Structure of QC$_2$D ground state fields at nonzero matter densities
A quantitative investigation into the modification of ground-state field structures in two-color QCD (QC$_2$D) is presented at finite chemical potential. Using lattice simulations with Wilson gauge and fermion actions, we explore the chromo-electromagnetic field strengths under varying matter densities. To ensure accurate measurements, we develop and calibrate two highly improved topological charge operators and evaluate four gradient flow actions. Our results reveal a finite-volume crossover in the regime of the anticipated phase boundary at $\mu = m_\pi/2$, with both chromo-electric and chromo-magnetic field strengths suppressed before recovering and exceeding vacuum values at higher chemical potentials. We find the difference between the squared chromo-electric and chromo-magnetic field strengths, $E^2-B^2$, to increase in magnitude monotonically with increasing chemical potential. At $a\mu=0.7$, we find an $11\%$ suppression of $E^2$, a relatively small effect. A systematic analysis using sigmoid fits of lattice simulations in the crossover regime is performed to confirm the critical chemical potential obtained from the field structure is in agreement with the phase boundary at $m_\pi / 2$. These findings provide new insight into non-Abelian ground-state vacuum field structures and offer a foundation for future studies in real QCD.