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Register a new userPerfect Integrability and Gaudin Models
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Authors
Kang Lu
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We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite types, with periodic and regular quasi-periodic boundary conditions.
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This thesis presents an introduction to the class of Richardson-Gaudin integrable models, with special focus on the Bethe ansatz wave function, and investigates ways of applying the properties of Richardson-Gaudin models both in and out of integrability. A framework is outlined for the numerical and theoretical treatment of these systems, exposing a duality allowing the Bethe equations to be solved numerically. This is extended to the calculation of inner products and correlation functions. Using this framework, the influence of particle exchange on the Bethe ansatz is discussed, after which it is shown how the Bethe ansatz is able to accurately model wave functions of non-integrable models in two different settings. First, a variational approach is outlined for stationary models where integrability-breaking perturbations are explicitly introduced. Second, an alternative way of breaking integrability is through the introduction of dynamics and periodic driving, where it is shown how integrability can be used to model the resulting Floquet many-body resonances. Throughout this work, it is shown how the clear-cut structure and relatively large freedom in Richardson-Gaudin models makes them ideal for an investigation of the general principles of integrability, as well as being a perfect testing ground for the development of new quantum many-body techniques beyond integrability.
We give a sufficient condition for quantising integrable systems.
We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group $U_q(mathcal L(mathfrak{sl}_2))$. We give a complete set of the functional relations correcting inexactitudes of the previous considerations. A special attention is given to the connection of the representations used to construct the universal transfer operators and $Q$-operators.
In this contribution, we discuss three situations in which complete integrability of a three dimensional classical system and its quantum version can be achieved under some conditions. The former is a system with axial symmetry. In the second, we discuss a three dimensional system without spatial symmetry which admits separation of variables if we use ellipsoidal coordinates. In both cases, and as a condition for integrability, certain conditions arise in the integrals of motion. Finally, we study integrability in the three dimensional sphere and a particular case associated with the Kepler problem in $S^3$.
Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation consisting entirely of invariant submanifolds. A regular leaf, known as a flux surface,of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact, smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves including elliptic and hyperbolic magnetic axes. In this paper, the existence of exact, smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydrodynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields; that they are presymplectic systems.
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