Gleeson, Patrick2024-12-112024-12-11https://hdl.handle.net/1885/733729404Spin chains — systems governed by spin-spin interactions between neighbouring sites — arise in condensed-matter physics, nonequilibrium transport, and quantum information. Boundary fields at the edges of these chains have particularly important consequences for their applications in these areas. Accurate investigation of the associated phenomena often necessitates exact analytical solutions. This thesis considers one of the simplest interesting spin chains: the one-dimensional open spin-1/2 XXZ model. Known analytical descriptions for the eigenspectrum of this model are not provably complete in the presence of completely general boundary fields. This motivates the extension of the provably-complete Q-operator technique to this case. In particular, this thesis demonstrates that a non-standard construction of the Q-operator — as developed for the open asymmetric simple exclusion process (ASEP) — can be comprehensively interpreted within the standard Q-operator framework. It is shown how to formulate the non-standard construction in terms of the standard Lax operators, R-matrix, and double-row trace. This formulation is made concrete through analytical and numerical comparisons in the spin-1/2 XXZ case. The key feature of the non-standard construction is its use of parameter-dependent boundary vectors in place of the standard reflection equation. Based on the formulation above, it is conjectured that these two approaches are equivalent. In particular, this thesis presents numerical and analytical evidence that the non-standard ASEP construction encodes the first known solution of the arbitrary-weight reflection equation for the XXZ chain with non-trivial boundary fields. Such a solution is necessary for the standard Q-operator construction. This suggests the power of the parameter-dependent boundary-vector approach, independently of its conjectured equivalence to the standard reflection equation. This thesis may enable the boundary-vector-based Q-operator to be extended to a wider range of systems, thereby obtaining provably-complete descriptions of their eigenspectra.enQ-operatorspin chainboundary fieldsXXZ modelmatrix product ansatzboundary vectorsasymmetric simple exclusion processASEPLazarescu and Pasquier 2014quantum integrabilityreflection equation10.25911/TWTP-HS54