Abstract
In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space \({\mathcal W_\alpha}\) of all quasi-local operators, which are products of the disorder field \({q^{\alpha \sum_{j=-\infty}^0\sigma ^3_j}}\) with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ), c(ζ) which mutually anti-commute and kill the “primary field”. Here we construct the creation counterpart b*(ζ), c*(ζ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t*(ζ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b*(ζ), c*(ζ), t*(ζ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y.: Hidden Grassmann structure in the XXZ model. Commun. Math. Phys. 272, 263–281 (2007)
Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Fermionic basis for space of operators in the XXZ model. SISSA Proceedings of Science (2007), Paper 015, 34 pp. (electronic)
Bazhanov V., Lukyanov S., Zamolodchikov A.: Integrable structure of conformal field theory III. The Yang-Baxter relation. Commun. Math. Phys. 200, 297–324 (1999)
Boos H., Göhmann F., Klümper A., Suzuki J.: Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field. J. Phys. A 40, 10699–10727 (2007)
Jimbo, M., Miwa, T.: Algebraic analysis of solvable lattice models. Reg. Conf. Ser. in Math. 85 , Providence RI: Amer. Math. Soc., 1995
Tolstoy V., Khoroshkin S.: The universal R-matrix for quantized affine Lie algebras. Funct. Anal. and Appl. 26, 69–71 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Takhtajan
Dedicated to the memory of Alexei Zamolodchikov
Membre du CNRS
On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992 Moscow, Russia
Rights and permissions
About this article
Cite this article
Boos, H., Jimbo, M., Miwa, T. et al. Hidden Grassmann Structure in the XXZ Model II: Creation Operators. Commun. Math. Phys. 286, 875–932 (2009). https://doi.org/10.1007/s00220-008-0617-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0617-z