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:1201.0627

Unraveling hidden hierarchies and dual structures in an integrable field model

Anjan Kundu

Published Jan 3, 2012• 5 citations
arXivHTMLPDF

Abstract

An integrable field theory, due to path-independence on the space-time plane, should yield together with an infinite set of independent conserved charges also similar dual charges determining the boundary and defect contributions. On the example of the nonlinear Schroedinger equation we unravel hidden hierarchies and dual structures and show the complete integrability through a novel Yang-Baxter equation at the classical and quantum level with exact solution.

DOI
10.48550/arxiv.1201.0627

Related papers

Conserved charges from self-duality

L. Dolan, Michael Grady

Given a simple self-dual quantum Hamiltonian $H=KB+\ensuremath{\Gamma}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}$, where $K$ and $\ensuremath{\Gamma}$ are coupling constants, and the condition that $[B,[B,[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]]]=16[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]$, then we construct an infinite set of conserved charges ${Q}_{2n}$; $[H,{Q}_{2n}]=0$. In simple models, like the two-dimensional Ising or Baxter eight-vertex, these charges appear in the associated quantum theories and are equivalent to those which result from the transfer-matrix formulation and exact quantum integrability of the system. The power of our result is that it is an operator statement and does not refer to the number of dimensions or the nature of the space-time manifold: lattice, continuum, or loop space. It is suggested how the establishment of this link between duality and integrability could be used to exploit the Kramers-Wannier-type self-duality of the four-dimensional $\mathrm{SU}(N)$ gauge theory to find hidden symmetry.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Mar 15, 1982•164 citations

Infinite Sets of Conserved Charges and Duality in Quantum Field Theory

Michael Grady

The thesis consists of nine chapters and two appendices. In chapter II I discuss the connection given by the transfer matrix formulation between a classical statistical mechanical system and an associated quantum Hamiltonian, using as an example the familiar two- 4 dimensional Ising model. Next, in chapter III, I show how this applies to Baxter's 8-vertex model and the XYZ Hamiltonian, and how an infinite set of conserved charges for the XYZ model arises from properties of the transfer matrix. In Chapter IV I discuss the meaning of self-duality as it applies to quantum Hamiltonians, and develop a general form for a self-dual Hamiltonian. In Chapter V I show how one can find the explicit form for conserved charges in the XY model by a heuristic technique which takes advantage of the models' self-duality. In chapter VI it is shown how the solution to the XYZ model by the quantum inverse scattering method leads to the infinite set of conserved charges associated with it. The thesis culminates in the statement and proof of the theorem mentioned above, which stipulates a condition under which self-dual theories possess an infinite set of conserved charges (Chapter VII), after which I briefly discuss applications of the theorem (VIII) and draw some conclusions (IX).

•Sep 8, 2025•0 citations

Introduction to classical and quantum integrability

Ana L. Retore

Abstract In these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability. Starting from Liouville integrability and passing through Lax pair and r-matrix we discuss the construction of the conserved charges for classical integrable models taking as example the harmonic oscillator. The construction of these charges for 2D integrable field theories is also discussed using a Lax connection and the sine-Gordon model as example. On the quantum side, the XXZ spin chain is used to explain the systematic construction of the conserved charges starting from a quantum R-matrix, solution of the quantum Yang–Baxter equation. The diagonalization of these charges is performed using the algebraic Bethe ansatz. At the end, the interpretation of the R-matrix as an S-matrix in a scattering process is also presented. These notes were written for the lectures delivered at the school ‘Integrability, Dualities and Deformations’, that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

arXiv:2109.14280•Journal of Physics A Mathematical and Theoretical•Mar 3, 2022•27 citations

Exact Integrability of the su (n) Hubbard Model

Z. Maassarani

The bosonic su (n) Hubbard model was recently introduced. The model was shown to be integrable in one dimension by exhibiting the infinite set of conserved quantities. I derive the R-matrix and use it to show that the conserved charges commute among themselves. This new matrix is a nonadditive solution of the Yang–Baxter equation. Some properties of this matrix are derived.

arXiv:cond-mat/9710083•Modern Physics Letters B•Feb 10, 1998•24 citations

THE YANG–BAXTER SYMMETRY IN FIELD THEORY

Hector J. de Vega

This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of $q-$deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of $S{\hat U}(2)_q$). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix $τ(ł)$ are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit $τ(ł)$ turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium].

arXiv:hep-th/9312084•Advanced series in mathematical physics•Feb 1, 1994•0 citations

YANG-BAXTER AND KAC-MOODY ALGEBRAS IN FIELD THEORY WITH SOLITONS

Hector J. de Vega

The integrability of some important physical systems with a finite number of degrees of freedom relies on purely algebraic properties. One can solve the Coulomb potential and the harmonic oscillator both classically and quantum mechanically, using the appropriate dynamical algebra only [1]. Integrable systems with an infinite number of degrees of freedom are those with an infinite number of conserved quantities. They exhibit loop algebras, Yang-Baxter algebras and Kac-Moody algebras as symmetries. The investigation of theories has shown in recent years that Yang-Baxter algebras are the basic concepts of integrability [2]. The commutativity of a family of transfer matrices both in field theory and statistical mechanics follows directly from the Yang-Baxter algebra of monodromy matrices. Since the transfer matrix is a generating functional of local and non-local charges (including the hamiltonian) one gets an infinite set of commuting and conserved quantities. So, we can say that the theory is integrable. We are using the word integrable in the large sense explained above. Moreover, theories can be often explicitly solved to a large extent [2, 3]. In this brief report I will concentrate on a specific class of field theories. A complete account can be found in refs. 4-7. This class

Talk given at•Jan 1, 1985•0 citations
An integrable field theory, due to path-independence on the space-time plane, should yield together with an infinite set of independent conserved charges also similar dual charges determining the boundary and defect contributions. On the example of the nonlinear Schroedinger equation we unravel hidden hierarchies and dual structures and show the complete integrability through a novel Yang-Baxter equation at the classical and quantum level with exact solution.
Given a simple self-dual quantum Hamiltonian $H=KB+\ensuremath{\Gamma}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}$, where KKK and $\ensuremath{\Gamma}$ are coupling constants, and the condition that $[B,[B,[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]]]=16[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]$, then we construct an infinite set of conserved charges Q2n{Q}_{2n}Q2n​; [H,Q2n]=0[H,{Q}_{2n}]=0[H,Q2n​]=0. In simple models, like the two-dimensional Ising or Baxter eight-vertex, these charges appear in the associated quantum theories and are equivalent to those which result from the transfer-matrix formulation and exact quantum integrability of the system. The power of our result is that it is an operator statement and does not refer to the number of dimensions or the nature of the space-time manifold: lattice, continuum, or loop space. It is suggested how the establishment of this link between duality and integrability could be used to exploit the Kramers-Wannier-type self-duality of the four-dimensional SU(N)\mathrm{SU}(N)SU(N) gauge theory to find hidden symmetry.
The thesis consists of nine chapters and two appendices. In chapter II I discuss the connection given by the transfer matrix formulation between a classical statistical mechanical system and an associated quantum Hamiltonian, using as an example the familiar two- 4 dimensional Ising model. Next, in chapter III, I show how this applies to Baxter's 8-vertex model and the XYZ Hamiltonian, and how an infinite set of conserved charges for the XYZ model arises from properties of the transfer matrix. In Chapter IV I discuss the meaning of self-duality as it applies to quantum Hamiltonians, and develop a general form for a self-dual Hamiltonian. In Chapter V I show how one can find the explicit form for conserved charges in the XY model by a heuristic technique which takes advantage of the models' self-duality. In chapter VI it is shown how the solution to the XYZ model by the quantum inverse scattering method leads to the infinite set of conserved charges associated with it. The thesis culminates in the statement and proof of the theorem mentioned above, which stipulates a condition under which self-dual theories possess an infinite set of conserved charges (Chapter VII), after which I briefly discuss applications of the theorem (VIII) and draw some conclusions (IX).
Abstract In these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability. Starting from Liouville integrability and passing through Lax pair and r-matrix we discuss the construction of the conserved charges for classical integrable models taking as example the harmonic oscillator. The construction of these charges for 2D integrable field theories is also discussed using a Lax connection and the sine-Gordon model as example. On the quantum side, the XXZ spin chain is used to explain the systematic construction of the conserved charges starting from a quantum R-matrix, solution of the quantum Yang–Baxter equation. The diagonalization of these charges is performed using the algebraic Bethe ansatz. At the end, the interpretation of the R-matrix as an S-matrix in a scattering process is also presented. These notes were written for the lectures delivered at the school ‘Integrability, Dualities and Deformations’, that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.
This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of q−q-q−deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of SU^(2)qS{\hat U}(2)_qSU^(2)q​). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix τ(ł)τ(ł)τ(ł) are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit τ(ł)τ(ł)τ(ł) turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium].
The integrability of some important physical systems with a finite number of degrees of freedom relies on purely algebraic properties. One can solve the Coulomb potential and the harmonic oscillator both classically and quantum mechanically, using the appropriate dynamical algebra only [1]. Integrable systems with an infinite number of degrees of freedom are those with an infinite number of conserved quantities. They exhibit loop algebras, Yang-Baxter algebras and Kac-Moody algebras as symmetries. The investigation of theories has shown in recent years that Yang-Baxter algebras are the basic concepts of integrability [2]. The commutativity of a family of transfer matrices both in field theory and statistical mechanics follows directly from the Yang-Baxter algebra of monodromy matrices. Since the transfer matrix is a generating functional of local and non-local charges (including the hamiltonian) one gets an infinite set of commuting and conserved quantities. So, we can say that the theory is integrable. We are using the word integrable in the large sense explained above. Moreover, theories can be often explicitly solved to a large extent [2, 3]. In this brief report I will concentrate on a specific class of field theories. A complete account can be found in refs. 4-7. This class