Influence of the exchange symmetry beyond the exclusion principle

Thesis

Since typically the physics of many body quantum systems is solely described and determined by pairwise interactions, the concept of reduced particle information is fundamentally relevant, particularly for indistinguishable particles. Determining the compatibility of r-particle reduced density operators (r-RDOs) with some N-particle state is known as the N-representability problem. The Pauli exclusion principle provides a set of necessary and sufficient conditions for the N-representability of fermionic 1-RDOs. However, pure N-representability requires a set of stricter conditions which are known as Generalised Pauli constraints (GPCs).

In this thesis we investigate the influence and significance of the GPCs beyond the well-explored and established relevance of the Pauli exclusion principle. In particular, we analytically study the (quasi-)pinning effect which corresponds to a saturation of the GPCs and imposes structural simplifications on the N-fermion state. So far, nothing is known about its origin and mechanisms. We will close this gap by analytically examining the dependence of quasipinning on the particle number, the dimensionality and the spin polarisation in a representative physical model system. Our results suggest the notion of a microscopic Pauli pressure which is a consequence of the competition between energy minimisation and antisymmetry, and drives the system towards the boundary of the allowed region. Furthermore, the relevance of quasipinning beyond the wellknown Pauli exclusion principle is evaluted by the Q-measure. The outcomes provide first direct evidence for the need to incorporate GPCs into the physical description.

To explore the role of the fermionic exchange symmetry beyond the Pauli exclusion principle from a more general perspective, we consider systems of hardcore bosons. Indeed, they exhibit a (lattice-based) exclusion principle, although their exchange symmetry is bosonic. Occupation numbers on oneparticle states other than the lattice basis can exceed one. We prove that the maximal occupation number is given by λ_max = N/d(d - N + 1) where N, d are the numbers of particles and lattice sites. In addition, we derive necessary conditions for the N-representability of hardcore boson 1-RDOs and investigate the one-particle entropy boundaries. Finally, we present one-dimensional physical systems which macroscopically saturate λ_max and exhibit (fractional) Bose-Einstein condensation.

Authors: Tennie, F

• DOI: 10.5287/ora-xox4jjkr1
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford