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arXiv:1005.4603 · Symmetry Integrability and Geometry Methods and Applications

Quantum Integrable 1D anyonic Models: Construction through Braided Yang-Baxter Equation

Anjan Kundu

Published Oct 9, 2010• 5 citations
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Abstract

Applying braided Yang-Baxter equation quantum integrable and Bethe ansatz solvable 1D anyonic lattice and field models are constructed.Along with known models we discover novel lattice anyonic and q-anyonic models as well as nonlinear Schrödinger equation (NLS) and the derivative NLS quantum field models involving anyonic operators, N -particle sectors of which yield the well known anyon gases, interacting through δ and derivative δ-function potentials.

DOI
10.3842/sigma.2010.080

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A scheme suitable for describing quantum non-ultralocal models, including supersymmetric ones, is proposed. Braided algebras are generalized to be used through Baxterization for constructing braided quantum Yang—Baxter equations. Supersymmetric and some known non-ultralocal models are derived in the framework of this approach. As further applications of the scheme, the construction of new quantum integrable non-ultralocal models, like mKdV and anyonic supersymmetric models including deformed anyonic superalgebra, is outlined.
Various aspects of the theory of quantum integrable systems are reviewed. Basic ideas behind the construction of integrable ultralocal and nonultralocal quantum models are explored by exploiting the underlying algebraic structures related to the Yang-Baxter equations. Physical meaning of abstract mathematical notions like universal R-matrix, quantized algebras, Sklyanin algebra, braided algebra, Hopf algebra etc. and the role played by them in integrable systems are highlighted. Systematic construction of quantum integrable lattice as well as field models and their exact excitation spectra are presented through examples. The coordinate and algebraic formulations of the Bethe ansatz are illustrated with comparison, along with the description of nested and functional Bethe ansatzes. The techniques for deriving quantum Hamiltonians from the Lax operators are demonstrated on concrete models. The exposition of this review is kept in a fairly elementary level with emphasis on the physical contents.
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