; the vertically symmetric symmetry class corresponds to this boundary). The bulk free energy per site is F=−lim⁡N→∞log⁡ZN/(2N2)F=-\lim_{N\to\infty}\log Z_N/(2N^2) (temperature set to 1), where λ\lambda and μ\mu are the homogeneous row and column spectral parameters.

Ribeiro and Korepin solved the disordered regime. In the homogeneous limit the Tsuchiya determinant τN\tau_N satisfies the bidimensional Toda equation −∂λμ2log⁡τN=τN+1τN−1/τN2-\partial^2_{\lambda\mu}\log\tau_N=\tau_{N+1}\tau_{N-1}/\tau_N^2; the ansatz τN=CN e2N2f(λ,μ)+O(N)\tau_N=C_N\,e^{2N^2 f(\lambda,\mu)+O(N)} reduces the thermodynamic limit to the Liouville equation −2 ∂λμ2f=e4f-2\,\partial^2_{\lambda\mu}f=e^{4f}, whose general solution e2f=−u′(λ)v′(μ)/(u(λ)+v(μ))e^{2f}=\sqrt{-u'(\lambda)v'(\mu)}/(u(\lambda)+v(\mu)) they fixed by matching exactly solvable special points (such as μ=±(λ±γ)\mu=\pm(\lambda\pm\gamma) and γ=π/4\gamma=\pi/4). For Δ>1\Delta>1 they could only evaluate the leading-order, dominant-state contribution — a frozen configuration that segregates vertex types above and below a diagonal — giving

e−2F(λ,μ)=sinh⁡(λ−∣μ∣+∣γ∣) sinh⁡(λ+∣μ∣−γ) sinh⁡(λ+∣μ∣+γ),e^{-2F(\lambda,\mu)}=\sinh(\lambda-|\mu|+|\gamma|)\,\sqrt{\sinh(\lambda+|\mu|-\gamma)\,\sinh(\lambda+|\mu|+\gamma)},

but, in their words, they were 'unable to fix a solution of Liouville equation which agrees with our formula', so this expression rests only on the leading-state picture; the antiferroelectric regime was not addressed ('we intend to address the other phases in the future'). The same work raises the analog of the arctic-circle phenomenon: in the domain-wall six-vertex model, typical large configurations separate spatially into frozen (ferroelectrically ordered) corner regions and a central disordered region, bounded by an arctic curve computed non-rigorously via the tangent method of Colomo and Sportiello (a technique that extracts the curve from the asymptotics of boundary one-point functions). For the reflecting end, this phase-separation question has since been advanced only inside the disordered phase: Monte Carlo evidence (Lyberg, Korepin, Ribeiro, Viti 2018), the exact free-fermion-point curve — a semicircle (Passos–Ribeiro 2019) — and tangent-method arctic curves for the disordered phase (Di Francesco 2021, building on Di Francesco–Lapa for the uniform case). Nothing is known for Δ>1\Delta>1 or Δ<−1\Delta<-1, and no rigorous asymptotic analysis of the Tsuchiya determinant exists in any regime, in contrast with the completed rigorous programs for DWBC, partial-DWBC, and half-turn boundaries.

Problem Statement

Consider the six-vertex model on an N×2NN\times 2N square lattice with domain-wall boundary conditions on three sides and one reflecting end (U-turn boundary) with diagonal reflection (KK-)matrix, whose partition function ZNZ_N is the Tsuchiya determinant; in the homogeneous limit the row and column spectral parameters are λ\lambda and μ\mu, and the bulk free energy per site is F=−lim⁡N→∞log⁡ZN/(2N2)F=-\lim_{N\to\infty}\log Z_N/(2N^2) (temperature set to 1). This free energy is known only in the disordered regime −1<Δ<1-1<\Delta<1. Determine the thermodynamic limit of this model in the remaining anisotropy regimes, the three requested results jointly completing the phase-dependent bulk thermodynamics and limit-shape theory of this one model:

(i) Ferroelectric regime Δ>1\Delta>1: in the hyperbolic parametrization a=sinh⁡(λ−γ)a=\sinh(\lambda-\gamma), b=sinh⁡(λ+γ)b=\sinh(\lambda+\gamma), c=sinh⁡(2∣γ∣)c=\sinh(2|\gamma|) (so Δ=cosh⁡2γ\Delta=\cosh 2\gamma), derive the large-NN free energy F(λ,μ)F(\lambda,\mu) — in particular establish, or correct, the leading-state expression

e−2F(λ,μ)=sinh⁡(λ−∣μ∣+∣γ∣) sinh⁡(λ+∣μ∣−γ) sinh⁡(λ+∣μ∣+γ),e^{-2F(\lambda,\mu)}=\sinh(\lambda-|\mu|+|\gamma|)\,\sqrt{\sinh(\lambda+|\mu|-\gamma)\,\sinh(\lambda+|\mu|+\gamma)},

either by fixing suitable homogeneous-limit boundary data for the bidimensional-Toda-to-Liouville continuum reduction so that the matched Liouville solution reproduces it, or by an alternative controlled asymptotic analysis of the Tsuchiya determinant.

(ii) Antiferroelectric regime Δ<−1\Delta<-1: determine the free energy, for which no expression of any kind is known in the literature.

(iii) Spatial phase separation: characterize the analog of the arctic curve for the reflecting-end boundary in these regimes — the separation between frozen (ferroelectrically or Néel ordered) and disordered regions in typical large configurations — including its contact with the frozen regions and its relation to the bulk free energy.

Any sound method is admissible (matched Liouville solution with verified boundary data, Riemann–Hilbert or orthogonal-polynomial asymptotics of the determinant, or another rigorous or controlled derivation). The known disordered-regime result, the exactly solvable special points of the partition function, and the free-fermion limit serve as mandatory consistency checks; regime-boundary limits such as Δ→1+\Delta\to1^+ must be consistent with the disordered-phase expression where both apply.

References

  1. G. A. P. Ribeiro and V. E. Korepin, 'Thermodynamic limit of the six-vertex model with reflecting end', J. Phys. A: Math. Theor. 48 (2015) 045205. DOI: 10.1088/1751-8113/48/4/045205; arXiv:1409.1212; https://arxiv.org/abs/1409.1212
  2. Osamu Tsuchiya, 'Determinant formula for the six-vertex model with reflecting end', J. Math. Phys. 39 (1998) 5946-5951. DOI: 10.1063/1.532606; https://doi.org/10.1063/1.532606
  3. Greg Kuperberg, 'Symmetry classes of alternating-sign matrices under one roof', Ann. of Math. (2) 156 (2002) 835-866. DOI: 10.2307/3597283; arXiv:math/0008184; https://arxiv.org/abs/math/0008184
  4. V. E. Korepin and P. Zinn-Justin, 'Thermodynamic limit of the six-vertex model with domain wall boundary conditions', J. Phys. A: Math. Gen. 33 (2000) 7053-7066. DOI: 10.1088/0305-4470/33/40/304; https://doi.org/10.1088/0305-4470/33/40/304
  5. Pavel M. Bleher and Vladimir V. Fokin, 'Exact solution of the six-vertex model with domain wall boundary conditions. Disordered phase', Comm. Math. Phys. 268 (2006) 223-284. DOI: 10.1007/s00220-006-0097-y; arXiv:math-ph/0510033; https://arxiv.org/abs/math-ph/0510033
  6. Pavel Bleher and Karl Liechty, 'Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase', Comm. Math. Phys. 286 (2009) 777-801. DOI: 10.1007/s00220-008-0709-9; https://doi.org/10.1007/s00220-008-0709-9
  7. Pavel Bleher and Karl Liechty, 'Exact solution of the six-vertex model with domain wall boundary conditions: Antiferroelectric phase', Comm. Pure Appl. Math. 63 (2010) 779-829. DOI: 10.1002/cpa.20311; https://doi.org/10.1002/cpa.20311
  8. Pavel Bleher and Karl Liechty, 'Six-vertex model with partial domain wall boundary conditions: Ferroelectric phase', J. Math. Phys. 56 (2015) 023302. DOI: 10.1063/1.4908227; https://doi.org/10.1063/1.4908227
  9. Pavel Bleher and Karl Liechty, 'Domain wall six-vertex model with half-turn symmetry', Constr. Approx. 47 (2018) 141-162. DOI: 10.1007/s00365-017-9405-3; arXiv:1702.01190; https://arxiv.org/abs/1702.01190
  10. I. Lyberg, V. Korepin, G. A. P. Ribeiro and J. Viti, 'Phase separation in the six-vertex model with a variety of boundary conditions', J. Math. Phys. 59 (2018) 053301. DOI: 10.1063/1.5018324; arXiv:1711.07905; https://arxiv.org/abs/1711.07905
  11. I. R. Passos and G. A. P. Ribeiro, 'Boundary correlations for the six-vertex model with reflecting end boundary condition', J. Stat. Mech. (2019) 083101. DOI: 10.1088/1742-5468/ab3113; arXiv:1904.06383; https://arxiv.org/abs/1904.06383
  12. I. R. Passos and G. A. P. Ribeiro, 'Arctic curve of the free-fermion six-vertex model with reflecting end boundary condition', J. Stat. Mech. (2019) 113103. DOI: 10.1088/1742-5468/ab4bbf; arXiv:1908.05773; https://arxiv.org/abs/1908.05773
  13. Philippe Di Francesco, 'Arctic curves of the reflecting boundary six vertex and of the twenty vertex models', J. Phys. A: Math. Theor. 54 (2021) 355201. DOI: 10.1088/1751-8121/ac17a6; arXiv:2106.02098; https://arxiv.org/abs/2106.02098
  14. Philippe Di Francesco and Matthew F. Lapa, 'Arctic curves in path models from the tangent method', J. Phys. A: Math. Theor. 51 (2018) 155202. DOI: 10.1088/1751-8121/aab3c0; arXiv:1711.03182; https://arxiv.org/abs/1711.03182
  15. Filippo Colomo and Andrea Sportiello, 'Arctic curves of the six-vertex model on generic domains: the tangent method', J. Stat. Phys. 164 (2016) 1488-1523. DOI: 10.1007/s10955-016-1590-0; arXiv:1605.01388; https://arxiv.org/abs/1605.01388
  16. Wen-Li Yang, Xi Chen, Jun Feng, Kun Hao, Bo-Yu Hou, Kang-Jie Shi and Yao-Zhong Zhang, 'Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end', Nucl. Phys. B 844 (2011) 289-307. DOI: 10.1016/j.nuclphysb.2010.11.008; arXiv:1107.5627; https://arxiv.org/abs/1107.5627