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Richardson-Gaudin models and broken integrability

Pieter Claeys (UGent)
(2018)
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Abstract
The fundamental object in quantum mechanics is the wave function, which can in principle be obtained as a direct solution to the Schrödinger equation. Unfortunately, exact solutions to this equation quickly prove to be impossible in physical systems consisting of many interacting particles. Integrable systems then present a special family of quantum many-body systems where the Schrödinger equation can be solved exactly using Bethe ansatz techniques. 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 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.

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Citation

Please use this url to cite or link to this publication:

Chicago
Claeys, Pieter. 2018. “Richardson-Gaudin Models and Broken Integrability”. Ghent, Belgium: Ghent University. Faculty of Sciences.
APA
Claeys, Pieter. (2018). Richardson-Gaudin models and broken integrability. Ghent University. Faculty of Sciences, Ghent, Belgium.
Vancouver
1.
Claeys P. Richardson-Gaudin models and broken integrability. [Ghent, Belgium]: Ghent University. Faculty of Sciences; 2018.
MLA
Claeys, Pieter. “Richardson-Gaudin Models and Broken Integrability.” 2018 : n. pag. Print.
@phdthesis{8564030,
  abstract     = {The fundamental object in quantum mechanics is the wave function, which can in principle be obtained as a direct solution to the Schr{\"o}dinger equation. Unfortunately, exact solutions to this equation quickly prove to be impossible in physical systems consisting of many interacting particles.
Integrable systems then present a special family of quantum many-body systems where the Schr{\"o}dinger equation can be solved exactly using Bethe ansatz techniques. 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 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.},
  author       = {Claeys, Pieter},
  language     = {eng},
  pages        = {IX, 160},
  publisher    = {Ghent University. Faculty of Sciences},
  school       = {Ghent University},
  title        = {Richardson-Gaudin models and broken integrability},
  year         = {2018},
}