Abstract
We derive spin operator matrix elements between general eigenstates of the superintegrable ℤ N -symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables method.
Similar content being viewed by others
References
Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)
Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85, 808–816 (1952)
Onsager, L.: Proc. IUPAP conference on statistical mechanics, discussione ed observazioni. Nuovo Cimento (Suppl.) Ser. 9 6, 261 (1949)
von Gehlen, G., Rittenberg, V.: Z n -symmetric quantum chains with an infinite set of conserved charges and Z n zero modes. Nucl. Phys. B 257(FS14), 351–370 (1985)
Baxter, R.J.: The superintegrable chiral Potts model. Phys. Lett. A 133, 185–189 (1988)
Baxter, R.J., Perk, J.H.H., Au-Yang, H.: New solutions of the star-triangle relations for the chiral Potts model. Phys. Lett. A 128, 138–142 (1988)
Au-Yang, H., Perk, J.H.H.: Onsager’s star-triangle equation: master key to integrability. Adv. Stud. Pure Math. 19, 57–94 (1989)
Albertini, G., McCoy, B.M., Perk, J.H.H., Tang, S.: Excitation spectrum and order parameter for the integrable N-state chiral Potts model. Nucl. Phys. B 314, 741–763 (1989)
Henkel, M., Lacki, J.: Integrable chiral Z n -quantum chains and a new class of trigonometric sums. Phys. Lett. A 138, 105–109 (1989)
Baxter, R.J.: The order parameter of the chiral Potts model. Phys. Rev. Lett. 94, 130602 (2005)
Baxter, R.J.: The order parameter of the chiral Potts model. J. Stat. Phys. 120, 1–36 (2005)
Baxter, R.J.: A conjecture for the superintegrable chiral Potts model. J. Stat. Phys. 132, 983–1000 (2008)
Baxter, R.J.: Some remarks on a generalization of the superintegrable chiral Potts model. arXiv:0906.3551 (2009)
Baxter, R.J.: Proof of the determinantal form of the spontaneous magnetization of the superintegrable chiral Potts model. arXiv:1001.0281 (2010)
Baxter, R.J.: Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D PQ . arXiv:0912.4549 (2009)
von Gehlen, G., Iorgov, N., Pakuliak, S., Shadura, V., Tykhyy, Y.: Form-factors in the Baxter-Bazhanov-Stroganov model II: Ising model on the finite lattice. J. Phys. A, Math. Theor. 41, 095003 (2008)
Bugrij, A., Lisovyy, O.: Correlation function of the two-dimensional Ising model on a finite lattice. II. Theor. Math. Phys. 140, 987–1000 (2004)
Iorgov, N.: Form-factors of the finite quantum XY-chain. arXiv:0912.4466 (2009)
Bazhanov, V.V., Stroganov, Y.G.: Chiral Potts model as a descendant of the six-vertex model. J. Stat. Phys. 59, 799–817 (1990)
Baxter, R.J., Bazhanov, V.V., Perk, J.H.H.: Functional relations for transfer matrices of the chiral Potts model. Int. J. Mod. Phys. B 4, 803–870 (1990)
Nishino, A., Deguchi, T.: The L(sl 2) symmetry of the Bazhanov-Stroganov model associated with the superintegrable chiral Potts model. Phys. Lett. A 356, 366–370 (2006)
Nishino, A., Deguchi, T.: An algebraic derivation of the eigenspaces associated with an Ising-like spectrum of the superintegrable chiral Potts model. J. Stat. Phys. 133, 587–615 (2008)
Au-Yang, H., Perk, J.H.H.: Eigenvectors in the superintegrable model I: sl 2-generators. J. Phys. A, Math. Theor. 41, 275201 (2008)
Au-Yang, H., Perk, J.H.H.: Eigenvectors in the superintegrable model II: ground state sector. J. Phys. A, Math. Theor. 42, 375208 (2009)
Au-Yang, H., Perk, J.H.H.: Identities in the superintegrable chiral Potts model. arXiv:0906.3153 (2009)
Au-Yang, H., Perk, J.H.H.: Quantum loop subalgebra and eigenvectors of the superintegrable chiral Potts transfer matrices. arXiv:0907.0362 (2009)
Albertini, G., McCoy, B.M., Perk, J.H.H.: Level crossing transitions and the massless phases of the superintegrable chiral Potts chain. Phys. Lett. A 139, 204–212 (1989)
Howes, L.P., Kadanoff, L.P., Den Nijs, M.: Quantum model for commensurate-incommensurate transitions. Nucl. Phys. B 215, 169 (1983)
Dolan, L., Grady, M.: Conserved charges from self-duality. Phys. Rev. D 25, 1587–1604 (1982)
Perk, J.H.H.: Star-triangle equations, quantum lax pairs and higher genus curves. Proc. Symp. Pure Math. 49, 341–354 (1989)
Davies, B.: Onsager’s algebra and superintegrability. J. Phys. A, Math. Gen. 23, 2245–2261 (1990)
Roan, S-S., Onsager’s algebra, loop algebra and chiral Potts model. Preprint Max-Planck-Institut für Mathematik Bonn, MPI/91-70 (1991)
Date, E., Roan, S.-S.: The algebraic structure of the Onsager algebra. Czechoslov. J. Phys. 50, 37–44 (2000)
Roan, S.-S.: The Onsager algebra symmetry of τ (j)-matrices in the superintegrable chiral Potts model. J. Stat. Mech. 0509, P007 (2005)
Albertini, G., McCoy, B.M., Perk, J.H.H.: Eigenvalue spectrum of the superintegrable chiral Potts model. Adv. Stud. Pure Math. 19, 1–55 (1989)
Baxter, R.J.: Chiral Potts model with skewed boundary conditions. J. Stat. Phys. 73, 461–495 (1993)
von Gehlen, G.: Integrable Z n -chiral Potts model: phase diagram and rapidity-momentum relation. In: Ge, M.L., Wu, F.Y. (eds.) Statistical Models, Yang-Baxter Equation and Related Topics, pp. 102–109. World Scientific, Singapore (1996). arXiv:hep-th/9601001
Fabricius, K., McCoy, B.M.: Correlation functions for the three state superintegrable chiral Potts spin chain of finite lengths. arXiv:1001.0614 (2010)
Baxter, R.: The superintegrable chiral Potts model: thermodynamic properties and “inverse” model, and a simple associated Hamiltonian. J. Stat. Phys. 57, 1–39 (1989)
von Gehlen, G.: Finite-Size Energy Levels of the Superintegrable Chiral Potts Model. Springer Lecture Notes in Physics, vol. 524, p. 307. Springer, Berlin (1999). arXiv:hep-th/9811123
Au-Yang, H., Perk, J.H.H.: Spontaneous magnetization of the Chiral Potts model. arXiv:1003.4805 [math-ph]
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Iorgov, N., Pakuliak, S., Shadura, V. et al. Spin Operator Matrix Elements in the Superintegrable Chiral Potts Quantum Chain. J Stat Phys 139, 743–768 (2010). https://doi.org/10.1007/s10955-010-9972-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-9972-1