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arXiv:1401.7477 · Journal of Physics A Mathematical and Theoretical

Iterative construction of eigenfunctions of the monodromy matrix for $SL(2,\mathbb {C})$ magnet

S É Derkachov, Alexander N Manashov

Published Jul 10, 2014• 29 citations
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Abstract

Eigenfunctions of the matrix elements of the monodromy matrix provide a convenient basis for studies of spin chain models. We present an iterative method for constructing the eigenfunctions in the case of the SL(2,C) spin chains. We derived an explicit integral representation for the eigenfunctions and calculated the corresponding scalar products (Sklyanin's measure).

DOI
10.1088/1751-8113/47/30/305204

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Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C)SL(2,\mathbb {C})SL(2,C) magnet
Eigenfunctions of the matrix elements of the monodromy matrix provide a convenient basis for studies of spin chain models. We present an iterative method for constructing the eigenfunctions in the case of the SL(2,C) spin chains. We derived an explicit integral representation for the eigenfunctions and calculated the corresponding scalar products (Sklyanin's measure).
We consider the problem of seeking the eigenvectors for a commuting family of quantum minors of the monodromy matrix for an SL(n,ℂ)-invariant inhomogeneous spin chain. The algebra generators and elements of the L-operator at each site of the chain are implemented as linear differential operators in the space of functions of n(n−1)/2 variables. In the general case, the representation of the sl n (ℂ) algebra at each site is infinite-dimensional and belongs to the principal unitary series. We solve this problem using a recursive procedure with respect to the rank n of the algebra. We obtain explicit expressions for the eigenvalues and eigenvectors of the commuting family. We consider the particular cases n = 2 and n = 3 and also the limit case of the one-site chain in detail.
We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin-1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F-basis and the usual basis.
We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin- 1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the Monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F- basis and the usual basis. 1. Introduction. One of the most important open problems in the theory of quantum integrable models is the calculation of the correlation functions. In the framework of the Algebraic Bethe Ansatz method [1] the problem is the combinatorial complexity of calculations due to the structure of Bethe eigenstates. Only the very limited number of physical results for the correlation
We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.
Separation of variables for the quantum SL(3,C)SL(3,\mathbb C)SL(3,C) spin magnet: eigenfunctions of Sklyanin BBB-operator