Arctic curves of the six-vertex model with partial domain-wall, reflecting-end, and half-turn boundary conditions: generic-weight determination and the domain-wall equivalence class

The six-vertex model with domain-wall boundary conditions (DWBC) on a square exhibits arctic phase separation, and its thermodynamic-limit arctic curve is known analytically. A 2018 Monte Carlo study by Lyberg, Korepin, Ribeiro and Viti showed that three DWBC variants — partial domain-wall boundary conditions (pDWBC), one reflecting end (RE), and half-turn boundary conditions (HTBC) — also phase-separate, and posed the problem of determining their arctic curves analytically and of classifying which boundary conditions reproduce the DWBC curve. Later work settled only two special cases: the RE (U-turn) curve at arbitrary weights in the disordered regime, and the pDWBC curve at the free-fermion point. The surviving problem asks for the analytic thermodynamic-limit arctic curves of pDWBC at generic weights (disordered beyond the free-fermion point, and antiferroelectric), of RE in the antiferroelectric regime where asymmetric weights create frozen corners absent from DWBC, and of HTBC (numerically the lower half of the DWBC curve), together with a characterization of the class of boundary conditions whose limit shapes coincide with (halves of) the DWBC arctic curve.

Background

The six-vertex model is defined on a square lattice whose edges carry arrows, with the ice rule imposed at every vertex: exactly two arrows point in and two point out at each vertex. Six local arrow configurations are allowed, grouped into three pairs with weights $a_1,a_2$, $b_1,b_2$, $c_1,c_2$. In the symmetric case $a_1=a_2=a$, $b_1=b_2=b$, $c_1=c_2=c$ the model has two parameters, conventionally combined into the anisotropy $\Delta=\frac{a^2+b^2-c^2}{2ab},$ which separates the ferroelectric ($\Delta>1$), disordered ($|\Delta|<1$) and antiferroelectric ($\Delta<-1$) regimes; $\Delta=0$ is the free-fermion point, where the model can be rewritten in terms of free lattice fermions and is equivalent to a dimer model.

With domain-wall boundary conditions (DWBC) on an $N\times N$ square — all boundary arrows fixed so that horizontal arrows enter on the left and leave on the right while vertical arrows enter at the bottom and leave at the top (configurations are in bijection with alternating sign matrices, that is, matrices with entries $0,\pm1$ whose nonzero entries alternate in sign along every row and column and whose rows and columns each sum to 1) — the model develops arctic phase separation: in the thermodynamic limit a typical configuration splits into frozen regions filled with a single vertex type and a central liquid region, separated by a deterministic curve called the arctic curve. For DWBC this curve is known: it was derived in the disordered and antiferroelectric regimes by the tangent method (Colomo–Pronko; Colomo–Pronko–Zinn-Justin; Colomo–Sportiello), and proved rigorously at the ice point $a=b=c$ on three-bundle domains, which generalize the square, by Aggarwal. The tangent method determines the arctic curve as the envelope of the straight segments followed by an outermost path of the configuration — drawn as a family of paths that may touch and bounce off one another — whose endpoint is displaced outside the domain; it requires the large-$N$ asymptotics of a boundary one-point function.

Lyberg, Korepin, Ribeiro and Viti (2018) studied numerically three DWBC variants, adapting a Monte Carlo algorithm originally devised for DWBC. (i) Partial domain-wall boundary conditions (pDWBC) on an $M\times N$ rectangle with $M\ge N$: domain-wall arrows on three sides and a free boundary on one long side; the free side introduces a third weight parameter $\eta$, through $w(a_1)=ae^{-\eta}$, $w(a_2)=ae^{\eta}$, $w(b_1)=be^{-\eta}$, $w(b_2)=be^{\eta}$, which was set to zero in their simulations. (ii) One reflecting end (RE) on a $2N\times N$ rectangle: three sides carry domain-wall-type arrows and the fourth consists of $N$ U-turn pairs pairing adjacent boundary edges, an integrable reflecting boundary implemented through the reflection algebra (the standard algebraic framework for integrable boundaries in the quantum inverse scattering method); the partition function is an explicit determinant and its thermodynamic free energy is known in the disordered regime (Ribeiro–Korepin), while at $a=b$ the model enumerates vertically symmetric alternating sign matrices (alternating sign matrices invariant under reflection about their vertical axis). (iii) Half-turn boundary conditions (HTBC) on a $2N\times N$ rectangle: configurations correspond exactly to DWBC configurations on a $2N\times 2N$ square that are invariant under a $180^\circ$ rotation (half-turn symmetric alternating sign matrices); as with pDWBC an extra boundary-weight parameter exists in principle and was set to zero in their simulations, and partition-function asymptotics are known in all regimes (Bleher–Liechty). Their simulations show that all three variants phase-separate, that RE with $a=b$ and HTBC appear to have arctic curves equal to the lower half of the DWBC curve, and that RE with $a\ne b$ (studied at $\Delta=-2$) develops frozen corners absent from DWBC. After 2018 two special cases were settled: Passos–Ribeiro derived the RE curve at the free-fermion point (a semicircle), and, more generally, Di Francesco derived tangent-method arctic curves for the reflecting (U-turn) model at arbitrary weights in the disordered regime; de Kemmeter, Debin and Ruelle derived the pDWBC curve at the free-fermion point for arbitrary aspect ratio. Everything else — pDWBC at generic weights, RE in the antiferroelectric regime, HTBC in any regime, and the classification of boundary conditions sharing the DWBC curve — remains open.

Problem Statement

For the six-vertex model with real positive weights $a,b,c$ and anisotropy $\Delta=(a^2+b^2-c^2)/(2ab)$, determine analytically the thermodynamic-limit arctic curves (the boundaries of the liquid regions) for the following domain-wall-type boundary conditions, in the regimes where they are not yet known:

1. Partial domain-wall boundary conditions (pDWBC) on an $M\times N$ rectangle with $M\ge N$ (domain-wall arrows on three sides, one long side free), at generic weights — arbitrary positive $a,b,c$, with the boundary-weight parameter $\eta$ set to zero as in the source's simulations — across the range $\Delta<1$ where phase separation is observed, beyond the free-fermion point $\Delta=0$ already treated, covering both the disordered regime (including $a\ne b$) and the antiferroelectric regime $\Delta<-1$.

2. The model with one reflecting end (RE) on a $2N\times N$ rectangle (the side with $N$ U-turn pairs, with equal U-turn weights as in the source's simulations), in the antiferroelectric regime $\Delta<-1$ — both for $a=b$, where numerics indicate the curve is the lower half of the domain-wall antiferroelectric curve, and for $a\ne b$, where numerics at $\Delta=-2$, $b=2a$ show frozen upper corners absent from DWBC and a curve qualitatively different from it.

3. Half-turn boundary conditions (HTBC), the $2N\times N$ fundamental domain of $180^\circ$-rotation-invariant DWBC configurations on a $2N\times 2N$ square, in both the disordered and antiferroelectric regimes — confirming analytically the numerically observed identity of its arctic curve with the lower half of the domain-wall arctic curve.

4. Building on these, characterize the class of boundary conditions whose thermodynamic-limit arctic curves coincide with (halves of) the domain-wall arctic curve. The classification runs over the integrable boundary conditions of domain-wall type: those that fix the edge arrows on some lattice sides, as the domain-wall boundary does, and equip the remaining sides with a reflecting (U-turn) end or a free side, with arbitrary boundary weights. The two named instances (RE with $a=b$ and HTBC) are illustrative, and a general characterization over this class is required.

All four parts are required for a complete resolution: settling any single part (or any proper subset) is partial progress toward, but not resolution of, the task. These four parts share one objective: to extend the exact limit-shape theory of the domain-wall six-vertex model to its natural boundary-condition variants and to determine which variants preserve the domain-wall limit shape. A complete proof of convergence for any of the families is not required.

References

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