Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Separation of variables for the quantum $SL(3,\mathbb C)$ spin magnet: eigenfunctions of Sklyanin $B$-operator

Sergey E. Derkachov, P. A. Valinevich

Published Jul 1, 2018• 4 citations
OpenAlex

Abstract

The quantum SL(3,C) invariant spin magnet with infinite-dimensional principal series representation in local spaces is considered. We construct eigenfunctions of Sklyanin B-operator which define the representation of separated variables of the model.

Related papers

Separation of variables for the quantum SL(3,C) spin magnet: eigenfunctions of Sklyanin B-operator

Sergey E. Derkachov, P. A. Valinevich

The quantum SL(3,C) invariant spin magnet with infinite-dimensional principal series representation in local spaces is considered. We construct eigenfunctions of Sklyanin B-operator which define the representation of separated variables of the model.

arXiv:1807.00302•Jul 1, 2018•13 citations

Separation of variables for the quantum SL(2, R) spin chain

Sergey E. Derkachov, Gregory P. Korchemsky, Alexander N Manashov

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.

•Jan 1, 2002•21 citations

R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain

Sergey E. Derkachov

The problem of constructing the SL(N, C) invariant solutions to the Yang-Baxter equation is considered.The solutions (R-operators) for arbitrarily principal series representations of SL(N, C) are obtained in an explicit form.We construct the commutative family of the operators Q k (u) which can be identified with the Baxter operators for the noncompact SL(N, C) spin magnet.

arXiv:nlin/0612003•Symmetry Integrability and Geometry Methods and Applications•Dec 2, 2006•37 citations

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

S É Derkachov, Alexander N Manashov

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).

arXiv:1401.7477•Journal of Physics A Mathematical and Theoretical•Jul 10, 2014•29 citations

Separation of variables for quantum integrable systems on elliptic curves

Giovanni Felder, Anke Schorr

We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group E , ( sl 2 ) by difference operators. One model of statistical mechanics to which this method applies is the interaction-round-a-face model with antiperiodic boundary conditions. The eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher-order theta functions.

arXiv:math/9905072•Journal of Physics A Mathematical and General•Nov 8, 1999•24 citations

Form-factors and complete basis of observables via separation of variables for higher rank spin chains

Nikolay Gromov, Nicolò Primi, Paul Ryan

Abstract Integrable $$ \mathfrak{sl} $$ sl (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for $$ \mathfrak{sl} $$ sl (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.

arXiv:2202.01591•Journal of High Energy Physics•Nov 9, 2022•13 citations
Separation of variables for the quantum SL(3,C)SL(3,\mathbb C)SL(3,C) spin magnet: eigenfunctions of Sklyanin BBB-operator
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.
Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C)SL(2,\mathbb {C})SL(2,C) magnet
We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group E , ( sl 2 ) by difference operators. One model of statistical mechanics to which this method applies is the interaction-round-a-face model with antiperiodic boundary conditions. The eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher-order theta functions.
Abstract Integrable sl \mathfrak{sl} sl sl (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for sl \mathfrak{sl} sl sl (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.