Five vertex model with fixed boundary conditions

N. M. Bogolyubov

Algebra i analiz, Tome 21 (2009) no. 3, pp. 58-78.

Résumé

The exactly solvable five-vertex model on a square lattice with fixed boundary conditions is considered. Application of the algebraic Bethe ansatz makes it possible to express the partition function and the boundary correlation functions of the nonhomogeneous model in the determinantal form. The relationship established between the homogeneous model and plane partitions helps to calculate its partition function.

Bibliographie

[1] Bekster R., Tochno reshaemye modeli v statisticheskoi mekhanike, Mir, M., 1985 | MR

[2] Lieb E. H., Wu F. Y., Phase transitions and critical phenomena, Vol. 1, Acad. Press, London, 1972 | Zbl

[3] Korepin V. E., “Calculation of norms of Bethe wave functions”, Comm. Math. Phys., 86 (1982), 391–418 | DOI | MR | Zbl

[4] Korepin V., Zinn-Justin P., “Thermodynamic limit of the six-vertex model with domain wall boundary conditions”, J. Phys. A, 33 (2000), 7053–7066 | DOI | MR | Zbl

[5] Bogoliubov N. M., Kitaev A. V., Zvonarev M. B., “Boundary polarization in the six-vertex model”, Phys. Rev. E (3), 65:2 (2002), 026126, 4 pp. | DOI | MR

[6] Bogoliubov N. M., Pronko A. G., Zvonarev M. B., “Boundary correlation functions of the six-vertex model”, J. Phys. A, 35:27 (2002), 5525–5541 | DOI | MR | Zbl

[7] Allison D., Reshetikhin N., “Numerical study of the 6-vertex model with domain wall boundary conditions”, Ann. Inst. Fourier (Grenoble), 55 (2005), 1847–1869 | MR | Zbl

[8] Syljuåsen O., Zvonarev M., “Directed-loop Monte Carlo simulations of vertex models”, Phys. Rev. E (3), 70 (2004), 016118 | DOI

[9] Makdonald I., Simmetricheskie funktsii i mnogochleny Kholla, Mir, M., 1985 | MR

[10] Bressoud D. M., Proofs and confirmations. The story of the alternating sign matrix conjecture, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[11] Kuperberg G., “Another proof of the alternating-sign matrix conjecture”, Int. Math. Res. Not., 1996:3 (1996), 139–150 | DOI | MR | Zbl

[12] Colomo F., Pronko A. C., “Square ice, alternating sign matrices, and classical orthogonal polynomials”, J. Stat. Mech. Theory Exp., 2005, no. 1, 005, 33 pp., (electronic) | MR

[13] Ferrari P. L., Spohn H., “Domino tilings and the six-vertex model at its free fermion point”, J. Phys. A, 39 (2006), 10297–10306 | DOI | MR | Zbl

[14] Bogoliubov N. M., “Boxed plane partitions as an exactly solvable boson model”, J. Phys. A, 38 (2005), 9415–9430 | DOI | MR | Zbl

[15] Bogolyubov N. M., “Perechislenie ploskikh razbienii i algebraicheskii anzats Bete”, Teor. i mat. fiz., 150:2 (2007), 193–203 | MR | Zbl

[16] Tsilevich N. V., “Kvantovyi metod obratnoi zadachi dlya $q$-bozonnoi modeli i simmetricheskie funktsii”, Funkts. anal. i ego pril., 40:3 (2006), 53–65 | MR | Zbl

[17] Shigechi K., Uchiyama M., “Boxed skew plane partition and integrable phase model”, J. Phys. A, 38 (2005), 10287–10306 | DOI | MR | Zbl

[18] Bogolyubov N. M., “Chetyrëkhvershinnaya model”, Zap. nauch. semin. POMI, 347, 2007, 34–55 | MR

[19] Bogolyubov N. M., “Chetyrëkhvershinnaya model i sluchainye ukladki”, Teor. i mat. fiz., 155:1 (2008), 25–38 | MR | Zbl

[20] Noh J. D., Kim D., “Interacting domain walls and the five-vertex model”, Phys. Rev. E, 49 (1994), 1943 | DOI

[21] Gwa L.-H., Spohn H., “Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian”, Phys. Rev. Lett., 68 (1992), 725–728 | DOI | MR | Zbl

[22] Rajesh R., Dhar D., “An exactly solvable anisotropic directed percolation model in three dimensions”, Phys. Rev. Lett., 81 (1998), 1646 | DOI

[23] Takhtadzhyan L. A., Faddeev L. D., “Kvantovyi metod obratnoi zadachi i $XYZ$ model Geizenberga”, Uspekhi mat. nauk, 34:5 (1979), 13–63 | MR

[24] Faddeev L. D., “Quantum completely integrable models in field theory”, Mathematical Physics Reviews, Vol. 1, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., 1, Harwood Acad., Chur, 1980, 107–155 | MR

[25] Kulish P. P., Sklyanin E. K., “Quantum spectral transform method. Recent developments”, Lecture Notes in Phys., 151, Springer, Berlin–New York, 1982, 61–119 | MR

[26] Bogolyubov N. M., Izergin A. G., Korepin V. E., Korrelyatsionnye funktsii integriruemykh sistem i kvantovyi metod obratnoi zadachi, Nauka, M., 1992 | MR | Zbl

[27] Korepin V. E., Bogoliubov N. M., Izergin A. G., Quantum inverse scattering method and correlation functions, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl

[28] Bogoliubov N. M., Nassar T., “On the spectrum of the non-Hermitian phase-difference model”, Phys. Lett. A, 234 (1997), 345–350 | DOI | MR | Zbl

[29] Kim D., “Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Kardar–Parisi–Zhang-type growth model”, Phys. Rev. E, 52 (1995), 3512 | DOI

[30] Izergin A. G., Coker D. A., Korepin V. E., “Determinant formula for the six-vertex model”, J. Phys. A, 25 (1992), 4315–4334 | DOI | MR | Zbl

[31] Gwa L.-H., Spohn H., “Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation”, Phys. Rev. A, 46 (1992), 844 | DOI

[32] Priezzhev V. B., “Exact nonstationary probabilities in the asymmetric exclusion process on a ring”, Phys. Rev. Lett., 91 (2003), 050601 | DOI

[33] Golinelli O., Mallick K., “The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics”, J. Phys. A, 39 (2006), 12679–12705 | DOI | MR | Zbl