The Interplay of Unitary and Permutation Symmetries in Composite Quantum Systems

• Main Author: Jakab, Dávid
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2022
• Subjects: Atomic physics; Electrons; Energy; High temperature; High Temperature Physics; Hilbert space; Lie groups; Low Temperature Physics; Magnetism; Mathematics; Phase transitions; Superconductivity; Symmetry; Thermodynamics
• ISBN: 9798384333029

We study three different topics linked together by an interplay of SU(d) and permutation symmetries. First, we study the spin-1 bilinear-biquadratic model on the complete graph of Nsites. Due to the complete permutation invariance, this Hamiltonian can be reexpressed as a linear combination of SU(2) and SU(3) quadratic Casimir operators. Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.In the second topic, we slightly relax the permutation symmetry, and study a bipartite collective spin-1 model with an exchange interaction that has different strength between and within the two subsystems. Such a setup is inspired by recent experiments with ultracold atoms. Using the SU(3) symmetry of the exchange interaction and the permutation symmetry within the subsystems, we can employ representation theoretic methods to diagonalize the Hamiltonian of the system in the entire parameter space of the two coupling strengths. These techniques then allow us to explicitly construct and explore the ground-state phase diagram.The third topic breaks with the investigation of spin models. Instead, we solve a version of the quantum marginal problem that is characterized by the same bipartite permutation symmetry as the previous spin model. This is the shareability, a.k.a. symmetric extendability problem. The question posed is, when can a given bipartite quantum state consistently arise as the reduced state of a larger composite system? The composite system is split into two parts, and we are only interested in the bipartite reduced states that overlap with both. We restrict the problem to Werner and isotropic states, the unitary symmetry of which allow us to use the same representation theoretic tools that we use during the study of our bipartite spin model. For both classes of states, we present necessary and sufficient conditions for shareability.