The algebraic Bethe ansatz and quantum integrable systems

N. A. Slavnov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 4, pp. 727-766

Résumé

Methods are considered for applying an algebra with bilinear commutation relations to the theory of quantum integrable systems. This survey describes most of the results obtained in this area over the last twenty years, mainly in connection with the computation of correlation functions of quantum integrable systems. Methods for constructing eigenfunctions of the quantum transfer matrix and computing inner products and correlation functions are presented in detail. An example of application of the general scheme to the model of the $X\!X\!Z$ Heisenberg chain is considered.

Bibliographie

[1] H. Bethe, “Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette”, Z. Phys., 71:3–4 (1931), 205–226 | DOI | Zbl

[2] L. Hulthén, “Über das Austauschproblem eines Kristalls”, Ark. Mat. Astron. Fysik, 26, no. 11, 1938, 1–106 | Zbl

[3] R. Orbach, “Linear antiferromagnetic chain with anisotropic coupling”, Phys. Rev., 112:2 (1958), 309–316 | DOI

[4] L. R. Walker, “Antiferromagnetic linear chain”, Phys. Rev., 116:5 (1959), 1089–1090 | DOI

[5] E. H. Lieb, W. Liniger, “Exact analysis of an interacting Bose gas. I. The general solution and the ground state”, Phys. Rev., 130:4 (1963), 1605–1616 | DOI | MR | Zbl

[6] E. H. Lieb, D. C. Mattis (eds.), Mathematical physics in one dimension: exactly soluble models of interacting particles, Academic Press, New York, 1966

[7] C. N. Yang, C. P. Yang, “One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe's hypothesis for ground state in a finite system”, Phys. Rev., 150:1 (1966), 321–327 ; “II. Properties of the ground state energy per lattice site in a finite system”, 327–339 | DOI | DOI | Zbl

[8] C. N. Yang, “Some exact results for the many-body problem in one dimension with repulsiv delta-function interaction”, Phys. Rev. Lett., 19:23 (1967), 1312–1315 | DOI | MR | Zbl

[9] J. des Cloizeaux, M. Gaudin, “Anisotropic linear magnetic chain”, J. Math. Phys., 7:8 (1966), 1384–1400 | DOI

[10] M. Gaudin, La fonction d'onde de Bethe, Collect. Commissariat Energ. Atom. Ser. Sci., Masson, Paris, 1983 ; M. Goden, Volnovaya funktsiya Bete, Mir, M., 1987 | MR | Zbl | MR

[11] F. A. Berezin, V. N. Sushko, “Relyativistskaya dvumernaya model samovzaimodeistvuyuschego polya nenulevoi massy pokoya”, ZhETF, 45 (1965), 1293–1306

[12] E. H. Lieb, F. Y. Wu, “Absence of Mott transition in an exact solution of the short range one band model in one dimension”, Phys. Rev. Lett., 20:25 (1968), 1445–1448 | DOI

[13] L. Onsager, “Cristal statistics. I. A two-dimensional model with an order-disorder transition”, Phys. Rev., 65:3–4 (1944), 117–149 | DOI | MR | Zbl

[14] T. T. Wu, B. M. McCoy, C. A. Tracy, E. Barouch, “Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region”, Phys. Rev. B, 13:1 (1976), 316–374 | DOI | MR

[15] B. M. McCoy, C. A. Tracy, T. T. Wu, “Two-dimensional Ising model as an exactly solvable relativistic quantum field theory: explicit formulas for $n$-point functions”, Phys. Rev. Lett., 38:15 (1977), 793–796 | DOI

[16] E. Lieb, T. Shultz, D. Mattis, “Two soluble models of an antiferromagnetic chain”, Ann. Physics, 16:3 (1961), 407–466 | DOI | MR | Zbl

[17] B. M. McCoy, “Spin correlation functions of the $X-Y$ model”, Phys. Rev., 173:2 (1968), 531–541 | DOI

[18] A. Lenard, “Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons”, J. Math. Phys., 5:7 (1964), 930–943 | DOI | MR

[19] A. Lenard, “One-dimensional impenetrable bosons in thermal equilibrium”, J. Math. Phys., 7:7 (1966), 1268–1272 | DOI | MR

[20] H. G. Vaidya, C. A. Tracy, “One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature”, Phys. Rev. Lett., 42:1 (1979), 3–6 | DOI

[21] F. Kolomo, A. G. Izergin, V. E. Korepin, V. Tognetti, “Temperaturnye korrelyatsionnye funktsii v $XXO$-tsepochke Geizenberga. I”, TMF, 94:1 (1993), 19–51 | MR

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

[23] E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi. I”, TMF, 40:2 (1979), 194–220 | MR

[24] L. D. Faddeev, “How algebraic Bethe ansatz works for integrable models”, Symétries quantiques (Les Houches, France, 1995), eds. A. Connes et al., North-Holland, Amsterdam, 1998, 149–219 | MR | Zbl

[25] N. Kitanine, J. M. Maillet, V. Terras, “Form factors of the $XXZ$ Heisenberg spin-$1/2$ finite chain”, Nuclear Phys. B, 554:3 (1999), 647–678 ; arXiv: math-ph/9807020 | DOI | MR | Zbl

[26] P. P. Kulish, E. K. Sklyanin, “Quantum spectral transform method. Recent developments”, Integrable quantum field theories (Tvärminne, Finland, 1981), Lecture Notes in Phys., 151, eds. J. Heitarinta, C. Montonen, Springer, Berlin–New York, 1982, 61–119 | MR | Zbl

[27] A. G. Izergin, V. E. Korepin, “The quantum inverse scattering method approach to correlation functions”, Comm. Math. Phys., 94:1 (1984), 67–92 | DOI | MR | Zbl

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

[29] V. E. Korepin, “Analiz bilineinogo sootnosheniya shestivershinnoi modeli”, DAN SSSR, 265:6 (1982), 1361–1364 | MR

[30] A. G. Izergin, “Statisticheskaya summa shestivershinnoi modeli v konechnom ob'eme”, Dokl. AN SSSR, 297:2 (1987), 331–333 | MR | Zbl

[31] L. D. Faddeev, “History and perspectives of quantum groups”, Milan J. Math., 74:1 (2006), 279–294 | DOI | MR | Zbl

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

[33] E. K. Sklyanin, “Metod obratnoi zadachi rasseyaniya i kvantovoe nelineinoe uravneniya Shrëdingera”, Dokl. AN SSSR, 244:6 (1979), 1337–1340 | MR

[34] E. K. Sklyanin, L. D. Faddeev, “Kvantovomekhanicheskii podkhod k vpolne integriruemym modelyam teorii polya”, Dokl. AN SSSR, 243:6 (1978), 1430–1433

[35] A. G. Izergin, V. E. Korepin, “Pauli principal for one-dimensional bosons and the algebraic Bethe anstaz”, Lett. Math. Phys., 6:4 (1982), 283–288 | DOI | MR

[36] V. Tarasov, A. Varchenko, “Bases of Bethe vectors and difference equations with regular singular points”, Internat. Math. Res. Notices, 1995, no. 13, 637–669 | DOI | MR | Zbl

[37] N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, “Master equation for spin-spin correlation functions of the $XXZ$ chain”, Nucl. Phys. B, 712:3 (2005), 600–622 ; arXiv: hep-th/0406190 | DOI | MR | Zbl

[38] N. A. Slavnov, “Vychislenie skalyarnykh proizvedenii volnovykh funktsii i formfaktorov v algebraicheskom anzatse Bete”, TMF, 79:2 (1989), 232–240 | MR

[39] N. A. Slavnov, “Ob odnom tozhdestve dlya dualnykh polei”, Zapiski nauch. sem. POMI, 245 (1997), 270–281 ; N. A. Slavnov, “On an identity for dual fields”, J. Math. Sci. (New York), 100:2 (2000), 2181–2188 | MR | Zbl | DOI

[40] N. Kitanine, K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras, “On correlation functions of integrable models associated with the six-vertex $R$-matrix”, J. Stat. Mech., 2007, no. 1, P01022 ; arXiv: hep-th/0611142 | DOI | MR

[41] W. Heisenberg, “Zur Theorie des Ferromagnetismus”, Z. Phys., 49:9–10 (1928), 619–636 | DOI | Zbl

[42] P. P. Kulish, E. K. Sklyanin, “Quantum inverse scattering method and the Heisenberg ferromagnet”, Phys. Lett., 70:5–6 (1979), 461–463 | DOI | MR

[43] P. P. Kulish, “Kvantovyi metod obratnoi zadachi i tochno reshaemye modeli statisticheskoi fiziki”, Trudy II Mezhdunarodnogo simpoziuma po izbrannym problemam statisticheskoi fiziki, Dubna, 1981, 147–157

[44] A. G. Izergin, V. E. Korepin, “Correlation functions for the Heisenberg $XXZ$-antiferromagnet”, Comm. Math. Phys., 99:2 (1985), 271–302 | DOI | MR | Zbl

[45] N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, “Spin-spin correlation functions of the $XXZ-1/2$ Heisenberg chain in a magnetic field”, Nucl. Phys. B, 641:3 (2002), 487–518 ; arXiv: hep-th/0201045 | DOI | MR | Zbl

[46] N. Kitanine, J. M. Maillet, V. Terras, “Correlation functions of the $XXZ$ Heisenberg spin-$1/2$ chain in a magnetic field”, Nucl. Phys. B, 567:3 (2000), 554–582 ; arXiv: math-ph/9907019 | DOI | MR | Zbl

[47] N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, “Emptiness formation probability of the $XXZ$ spin-$1/2$ Heisenberg chain at $\Delta=1/2$”, J. Phys. A, 35:27 (2002), L385–L388 ; arXiv: hep-th/0201134 | DOI | MR | Zbl

[48] G. Kuperberg, “Another proof of the alternating sign matrix conjecture”, Internat. Math. Res. Notices, 1996, no. 3, 139–150 ; arXiv: math.CO/9712207 | DOI | MR | Zbl

[49] A. Razumov, Yu. Stroganov, “Spin chains and combinatorics”, J. Phys. A, 34:14 (2001), 3185–3190 ; arXiv: cond-mat/0012141 | DOI | MR | Zbl

[50] R. J. Baxter, “Partition function of the eight-vertex lattice model”, Ann. Physics, 70:1 (1972), 193–228 | DOI | MR | Zbl

[51] R. J. Baxter, “Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I. Some fundamental eigenvectors”, Ann. Physics, 76:1 (1973), 1–24 ; “II. Equivalence to a generalized ice-type lattice model”, 25–47 ; “III. Eigenvectors of the transfer matrix and hamiltonian”, 48–71 | DOI | Zbl | DOI | Zbl | DOI | Zbl