Abstract
We consider an \(XYZ\) spin chain within the framework of the generalized algebraic Bethe ansatz. We study scalar products of the transfer matrix eigenvectors and arbitrary Bethe vectors. In the particular case of free fermions, we obtain explicit expressions for the scalar products with different number of parameters in two Bethe vectors.
Similar content being viewed by others
Notes
The question of the existence of solutions of inhomogeneous systems does not arise because scalar products with imbalance \(\varkappa=\pm1\) obviously exist.
References
E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, “Quantum inverse problem method. I,” Theoret. and Math. Phys., 40, 688–706 (1979).
L. A. Takhtadzhyan and L. D. Faddeev, “The quantum method of the inverse problem and the Heisenberg \(XYZ\) model,” Russian Math. Surveys, 34, 11–68 (1979).
L. D. Faddeev, “How the algebraic Bethe ansatz works for integrable models,” in: Symmétries quantiques [Quantum Symmetries] (Proceedings of the Les Houches Summer School, Session LXIV, Les Houches, France, August 1 – September 8, 1995, A. Connes, K. Gawedzki, and J. Zinn-Justin, eds.), North-Holland, Amsterdam (1998), pp. 149–219; arXiv: hep-th/9605187.
A. G. Izergin and V. E. Korepin, “The quantum inverse scattering method approach to correlation functions,” Commun. Math. Phys., 94, 67–92 (1984).
V. E. Korepin, “Dual field formulation of quantum integrable models,” Commun. Math. Phys., 113, 177–190 (1987).
T. Kojima, V. E. Korepin, and N. A. Slavnov, “Determinant representation for dynamical correlation function of the quantum nonlinear Schrödinger equation,” Commun. Math. Phys., 188, 657–689 (1997); arXiv: hep-th/9611216.
M. Jimbo, K. Miki, T. Miwa, and A. Nakayashiki, “Correlation functions of the \(XXZ\) model for \(\Delta<-1\),” Phys. Lett. A, 168, 256–263 (1992); arXiv: hep-th/9205055.
N. Kitanine, J. M. Maillet, and V. Terras, “Correlation functions of the \(XXZ\) Heisenberg spin-\(1/2\) chain in a magnetic field,” Nucl. Phys. B, 567, 554–582 (2000); arXiv: math-ph/9907019.
F. Göhmann, A. Klümper, and A. Seel, “Integral representations for correlation functions of the \(XXZ\) chain at finite temperature,” J. Phys. A: Math. Gen., 37, 7625–7652 (2004); arXiv: hep-th/0405089.
N. Kitanine, J. M. Maillet, N. A. Slavnov, and V. Terras, “Master equation for spin-spin correlation functions of the \(XXZ\) chain,” Nucl. Phys. B, 712, 600–622 (2005); arXiv: hep-th/0406190; N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, and V. Terras, “Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions,” J. Stat. Mech., 2009, P04003, 66 pp. (2009), arXiv: 0808.0227; “A form factor approach to the asymptotic behavior of correlation functions,” 2011, P12010, 28 pp. (2011), arXiv: 1110.0803.; “Form factor approach to dynamical correlation functions in critical models,” 2012, P09001, 33 pp. (2012), arXiv: 1206.2630.
J. S. Caux and J. M. Maillet, “Computation of dynamical correlation functions of Heisenberg chains in a magnetic field,” Phys. Rev. Lett., 95, 077201, 3 pp. (2005); arXiv: cond-mat/0502365.
R. G. Pereira, J. Sirker, J. S. Caux, R. Hagemans, J. M. Maillet, S. R. White, and I. Affleck, “Dynamical spin structure factor for the anisotropic spin-\(1/2\) Heisenberg chain,” Phys. Rev. Lett., 96, 257202, 4 pp. (2006), arXiv: cond-mat/0603681; “Dynamical structure factor at small \(q\) for the \(XXZ\) spin-\(1/2\) chain,” J. Stat. Mech., 2007, P08022, 64 pp. (2007), arXiv: 0706.4327.
J. S. Caux, P. Calabrese, and N. A. Slavnov, “One-particle dynamical correlations in the one- dimensional Bose gas,” J. Stat. Mech., 2007, P01008, 21 pp. (2007); arXiv: cond-mat/0611321.
V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, Cambridge (1993).
N. A. Slavnov, Algebraic Bethe Ansatz and Correlation Functions: An Advanced Course, World Sci., Singapore (2022).
W. Heisenberg, “Zur Theorie des Ferromagnetismus,” Z. Phys., 49, 619–636 (1928).
B. Sutherland, “Two-dimensional hydrogen bonded crystals without the ice rule,” J. Math. Phys., 11, 3183–3186 (1970).
C. Fan and F. Y. Wu, “General lattice model of phase transitions,” Phys. Rev. B, 2, 723–733 (1970).
R. J. Baxter, “Eight-vertex model in lattice statistics,” Phys. Rev. Lett., 26, 832–833 (1971).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982).
S. Belliard and N. A. Slavnov, “Why scalar products in the algebraic Bethe ansatz have determinant representation,” JHEP, 10, 103, 16 pp. (2019); arXiv: 1908.00032.
N. Slavnov, A. Zabrodin, and A. Zotov, “Scalar products of Bethe vectors in the 8-vertex model,” JHEP, 06, 123, 53 pp. (2020); arXiv: 2005.11224.
N. Kitanine, J.-M. Maillet, and V. Terras, “Form factors of the XXZ Heisenberg spin-\(1/2\) finite chain,” Nucl. Phys. B, 554, 647–678 (1999); arXiv: math-ph/9807020.
F. Göhmann and V. E. Korepin, “Solution of the quantum inverse problem,” J. Phys. A: Math. Gen., 33, 1199–1220 (2000); arXiv: hep-th/9910253.
J. M. Maillet and V. Terras, “On the quantum inverse scattering problem,” Nucl. Phys. B, 575, 627–644 (2000); arXiv: hep-th/9911030.
G. Kulkarni and N. A. Slavnov, “Action of the monodromy matrix entries in the generalized algebraic Bethe ansatz,” Theoret. and Math. Phys., to appear; arXiv: 2303.02439.
E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain,” Ann. Phys., 16, 407–466 (1961).
B. M. McCoy, “Spin correlation functions of the \(X\)–\(Y\) model,” Phys. Rev., 173, 531–541 (1968).
Th. Niemeijer, “Some exact calculations on a chain of spins 1/2,” Physica, 36, 377–419 (1967).
S. Katsura, T. Horiguchi, and M. Suzuki, “Dynamical properties of the isotropic \(XY\) model,” Physica, 46, 67–86 (1970).
J. H. H. Perk and H. W. Capel, “Time-dependent \(xx\)-correlation functions in the one dimensional \(XY\)-model,” Phys. A, 89, 265–303 (1977).
H. G. Vaidya and C. A. Tracy, “Crossover scaling function for the one-dimensional \(XY\) model at zero temperature,” Phys. Lett. A, 68, 378–380 (1978).
T. Tonegawa, “Transverse spin correlation function of the one-dimensional spin-\(1/2\) \(XY\) model,” Solid State Comm., 40, 983–986 (1981).
M. D’lorio, U. Glaus, and E. Stoll, “Transverse spin dynamics of a one-dimensional \(XY\) system: A fit to spin-spin relaxation data,” Solid State Commun., 47, 313–315 (1983).
A. G. Izergin, N. A. Kitanin, and N. A. Slavnov, “On correlation functions of the \(XY\) model,” J. Math. Sci. (N. Y.), 88, 224–232 (1998).
K. Fabricius and B. M. McCoy, “New developments in the eight vertex model,” J. Stat. Phys., 111, 323–337 (2003), arXiv: cond-mat/0207177; “New developments in the eight vertex model II. Chains of odd length,” test, 120, 37–70 (2005), arXiv: cond-mat/0410113; “Functional equations and fusion matrices for the eight-vertex model,” Publ. Res. Inst. Math. Sci., 40, 905–932 (2004); arXiv: cond-mat/0311122.
K. Fabricius and B. M. McCoy, “An elliptic current operator for the 8 vertex model,” J. Phys. A: Math. Gen., 39, 14869–14886 (2006); arXiv: cond-mat/0606190.
T. Deguchi, “The 8V CSOS model and the \(sl_2\) loop algebra symmetry of the six-vertex model at roots of unity,” Internat. J. Modern Phys. B, 16, 1899–1905 (2002), arXiv: cond-mat/0110121; “Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix,” J. Phys. A: Math. Gen., 35, 879–895 (2002); arXiv: cond-mat/0109078.
K. Fabricius, “A new \(Q\)-matrix in the eight vertex model,” J. Phys. A: Math. Theor., 40, 4075–4086 (2007); arXiv: cond-mat/0610481.
S. Kharchev and A. Zabrodin, “Theta vocabulary I,” J. Geom. Phys., 94, 19–31 (2015); arXiv: 1502.04603.
Acknowledgments
We are grateful to A. Zabrodin and A. Zotov for the numerous and fruitful discussions.
Funding
The work of G. Kulkarni was supported by the SIMC postdoctoral grant of the Steklov Mathematical Institute. Section 4 of the paper represents the work of N. A. Slavnov. The work of N. A. Slavnov was supported by the Russian Science Foundation under grant no. 19-11-00062, https://rscf.ru/en/project/19-11-00062/, and performed at the Steklov Mathematical Institute, Russian Academy of Sciences.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflicts of interest.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 179–203 https://doi.org/10.4213/tmf10572.
Appendix A. Jacobi theta functions
Here, we only give some basic properties of Jacobi theta functions used in this paper (see [40] for more details).
The Jacobi theta functions are defined as
They have the following shift properties:
Appendix B. Zero eigenvectors
Let \(M\) be an \(n\times n\) matrix. Let its matrix elements admit a representation
We find zero eigenvectors of \(M\). For this, we extend the matrix \(B\) by adding \(n-m\) rows with elements \(B_{m+1,k},B_{m+2,k},\ldots,B_{n,k}\). We let \(\widetilde B\) denote this extended \(n\times n\) matrix. We require this extended matrix \(\widetilde B\) to be invertible. Then the zero eigenvectors \(\Psi^{(a)}\), \(a=1,\ldots,n-m\) have the components
In particular, let \(B\) be a rectangular Cauchy matrix
Appendix C. Contour integral method
A contour integral method allows one to calculate or transform sums of a special form containing the Jacobi theta functions. To illustrate this method, we consider two examples.
The first example is related to the transformation of the matrix product \(\mathbf\Omega^1\mathbf\Omega^0\). It follows from (4.12) that the product \(\mathbf\Omega^1\mathbf\Omega^0\) has the form
Thus, we obtain
In the second example, we calculate the coefficients \(\mathcal A_j\) in (4.27). We have
Rights and permissions
About this article
Cite this article
Kulkarni, G., Slavnov, N.A. Scalar products of Bethe vectors in the generalized algebraic Bethe ansatz. Theor Math Phys 217, 1574–1594 (2023). https://doi.org/10.1134/S0040577923100100
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577923100100