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

arXiv:1003.3071 · New Trends in Quantum Integrable Systems

KP AND TODA TAU FUNCTIONS IN BETHE ANSATZ

Kanehisa Takasaki

Published Oct 1, 2010• 11 citations
arXivHTMLPDF

Abstract

Recent work of Foda and his group on a connection between classical integrable hierarchies (the KP and 2D Toda hierarchies) and some quantum integrable systems (the 6-vertex model with DWBC, the finite XXZ chain of spin 1/2, the phase model on a finite chain, etc.) is reviewed. Some additional information on this issue is also presented.
DOI
10.1142/9789814324373_0019

Related papers

More on Slavnov Products of Spin Chains and KP Hierarchy Tau Functions

Thiago Araujo

Connections between classical and quantum integrable systems are analyzed from the viewpoint of Slavnov products of Bethe states. It is well known that, modulo model dependent aspects, the functional structure of Slavnov products generally takes the form of determinants. Building on recent results on the structure of rational and trigonometric models, we show that, provided certain conditions are satisfied, the Slavnov product of a given model can be interpreted as a tau function of the KP hierarchy, thus extending known results in a more general setting. Moreover, we show that Slavnov products can be expanded in terms of other tau functions. We also prove that their homogeneous limit can be systematically expressed as a Wronskian of functions related to the eigenvalues of the transfer matrices. Finally, we compute the Baker-Akhiezer functions associated with these Slavnov products and show that, apart from a universal multiplicative factor, they admit a closed determinantal representation.

arXiv:2505.21166•May 27, 2025•0 citations

XXZ SCALAR PRODUCTS, MIWA VARIABLES AND DISCRETE KP

OMAR FODA, GUS SCHRADER

We revisit the quantum/classical integrable model correspondence in the context of inhomogeneous finite length XXZ spin-1/2 chains with periodic boundary conditions and show that the Bethe scalar product of an arbitrary state and a Bethe eigenstate is a discrete KP tau-function. The continuous Miwa variables of discrete KP are the rapidities of the arbitrary state.

arXiv:1003.2524•New Trends in Quantum Integrable Systems•Oct 1, 2010•10 citations

Free fermions in classical and quantum integrable models

Wheeler, M.

This thesis studies the role of free fermions in certain classical and quantum integrable models. The classical models studied are the KP and BKP hierarchies of partial differential equations. We review the presence of free fermions in the theory of these hierarchies, a topic which was established in the work of the Kyoto school. The quantum models studied are descendents and relatives of the XYZ model, or its lattice equivalent, the eight-vertex model. We give a number of new results in the context of these models, revealing the presence of fermions in typical quantities such as Bethe eigenvectors, partition functions and scalar products.

arXiv:1110.6703•Oct 31, 2011•6 citations

THERMODYNAMICS OF THE TODA CHAIN

Alessandro Cuccoli, Roberto Livi, M. Spicci +2 more

A review of the classical and quantum thermodynamic properties of the Toda chain is provided, together with a survey of the techniques that have been used to work them out, i.e., the bilateral Laplace transform method for the classical system, the Bethe–Ansatz and the variational path-integral approach for the quantum one. For the classical Toda chain we also recall some of the main dynamical features, i.e. the integrability of the model and the soliton-like solutions of the equations of motion. In the quantum case a comparison between the Bethe–Ansatz and the variational method is made. In particular it is shown as the latter offers the possibility of also evaluating static correlation functions.

International Journal of Modern Physics B•Aug 15, 1994•12 citations

Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz

Christian Korff

We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q -operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q -operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyse the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.

arXiv:math-ph/0404028•Journal of Physics A Mathematical and General•Jul 8, 2004•15 citations

Aspectos clássicos e quânticos de modelos integráveis

Ana L. Retore

Aspects of classical and quantum integrability are explored. Gauge transformations play a fundamental role in both cases. Classical integrable hierarchies have an underlying algebraic structure which brings universality for the solutions of all the equations belonging to a hierarchy. Such universality is explored together with the gauge invariance of the zero curvature equation to systematically construct the Bäcklund transformations for the mKdV hierarchy, as well as to relate it with the KdV hierarchy. As a consequence the defect-matrix for the KdV hierarchy is obtained and a few explicit Bäcklund transformations are computed for both Type-I and Type-II. The generalization for super mKdV hierarchy is also explored. We studied symmetries and degeneracies of families of integrable quantum open spin chains with finite length associated to affine Lie algebras \hat{g} = A^{(2)}_{2n} , A^{(2)}_{2n−1}, B^{(1)}_n , C^{(1)}_ n , D^{(1)}_n whose K-matrices depend on a discrete parameter p (p = 0, ...,n). We show that all these transfer matrices have quantum group symmetry corresponding to removing the p^{th} node of the Dynkin diagram of \hat{g}. We also show that the transfer matrices for C^{(1)}_n and D^{(1)}_n also have duality symmetry and the ones for A^{(2)}_{2n−1}, B^{(1)}_n and D^{(1)}_n have Z_2 symmetries that map complex representations into their conjugates. Gauge transformations simplify considerably the proofs by allowing us to work in a way that only unbroken generators appear.The spectrum of the same integrable spin chains with the addition of D(2) n+1 is then determined using analytical Bethe ansatz. We conjecture a generalization for open chains for the Bethe ansatz Reshetikhin’s general formula and propose a formula relating the Dynkin labels of the Bethe states with the number of Bethe root sof each type.

•Aug 5, 2019•0 citations
Connections between classical and quantum integrable systems are analyzed from the viewpoint of Slavnov products of Bethe states. It is well known that, modulo model dependent aspects, the functional structure of Slavnov products generally takes the form of determinants. Building on recent results on the structure of rational and trigonometric models, we show that, provided certain conditions are satisfied, the Slavnov product of a given model can be interpreted as a tau function of the KP hierarchy, thus extending known results in a more general setting. Moreover, we show that Slavnov products can be expanded in terms of other tau functions. We also prove that their homogeneous limit can be systematically expressed as a Wronskian of functions related to the eigenvalues of the transfer matrices. Finally, we compute the Baker-Akhiezer functions associated with these Slavnov products and show that, apart from a universal multiplicative factor, they admit a closed determinantal representation.
This thesis studies the role of free fermions in certain classical and quantum integrable models. The classical models studied are the KP and BKP hierarchies of partial differential equations. We review the presence of free fermions in the theory of these hierarchies, a topic which was established in the work of the Kyoto school. The quantum models studied are descendents and relatives of the XYZ model, or its lattice equivalent, the eight-vertex model. We give a number of new results in the context of these models, revealing the presence of fermions in typical quantities such as Bethe eigenvectors, partition functions and scalar products.
A review of the classical and quantum thermodynamic properties of the Toda chain is provided, together with a survey of the techniques that have been used to work them out, i.e., the bilateral Laplace transform method for the classical system, the Bethe–Ansatz and the variational path-integral approach for the quantum one. For the classical Toda chain we also recall some of the main dynamical features, i.e. the integrability of the model and the soliton-like solutions of the equations of motion. In the quantum case a comparison between the Bethe–Ansatz and the variational method is made. In particular it is shown as the latter offers the possibility of also evaluating static correlation functions.
We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q -operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q -operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyse the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.
Aspects of classical and quantum integrability are explored. Gauge transformations play a fundamental role in both cases. Classical integrable hierarchies have an underlying algebraic structure which brings universality for the solutions of all the equations belonging to a hierarchy. Such universality is explored together with the gauge invariance of the zero curvature equation to systematically construct the Bäcklund transformations for the mKdV hierarchy, as well as to relate it with the KdV hierarchy. As a consequence the defect-matrix for the KdV hierarchy is obtained and a few explicit Bäcklund transformations are computed for both Type-I and Type-II. The generalization for super mKdV hierarchy is also explored. We studied symmetries and degeneracies of families of integrable quantum open spin chains with finite length associated to affine Lie algebras \hat{g} = A^{(2)}_{2n} , A^{(2)}_{2n−1}, B^{(1)}_n , C^{(1)}_ n , D^{(1)}_n whose K-matrices depend on a discrete parameter p (p = 0, ...,n). We show that all these transfer matrices have quantum group symmetry corresponding to removing the p^{th} node of the Dynkin diagram of \hat{g}. We also show that the transfer matrices for C^{(1)}_n and D^{(1)}_n also have duality symmetry and the ones for A^{(2)}_{2n−1}, B^{(1)}_n and D^{(1)}_n have Z_2 symmetries that map complex representations into their conjugates. Gauge transformations simplify considerably the proofs by allowing us to work in a way that only unbroken generators appear.The spectrum of the same integrable spin chains with the addition of D(2) n+1 is then determined using analytical Bethe ansatz. We conjecture a generalization for open chains for the Bethe ansatz Reshetikhin’s general formula and propose a formula relating the Dynkin labels of the Bethe states with the number of Bethe root sof each type.