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Aspectos clássicos e quânticos de modelos integráveis

Ana L. Retore

Published Aug 5, 2019• 0 citations
OpenAlex

Abstract

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.

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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.
We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.
NONLINEAR SYSTEMS AND CLASSICAL IST Introduction Definition of Integrability Lax Pair Technique Inverse Scattering Transform Hamiltonian Structure COORDINATE BETHE ANSATZ Introduction Nonlinear Systems and the CBA Fermionic System Boundary Condition in Bethe Ansatz Heisenberg Spin Chain Spin of the Bethe Ansatz State Other Integrable Models YANG-BAXTER EQUATION Introduction General Description Factorized Scattering Baxter's Star Triangle Relation Vertex Models Reflection Equation Algebra CONTINUOUS INTEGRABLE SYSTEMS Introduction Quantum Continuous Integrable Systems Conserved Quantities Nonultralocal systems and the YBE Operator Product Expansion and YBE Finite Boundary Conditions Modified Classical Yang-Baxter Equation ALGEBRAIC BETHE ANSATZ Introduction Discrete Self Trapping Model Asymmetric XXZ Model in a Magnetic Field Analytical Bethe Ansatz Off-Shell Bethe Ansatz Nested Bethe Ansatz Fusion Procedure Fusion Procedure for Open chains Fusion Procedure for Transfer Matrices Application of Fusion Procedure INTEGRABLE LONG-RANGE MODELS Introduction Long-Range Models from the ABA Symmetry Transformation Calogero-Moser Models SEPARATION OF VARIABLES Introduction Hamilton-Jacobi Equation Sklyanin's Method for SoV Goryachev-Chaplygin Top Quantum Case and the Role of Lie Algebra Bi-Hamiltonian Structure and SoV SoV for GCM Model SoV and Boundary Conditions BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanin's Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund Transformations Method of Projection Operators QUANTUM GLM EQUATION Introduction Quantum GLM Equation Quantum Floquet Function Exact Quantization Quantum GLM Equation in a Continuous System Bound States and an Alternative Approach APPENDICES Direct Product Calculus Grassman Algebra Bethe Ansatz Equation AKNS Problem BIBLIOGRAPHY INDEX
Integrable open quantum spin-chain transfer matrices constructed from trigonometric R-matrices associated to affine Lie algebras g ˆ , and from certain K-matrices (reflection matrices) depending on a discrete parameter p , were recently considered in arXiv:1802.04864 and arXiv:1805.10144 . It was shown there that these transfer matrices have quantum group symmetry corresponding to removing the p th node from the g ˆ Dynkin diagram. Here we determine the spectrum of these transfer matrices by using analytical Bethe ansatz, and we determine the dependence of the corresponding Bethe equations on p . We propose formulas for the Dynkin labels of the Bethe states in terms of the numbers of Bethe roots of each type. We also briefly study how duality transformations are implemented on the Bethe ansatz solutions.
Using anisotropic R-matrices associated with affine Lie algebras gˆ (specifically, A2n(2), A2n−1(2), Bn(1), Cn(1), Dn(1)) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of gˆ. We show that these transfer matrices also have a duality symmetry (for the cases Cn(1) and Dn(1)) and additional Z2 symmetries that map complex representations to their conjugates (for the cases A2n−1(2), Bn(1) and Dn(1)). A key simplification is achieved by working in a certain “unitary” gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.
This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational GL(n)GL(n)GL(n)-invariant RRR-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.
The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.