Explicit solution of the six-vertex-model Bethe-ansatz integral equations in arbitrary external electric fields

The exact thermodynamics of the six-vertex model in external electric fields $(h,v)$ — the 'asymmetric' six-vertex model — is known only implicitly: Sutherland, Yang and Yang (1967) reduced the free energy $F(h,v)$ to singular integral equations for the Bethe-ansatz root density on a support whose complex endpoints $\zeta$ are fixed by the fields through normalization conditions. Following Gaudin's Bethe-wavefunction monograph, these equations have been solved explicitly only for symmetric endpoint configurations (e.g. $\Re\zeta=\pi$) and perturbatively near special loci; the 1990s literature (Bukman–Shore; Albertini–Dahmen–Wehefritz; Albertini; Dahmen–Wehefritz-Kaufmann) later fixed the phase diagram in fields and the leading singularity exponents along the whole phase boundary, but the generic in-field solution remains a numerical problem. The open problem is to solve the in-field root-density integral equations explicitly for arbitrary endpoints $\zeta$ in every anisotropy regime, and thereby complete the analytic characterization of $F(h,v)$: singularity amplitudes, the crossover between the non-tangential exponent $3/2$ and the tangential exponent $3$, and the correlation-length behavior at the special boundary points where a transition not classified by conformal field theory has been conjectured.

Background

The six-vertex model places arrows on the edges of a square lattice subject to the ice rule — at each vertex exactly two arrows point in and two point out, as in Pauling's description of hydrogen positions in hydrogen-bonded crystals such as KH$_2$PO$_4$ (KDP). Six vertex configurations are allowed, with Boltzmann weights $\omega_1,\dots,\omega_6$, and the model's anisotropy is measured by $\Delta=(\omega_1\omega_2+\omega_3\omega_4-\omega_5\omega_6)/(2\sqrt{\omega_1\omega_2\omega_3\omega_4})$: $\Delta>1$ is the ferroelectric (KDP-type) regime, $-1\le\Delta\le1$ the disordered (critical) regime, and $\Delta<-1$ the antiferroelectric regime (realized by the F model, the standard antiferroelectric member of the six-vertex family). Coupling the model to external electric fields $h$ and $v$ (dimensionless, $\beta$ times energy) that favor right over left and up over down arrows breaks the arrow-reversal symmetries; the resulting model is conventionally called the asymmetric six-vertex model. Its thermodynamics is carried by the free energy per site $F(h,v)$ and the polarizations $x$ and $y$, the net fractions of right-pointing and up-pointing arrows, related by a two-dimensional Legendre transform $F(h,v)=\min_{x,y}\{F(x,y)-hx-vy\}$.

The row-to-row transfer matrix — the operator that advances the lattice configuration by one row — is diagonalized by the Bethe ansatz, an exact method that builds eigenstates from superpositions of free-particle waves with self-consistently determined momenta. In the thermodynamic limit the free energy is expressed through a root density $\rho$ — the density distribution of Bethe-ansatz roots on a contour in the complex spectral plane — which satisfies a linear singular integral equation whose driving terms depend on the anisotropy regime. The crucial in-field difficulty is that the endpoints $\zeta$ of the root-distribution support are not free parameters: they are determined implicitly by normalization conditions fixing the vertical polarization $y$ and the field $h$, so that solving for $F(h,v)$ at generic $(h,v)$ means solving for the density and the endpoints simultaneously. At zero field, where the support sits on the symmetric locus $\Re\zeta=\pi$, the integral equations collapse to convolutions and are solvable in closed form in terms of dilogarithms in all three regimes. In fields, closed forms are known only on symmetric or otherwise special loci: inside the frozen or zero-magnetization regions, where $F$ is linear or flat; at the ferroelectric conical points; and at the special antiferroelectric boundary points $(h_c,v_c)$.

The qualitative in-field phase structure is nevertheless fully established. For $\Delta>1$ at low temperature there are four frozen, completely polarized phases separated from two incompletely polarized phases by second-order boundaries of the Pokrovsky–Talapov (PT) type — a commensurate–incommensurate transition, generic for situations where a density of domain walls depins from zero, with free-energy exponent $3/2$ and polarization exponent $1/2$ — and by a first-order line along $h=-v$; Bukman and Shore showed that the points where PT boundaries meet the first-order line are conical points, loci where a one-parameter family of polarizations coexists. For $\Delta<-1$ a zero-magnetization region containing the origin is bounded by a curve $\Gamma$; outside it an incommensurate phase sets in, in which the polarizations vary continuously with the fields, and $\Gamma$ carries two special points $(\pm h_c,\pm v_c)$ with $\cosh h_c=1/k$, where $k$ is the elliptic modulus (the parameter of the elliptic functions appearing in the zero-field solution). Via the Legendre construction $F$ also gives the equilibrium crystal shape in the Andreev sense — the Wulff shape of a crystal facet described by a restricted solid-on-solid model — which is how van Beijeren's body-centered solid-on-solid (BCSOS) mapping, an exact correspondence between a crystal-surface model without overhangs or voids and the six-vertex model, and the Jayaprakash–Saam analysis of fcc crystals made the six-vertex free energy in fields the microscopic basis of facet-rounding theory.

What has remained missing since Gaudin's chapter is the explicit analytic object behind all of this: a solution of the in-field integral equations usable at generic $(h,v)$ rather than case-by-case numerics. The leading singularity exponents along the whole phase boundary are known — $3/2$ for every non-tangential approach direction and $3$, with different amplitudes on the two sides, tangentially — but the amplitudes in general form, the crossover between the two regimes, the simultaneous two-component singularity structure of $F$ in $(h,v)$, and the correlation-length behavior at $(h_c,v_c)$ are not.

Problem Statement

Solve, explicitly, the Bethe-ansatz integral equations for the root densities of the six-vertex model in external electric fields $(h,v)$ for arbitrary values of the support endpoints $\zeta$ (equivalently, for arbitrary $(h,v)$ at fixed anisotropy $\Delta$), in each of the regimes $\Delta>1$, $-1<\Delta<1$, and $\Delta<-1$ (the degenerate boundary values $\Delta=\pm1$ are treated separately and are not part of the demanded scope); and from this solution complete the analytic characterization of the free energy $F(h,v)$ over the whole field plane.

Concretely, an answer should provide:

1. A closed-form expression, or a representation with proven uniform convergence and analyticity control, for the root density and the free energy at generic fields — inside the incompletely polarized phases of the ferroelectric and disordered regimes and the incommensurate phase of the antiferroelectric regime — reducing to the known results on the symmetric locus $\Re\zeta=\pi$, at zero field, at the ferroelectric conical points, and at the special antiferroelectric points $(h_c,v_c)$. A proof that no representation exists within the class of functions generated from the elementary functions, the dilogarithm and the elliptic functions of the zero-field solution by finitely many algebraic operations, differentiations, integrations and compositions, with a characterization of the obstruction, would equally resolve this part.

2. The full analytic structure of $F(h,v)$: the non-analytic loci; the amplitudes (not only the leading exponents, which are known) of the singularities along the phase boundaries in general form; and the crossover scaling function interpolating between the non-tangential exponent $3/2$ and the tangential exponent $3$ as the direction of approach to the boundary becomes tangential.

3. The correlation-length behavior and the nature of the transition at the special boundary points $(h_c,v_c)$ of the antiferroelectric model, where a transition with diverging correlation length not classified by conformal field theory (the standard framework for classifying two-dimensional critical behavior) has been conjectured but not established.

These three deliverables express one objective — turning the implicit in-field exact solution into an explicit analytic object — and the first is the technical heart of it; all three are required in every case, and if the first is settled by the nonexistence branch then the second and third are to be determined directly from the integral equations.

References

1. M. Gaudin, "The six-vertex model", Chapter 7 of The Bethe Wavefunction (translated by J.-S. Caux), Cambridge University Press, 2014; French original La fonction d'onde de Bethe, Masson, 1983. DOI 10.1017/cbo9781107053885.010, https://doi.org/10.1017/cbo9781107053885.010
2. B. Sutherland, C. N. Yang, C. P. Yang, "Exact Solution of a Model of Two-Dimensional Ferroelectrics in an Arbitrary External Electric Field", Physical Review Letters 19, 588-591 (1967). DOI 10.1103/PhysRevLett.19.588, https://doi.org/10.1103/PhysRevLett.19.588
3. C. P. Yang, "Exact Solution of a Model of Two-Dimensional Ferroelectrics in an Arbitrary External Electric Field", Physical Review Letters 19, 586-588 (1967). DOI 10.1103/PhysRevLett.19.586, https://doi.org/10.1103/PhysRevLett.19.586
4. E. H. Lieb and F. Y. Wu, "Two-Dimensional Ferroelectric Models", in Phase Transitions and Critical Phenomena, Vol. 1, C. Domb and M. S. Green (eds.), Academic Press, London, 1972, pp. 331-490. INSPIRE record 157053, https://inspirehep.net/literature/157053
5. I. M. Nolden, "The asymmetric six-vertex model", Journal of Statistical Physics 67, 155-201 (1992). DOI 10.1007/BF01049030, https://doi.org/10.1007/BF01049030
6. J. D. Shore and D. J. Bukman, "Coexistence point in the six-vertex model and the crystal shape of fcc materials", Physical Review Letters 72, 604-607 (1994). DOI 10.1103/PhysRevLett.72.604, https://doi.org/10.1103/PhysRevLett.72.604
7. D. J. Bukman and J. D. Shore, "The conical point in the ferroelectric six-vertex model", Journal of Statistical Physics 78, 1277-1309 (1995). arXiv:cond-mat/9406060, DOI 10.1007/BF02180132, https://doi.org/10.1007/BF02180132
8. G. Albertini, S. R. Dahmen, B. Wehefritz, "The free energy singularity of the asymmetric six-vertex model and the excitations of the asymmetric XXZ chain", Nuclear Physics B 493, 541-570 (1997). arXiv:cond-mat/9606137, DOI 10.1016/S0550-3213(97)00078-3, https://doi.org/10.1016/S0550-3213(97)00078-3
9. G. Albertini, "Direction-Dependent Free Energy Singularity of the Antiferroelectric Asymmetric Six-Vertex Model", Journal of Statistical Physics 90, 853-871 (1998). arXiv:cond-mat/9706121, DOI 10.1023/A:1023285222161, https://doi.org/10.1023/A:1023285222161
10. J. D. Noh and D. Kim, "Finite-size scaling and the toroidal partition function of the critical asymmetric six-vertex model", Physical Review E 53, 3225-3239 (1996). DOI 10.1103/PhysRevE.53.3225, https://doi.org/10.1103/PhysRevE.53.3225
11. S. R. Dahmen, B. Wehefritz, G. Albertini, "A novel exponent in the Equilibrium Shape of Crystals", preprint cond-mat/9802152 (1998). arXiv:cond-mat/9802152, https://arxiv.org/abs/cond-mat/9802152
12. S. R. Dahmen and B. Wehefritz-Kaufmann, "A new exponent in the equilibrium shape of crystal surfaces", Journal of Statistical Mechanics: Theory and Experiment 2019, 104005 (2019). DOI 10.1088/1742-5468/ab2ad2, https://doi.org/10.1088/1742-5468/ab2ad2
13. R. J. Baxter, "Eight-Vertex Model in Lattice Statistics", Physical Review Letters 26, 832-833 (1971). DOI 10.1103/PhysRevLett.26.832, https://doi.org/10.1103/PhysRevLett.26.832
14. H. van Beijeren, "Exactly Solvable Model for the Roughening Transition of a Crystal Surface", Physical Review Letters 38, 993-996 (1977). DOI 10.1103/PhysRevLett.38.993, https://doi.org/10.1103/PhysRevLett.38.993
15. C. Jayaprakash, W. F. Saam, S. Teitel, "Roughening and Facet Formation in Crystals", Physical Review Letters 50, 2017-2020 (1983). DOI 10.1103/PhysRevLett.50.2017, https://doi.org/10.1103/PhysRevLett.50.2017
16. C. Jayaprakash and W. F. Saam, "Thermal evolution of crystal shapes: The fcc crystal", Physical Review B 30, 3916-3928 (1984). DOI 10.1103/PhysRevB.30.3916, https://doi.org/10.1103/PhysRevB.30.3916
17. A. Aggarwal, "Nonexistence and uniqueness for pure states of ferroelectric six-vertex models", Proceedings of the London Mathematical Society 124, 387-425 (2022). DOI 10.1112/plms.12430, https://doi.org/10.1112/plms.12430
18. H. Duminil-Copin, K. K. Kozlowski, D. Krachun, I. Manolescu, T. Tikhonovskaia, "On the Six-Vertex Model's Free Energy", Communications in Mathematical Physics 395, 1383-1430 (2022). arXiv:2012.11675, DOI 10.1007/s00220-022-04459-x, https://doi.org/10.1007/s00220-022-04459-x
19. J. Herzog-Arbeitman, S. Mantilla, I. Sodemann, "Solving the quantum dimer and six-vertex models one electric field line at a time", Physical Review B 99, 245108 (2019). arXiv:1902.01858, DOI 10.1103/PhysRevB.99.245108, https://doi.org/10.1103/PhysRevB.99.245108