One of the most intriguing phenomena in many-particle physics is the appearance of complexity and collectivity. In collective systems it's not just the interaction between single particles, but the synergy of all interactions between the different particles that determines the complexity of the system. The whole system is greater than the sum of its parts, or in Anderson's words: “More is different.” One of the most well-known examples of this is superconductivity, caused by the electromagnetic interaction between the electrons and the atoms in the crystal.

Many-particle physics is tasked with understanding the appearance and evolution of these collective phenomena for systems with a large system size. However, we are confronted by the fact that the Hilbert space of the fully-interacting quantum system increases exponentially with increasing system size. Powerful approximation techniques have to be developed that accurately model the correlation in order to obtain a feasible description of these systems . However, there exists a special class of so-called integrable systems where the eigenstates can be written as a Bethe Ansatz wavefunction due to a special underlying symmetry. For these systems, the wavefunction is factorisable and the elementary excitations depend on a set of so-called Richardson-Gaudin variables, which can be obtained by solving a set of non-linear algebraic 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 algebraic formulation of integrable systems these appear in many branches of physics. Some examples are

• Superconductivity in crystals and nanoparticles,
• Magnetism in spin chains,
• Atoms in a cavity interacting with an electromagnetic field,
• Proton-neutron isovector and isoscalar interactions in nuclei,
• Atomic gases interacting through a p-wave interaction.

Recent numerical developments in solving the non-linear equations allow for large-scale theoretical studies of these integrable systems. Some of the possible projects for thesis students are

• Searching for new integrable systems using the Algebraic Bethe Ansatz method,
• Determining the exact quantum phase diagram of a specific integrable system of interest,
• Studying quantum dynamics and evolution of systems out-of-equilibrium.

Goal Moving away from exactly-solvable systems, the properties of these systems can also be used to model non-integrable systems. The Bethe Ansatz is intricately connected to the phenomenon of fermion-pairing and can be generalized to describe pairing in more general systems such as molecules. Although the exact-solvability is lost for non-integrable systems, the Ansatz still has many beneficial properties that can be used to optimally capture the quantum correlations in many systems. This is an active area of research in our group, in close collaboration with the quantum chemistry group McMaster University (Canada) and the quantum chemistry group in Ghent. This topic offers many possibilities for thesis students, such as

• capturing pairing structures in mesoscopic structures, tightly connected to the Lewis picture of electronic pairing in molecules,
• a theoretical search for novel Bethe Ansatze for an accurate description of exotic pairing,
• combining a pairing Ansatz with known quantum-many body techniques, such as those developed in our group.

These thesis subjects contain a combination of mathematics, physics and computational work, with a different focus depending on the interests of the student.