The six-vertex model

This collection grew out of references recorded randomly over the last few years. This is an eclectic collection which is meant to be the source of information only. It is incomplete, if you will find an omission, please send me an email and I will insert the appropriate citation.

Books and reviews Some of these papers have original results. A number of them are thesis works. All papers in this section can be used as an introductory material for various aspects of the 6-vertex model.

E.H. Lieb and F.Y. Wu, Two Dimensional Ferroelectric Models, in Phase Transitions and Critical Phenomena, Vol. 1, ed. by C.Domb and M.S. Green, 321, Academic Press, London, 1972.
R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, Inc., London, 1982.
L.A. Takhtadzhan, L.D. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ model, Russian Math. Surveys {\bf 34} (1979), no. 5, 11--68.
V.E. Korepin, N.M. Bogolyubov, and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, 1993. M. Jimbo and T. Miwa, Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conference Series in Math. , v. 85, 1993.
P. Zinn-Justin, Six-vertex, loops and tiling models: integrability and combinatorics, http://www.lpthe.jussieu.fr/~pzinn/publi/hdr.pdf; http://arxiv.org/pdf/0901.0665.pdf
N. Reshetikhin, "Lectures on integrable models in statistical mechanics", In: "Exact methods in low-dimensional statistical physics and quantum computing", Proceedings of Les Houches School in Theoretical Physics, Oxford University Press, 2010.

Classics: Here is a collection of some classical papers on the 6-vertex model, XXZ model, and related topics.

C. N. Yang, C. P. Yang: One-Dimensional Chain of Anisotropic Spin- Spin Interactions. I. Proof of Bethe Hypothesis for Ground State in a Finite System. Phys. Rev. 150, 321-327 (1966).
C. N. Yang, C. P. Yang: v. 150, 327, 1966
B. Sutherland, C.N. Yang, and C.P. Yang, Exact Solution of a Model of Two- Dimensional Ferroelectrics in an Arbitrary External Electric Field, Phys. Rev. Letters , v. 19, 588, 1967.
B. Sutherland, Phys. Rev. Lett. , v. 19, 103, 1967;
C.P. Yang, Phys. Rev. Lett., v. 19, 586, 1967

Various aspects of the thermodynamical limit The following references concern various aspects of the thermodynamical limit in the 6-vertex model.

I. M. Nolden, The Asymmetric Six-Vertex Model, J. Statist. Phys., v. 67, 155, 1992; Ph.D. thesis, University of Utrecht, 1990.
Blecher, domain wall, http://www.math.iupui.edu/~bleher/Papers/2009_Exact%20solution%20ferroelectric.pdf
J.D. Noh and D. Kim, Finite-Size Scaling and the Toroidal Partition Function of the Critical Asymmetric Six-Vertex Model, cond-mat/9511001.
D.J. Bukman and J.D. Shore, The Conical Point in the Ferroelectric Six-Vertex Model, J. Stat. Phys. v. 78, 1277--1309, 1995.
D. Allison and N. Reshetikhin, Numerical Study of the 6-Vertex Model with Domain Wall Boundary Conditions, Ann. Inst. Fourier (Grenoble) v. 55, 1847--1869, 2005.
K. Eloranta, Diamond Ice, J. Statist. Phys. , v. 96, 1091--1109, 1999.
O.F. Syljuasen and M.B. Zvonarev, Directed-Loop Monte Carlo Simulations of Vertex Models, cond-mat/0401491.

The 6-vertex model and SOS and RSOS models. The 6-vertex model is closely related to SOS (Solid-on-Solid) models also know as models with dynamical R-matrices. When state configurations are restricted such models are known as Restricted-Solid-on-Solid models. Some papers on combinatorics and algebraic structure of SOS models and dynamical R-matrices.

Niccoli, http://arxiv.org/pdf/1207.1928.pdf, dynamical 6v
Partition function of the trigonometric SOS model with reflecting end, G. Filali, N. Kitanine, arXiv: 1004.1015v2

Reflecting boundary conditions: The systematic study of reflecting boundary conditions started with the work of I. Cherednik and E. Sklyanin (first reference below). These boundary conditions are integrable and have deep algebraic meaning. In the classical case they are related to symmetric spaces and in the quantum case they are related to reflection algebras and to co-ideal subalgebras in quantum affine algebras. They are also related to conformally invariant boundary conditions in conformal field theory.

I. Cherednik, Factorizing particles on a half line, and root systems, Theoret. and Math. Phys., v. 61 (1984), no. 1, 977-983.
E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen., v.21 (1988), 2375--2389.

One of the important tools to study correlation functions in the 6-vertex model with reflecting boundary conditions is the boundary q-KZ equation. This equation for correlation functions was found and was first used for computation of correlators in the following papers. An important part of the construction is the formula for solutions to the boundary q-KZ equation as an expectation value of vertex operators represented Heisenberg algebra and a Heisenberg representation of boundary states.

M. Jimbo, R. Kedem, T. Kojima, H. Konno, T. Miwa, XXZ chain with a boundary. Nucl. Phys. B , v.441 (1995), no. 3, 437--470.
M. Jimbo, R. Kedem, H. Konno, T. Miwa, R. Weston, Difference equations in spin chains with a boundary, Nucl. Phys. B, v. 448 (1995), no. 3, 429--456.
T. Kojima, Free field approach to diagonalization of boundary transfer matrix: recent advances, arXiv:1103.5526.
R. Weston, Correlation functions and the boundary qKZ equation in a fractured XXZ chain, J. Stat. Mech. Theory Exp. 2011, no. 12, P12002.
Ph. Di Francesco, P. Zinn-Justin, Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics, J. Stat. Mech. Theory Exp. , v. 2007, no. 12, P12009.

Here are some references on the 6-vertex model with mixed boundary conditions, including reflecting b.c. along part of the boundary.

Kohei Motegi, Two point functions for the six vertex model with reflecting end, arXiv:1006.4692.
The two point functions, which give the probability that the spins turn down at the boundaries, are studied for the six vertex model on a 2N×N lattice with domain wall boundary condition and left reflecting end. Two types of two point functions are expressed in terms of determinants.
Kohei Motegi, A note on a boundary one point function for the six vertex model with reflecting end, arXiv:1005.5037 Rep. Math. Phys. 67 (2011) 87

Boundary conditions for the 6-vertex models and their generalizations were studied in numerous papers. Here are some references.

L. Mezincescu, R.I. Nepomechie, Integrable open spin chains with non-symmetric R-matrices, J. Phys. A: Math. Gen. , v. 24, (1991), L17--L23.
L. Mezincescu, R.I. Nepomechie, Fusion procedure for open chains, J. Phys. A: Math. Gen., v. 25 (1992), 2533--2543.
R. Nepomechie, Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices.
R. Nepomechie, Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms.
Y. Zhou, Fusion hierarchies with open boundaries and exactly solvable models, in "Statistical models, Yang-Baxter equation and related topics, and Symmetry, statistical mechanical models and applications" (Tianjin, 1995), 351- 358, World Sci. Publ., River Edge, NJ, 1996.
Junpeng Cao, Hai-Qing Lin, Kang-jie Shi, Yupeng Wang, Exact solutions and elementary excitations in the XXZ spin chain with unparallel boundary fields. http://arxiv.org/pdf/cond-mat/0212163v1.pdf
Wen-Li Yang, Xi Chen, Jun Feng, Kun Hao, Bo-Yu Hou, Kang-Jie Shi, Yao-Zhong Zhang, Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end. arXiv:1107.5627
Guang-Liang Li, Kang-Jie Shi, The algebraic Bethe ansatz for open vertex models. http://arxiv.org/pdf/hep-th/0611127v2.pdf
P. Di Francesco Boundary qKZ equation and generalized Razumov-Stroganov sum rules for open IRF models arXiv:math-ph/0509011
Jan de Gier, Vladimir Rittenberg, Refined Razumov-Stroganov conjectures for open boundaries arXiv:math-ph/0408042 J de Gier, P. Pyatov, Bethe Anstaz for the Temperley-Lieb loop model with open boundaries. http://arxiv.org/pdf/hep-th/0312235v2.pdf
N. Kitanine, J.-M. Maillet, G. Niccoli, Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV(separation of variables), arXiv:1401.4901
S. Faldella, N. Kitanine, G. Niccoli, Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms, arXiv:1307.3960
Ghali Filali, Nikolai Kitanine , Spin Chains with Non-Diagonal Boundaries and Trigonometric SOS Model with Reflecting End arXiv:1011.0660
Ghali Filali (LPTM), Nikolai Kitanine (IMB), Partition function of the trigonometric SOS model with reflecting end, arXiv:1004.1015

Domain wall boundary condition: The remarkable property of the 6-vertex modle with Domain wall (DW) boundary conditions is that its partition function is a determinant.

A.G. Izergin, Partition Function of the 6-Vertex Model in a Finite Volume, (Russian) Dokl. Akad. Nauk USRR, v. 297, 331-333, 1987.
V. Korepin and P. Zinn-Justin, Thermodynamic Limit of the Six-Vertex Model with Domain Wall Boundary Conditions, J. Phys. A, {\bf 33}, 7053-7066, 2000.
V. Korepin and P. Zinn-Justin, Inhomogeneous Six-Vertex Model with Domain Wall Boundary Conditions and Bethe Ansatz, J. Math. Phys. {\bf 43}, 3261-3267, 2002.
P. Zinn-Justin, Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model, Phys. Rev. E {\bf 62}, 3411-3418, 2000.
F. Colomo, V. Noferini, A. G. Pronko, Algebraic arctic curves in the domain-wall six-vertex model, arXiv:1012.2555

Correlation functions:Roughly, there are two approaches to the computation of correlation functions. One is based on direct computations based on algebraic Bethe ansatz. The other is based on ideas coming from the computation of form-factors. Here are few (very few) references on this very important subject. The list of references and comments will grow. V.E. Korepin, Calculation of Norms of Bethe Wave Functions, Commun. Math. Phys., v.86, 391-418, 1982.
In the following papers a hidden fermionic structure was found for the XXZ spin chain and 6-vertex models. It gives a power method for computation of correlation functions in these models. Comment: The hidden fermionic structure is most likely related to the exterior algebra of the Chevalley-Eilenberg complex of the action of the torus generated by integrals of motion (see papers by Najayashiki and Smirnov).

Boos, H.; Jimbo, M.; Miwa, T.; Smirnov, F. Completeness of a fermionic basis in the homogeneous XXZ model. J. Math. Phys. 50 (2009), no. 9, 095206 Jimbo, Michio; Miwa, Tetsuji; Smirnov, Fedor Hidden Grassmann structure in the XXZ model V: sine-Gordon model. Lett. Math. Phys. 96 (2011), no. 1-3, 325-365.
Boos, H.; Jimbo, M.; Miwa, T.; Smirnov, F. Hidden Grassmann structure in the XXZ model IV: CFT limit. Comm. Math. Phys. 299 (2010), no. 3, 825-866.
Jimbo, M.; Miwa, T.; Smirnov, F. Hidden Grassmann structure in the XXZ model III: introducing the Matsubara direction. J. Phys. A 42 (2009), no. 30, 304018,br> Boos, H.; Jimbo, M.; Miwa, T.; Smirnov, F.; Takeyama, Y. Hidden Grassmann structure in the XXZ model. II. Creation operators. Comm. Math. Phys. 286 (2009), no. 3, 875-932.
Boos, H.; Jimbo, M.; Miwa, T.; Smirnov, F.; Takeyama, Y. Hidden Grassmann structure in the XXZ model. Comm. Math. Phys. 272 (2007), no. 1, 263-281.
N.M. Bogoliubov, A.G. Pronko, and M.B. Zvonarev, Boundary Correlation Functions of the Six-Vertex Model, J. Phys. A, v. 35, 5525-5541, 2002.

The 6-vertex model orthogonal polynomial, combinatorics: Here is a collection of some references on the 6-vertex model on finite domains. In these papers various combinatorial aspects of these models, relation to orthogonal polynomials and other related topics are investigated.

Martin Bender, Steven Delvaux, Arno B.J. Kuijlaars, Multiple Meixner-Pollaczek polynomials and the six-vertex model, arXiv:1101.2982.
Peter J. McNamara, Factorial Schur functions via the six vertex model, arXiv:0910.5288
Tiago Fonseca, Ferenc Balogh, The higher spin generalization of the 6-vertex model with domain wall boundary conditions and Macdonald polynomials, http://arxiv.org/pdf/1210.4527.pdf,
G. Kuperberg, Another Proof of the Alternating Sign Matrix Conjecture, Internat. Math. Res. Notices 3, 139--150, 1996. D.M. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge University Press, 1999.
A.V. Razumov and Yu. Stroganov, Combinatorial Structure of the Ground State of O(1) Loop Model, math.CO/0104216.

The 5-vertex model: H.Y. Huang, F.Y. Wu, H. Kunz, D. Kim, Interacting Dimers on the Honeycomb Lattice: an Exact Solution of the Five-Vertex Model, Physica A, v. 228, 1--32, 1996.