Background and problem

One of the most intriguing phenomena in quantum many-particle physics is the appearance of strong quantum correlations and complexity.  In these systems it's not just the interaction between single particles, but the synergy of all interactions together that gives rise to many different phases of matter.  The whole system is greater than the sum of its parts, or in Nobel laureate P. W. Anderson's words: “More is different!”

Theoretical quantum many-particle physics is tasked with explaining, understanding, and even predicting the appearance and evolution of these complex phenomena.  For instance, it is known that many quantum systems undergo a quantum phase transition when the density (or number of particles) reaches a certain threshold.   However, we are confronted by the fact that the Hilbert space of the fully-interacting quantum system increases exponentially with increasing system size.  This means that the exact theoretical investigation of these interesting systems is simply impossible, and one must usually resort to approximation techniques.  Interestingly, there exists a special class of so-called integrable systems, where the system can be solved exactly despite the strong quantum correlations and exponential size-explosion of the Hilbert space.  In these systems, the wave function is given by a factorisable Bethe Ansatz and the correlations are encoded explicitly by means of a set of generalized momenta, or rapidities.  Apart from the nice physical interpretation, these rapidities can also be efficiently computed by solving a set of non-linear algebraic (Bethe) equations. The major importance of this result is that the diagonalization of a Hamiltonian matrix in an exponentially scaling Hilbert space has been reduced to solving a set of non-linear equations scaling linearly with system size.  This allows for an exact study of systems far out of the reach of conventional many-body techniques.

Due to the purely algebraic formulation of integrable systems, they appear in many branches of physics, such as

• Superconductivity in crystals and nanoparticles,

• Magnetism in spin chains,

• Atoms in a cavity interacting with an electromagnetic field,

• Pairing interactions in atomic nuclei,

• Hyperfine interactions in condensed matter systems.

Goal

Recently, there has also been interest in the application of concepts from integrability in non-integrable systems, using the Bethe Ansatz wave function as an approximate (variational) wave function in e.g. quantum chemistry. Here, the theoretical and numerical toolbox provided by integrability allows the optimization of  such a wave function to be performed in an efficient way. The factorizable form of the wave function can here be interpreted as a straightforward extension of the Slater determinant, providing guaranteed improvements on Hartree-Fock theory. This similarity then allows corrections to be added to the Bethe Ansatz in a systematic way, combining concepts from quantum chemistry and perturbation theory in order to obtain a more accurate description of non-integrable models. Due to the large variety of quantum many-body systems described by integrable models, such an approach is similarly expected to be applicable in various physical contexts.

The development and implementation of such methods is then the main goal of this thesis, combining mathematics, physics and computational work, with a different focus depending on the interests of the student.