Phase-Space Approach to Wannier Pairing and Bogoliubov Orbitals in Square-Octagon Lattices

Abstract: Low-energy lattice models are the cornerstone for studying many-body physics and interactions between the system and measurement fields. A key challenge is identifying appropriate quasiparticle states that canonically transform between momentum and real space while retaining the correlation, entanglement, and geometric properties - generally called the Wannier obstruction. Here, we introduce a phase-space approach to bypass these obstructions. Instead of treating the phase space as a manifold, we embed a real space through a Bloch vector space at each momentum. Orbital and spin states are introduced through product states with the Bloch vector, while quantum statistics, correlations, topology, and entanglements are inherited from the Hamiltonian. We apply this framework to explore the unconventional pairing symmetry and the Bogoliubov-de Gennes (BdG) equation in phase space. Our findings demonstrate that while superconductivity exhibits global coherence, the local Wannier orbital symmetry primarily determines the pairing symmetry. We analytically solve the spin-fluctuation-mediated pairing symmetry on the phase space by engineering a flat band with artificial gauge fields. We validate the model on the square-octagon superconductor Lu$2$Fe$_3$Si$_5$ using density functional theory (DFT), revealing the coexistence of nodeless $s^{\pm}$ and nodal $s{z^2}$ pairing symmetries. This phase-space framework provides a robust, obstruction-free lattice model for complex many-body systems and their exotic excitations.