Abstract
We demonstrate that the six vertex model (XXZ spin chain) with Δ=(q+q-1)/2 and q2N=1 has an invariance under the loop algebra of sl2 which produces a special set of degenerate eigenvalues. For Δ=0 we compute the multiplicity of the degeneracies using Jordan–Wigner techniques.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
E. H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Letts. 18:692 (1967).
E. H. Lieb, Exact solution of the F model of an antiferroelectric, Phys. Rev. Lett. 18:1046 (1967).
E. H. Lieb, Exact solution of the two-dimensional Slater KDP model of a ferroelectric, Phys. Rev. Lett. 19:108 (1967).
E. H. Lieb, Residual entropy of square ice, Phys. Rev. 162:162 (1967).
B. Sutherland, Exact solution of a two-dimensional model for hydrogen bonded crystals, Phys. Rev. Lett. 19:103 (1967).
C. P. Yang, Exact solution of two dimensional ferroelectrics in an arbitrary external field, Phys. Rev. Lett. 19:586 (1967).
B. Sutherland, C. N. Yang, and C. P. Yang, Exact solution of two dimensional ferroelectrics in an arbitrary external field, Phys. Rev. Lett. 19:588 (1967).
H. A. Bethe, Zur Theorie der Metalle, Z. Physik 71:205 (1931).
E. Hulthen, Über das Austauschproblem eines Kristalles, Arkiv. Mat. Astron. Fysik 26A, No. 11 (1938).
R. Orbach, Linear antiferromagnetic chain with anisotropic coupling, Phys. Rev. 112:309 (1958).
L. R. Walker, Antiferromagnetic linear chain, Phys. Rev. 116:1089 (1959).
J. des Cloizeaux and M. Gaudin, Anisotropic linear magnetic chain, J. Math. Phys. 7:1384 (1966).
C. N. Yang and C. P. Yang, One-dimensional chain of anisotropic spin-spin interactions, Phys. Rev. 150:321 (1966); 327 (1966).
R. J. Baxter, Eight-vertex model in lattice statistics, Phys. Rev. Lett. 26:832 (1971); One-dimensional anisotropic Heisenberg chain, Phys. Rev. Lett. 26:834 (1971).
R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys. 70:193 (1972).
R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain I: Some fundamental eigenvectors, Ann. Phys. 76:1 (1973).
R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain II: Equivalence to a generalized ice-type lattice model, Ann. Phys. 76:25 (1973).
R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain III: Eigenvectors of the transfer matrix and the Hamiltonian, Ann. Phys. 76:48 (1973).
G. E. Andrews, R. J. Baxter, and P. J. Forrester, Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Stat. Phys. 35:193 (1984).
L. Onsager, Crystal statistics I: A two dimensional model with an order disorder transition, Phys. Rev. 65:117 (1944).
E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. 16:407 (1961).
P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Z. Physik 47:631 (1928).
V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theory through quantum groups, Nucl. Phys. B 330:523 (1990).
M. Jimbo, A q-analogue of U(gl(N + 1)), Hecke algebra, and the Yang_Baxter equation, Lett. Math. Phys. 11:247 (1986).
G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70:237 (1988).
G. Lusztig, Modular representations and quantum groups, Contemp. Math. 82:59 (1989).
C. de Concini and V. Kac, Representations of quantum groups at roots of 1, in Operator Algebras, Unitary Representation, Enveloping Algebras, and Invariant Theory, A. Connes et al., eds. (Progress in Mathematics 92, Birkhäuser, Basel 1990), p. 471.
G. Lusztig, Introduction to Quantum Groups (Birkäuser, 1993).
V. G. Kac, Infinite Dimensional Lie Algebras, 3rd ed. (Cambridge University Press, 1990).
A. B. Zamolodchikov and V. A. Fateev, Nonlocal (parafermion) currents in two-dimensional quantum field theory and self dual critical points in Z N symmetric statistical mechanics, Sov. Phys. JETP 62:215 (1985).
Y. Akutsu and M. Wadati, Exactly solvable models and new link polynomials I. N-State vertex models, J. Phys. Soc. Jpn. 56:3039 (1987).
T. Deguchi, M. Wadati, and Y. Akutsu, Exactly solvable models and new link polynomials V. Yang_Baxter operators and braid-monoid algebra, J. Phys. Soc. Jpn. 57:1905 (1988).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Deguchi, T., Fabricius, K. & McCoy, B.M. The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity. Journal of Statistical Physics 102, 701–736 (2001). https://doi.org/10.1023/A:1004894701900
Issue Date:
DOI: https://doi.org/10.1023/A:1004894701900