Structure of QC$_2$D ground state fields at nonzero matter densities

Abstract

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 $μ= m_π/2$, 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 $aμ=0.7$, 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 $m_π/ 2$. These findings provide new insight into non-Abelian ground-state vacuum field structures and offer a foundation for future studies in real QCD.

Figures (14)

• Figure 1: Comparison of the action density (left column) and topological charge density (right column) illustrated for chemical potentials of $a \mu = 0$ (top), $a \mu = 0.4$ (middle) and $a \mu = 0.7$ (bottom). For the action density in the left column, red/yellow/green/blue shading indicates high to low local action density. Similarly, for the topological charge density in the right column, violet/blue indicates negative topological charge density and red/yellow indicates positive topological charge density, with violet and red illustrating the largest magnitudes. The lowest action and topological-charge density regions below their respective thresholds are not rendered as described in the text.
• Figure 2: Comparison of squared chromo-electric field, $E^2(x)$ (left column), the squared chromo-magnetic field, $B^2(x)$ (middle column), and the difference, $E^2(x) - B^2(x)$ (right column) for chemical potentials of $a \mu = 0$ (top row), $a \mu = 0.4$ (middle row) and $a \mu = 0.7$ (bottom row). Color shadings for the positive field strengths, $E^2(x)$ and $B^2(x)$, are as for the action density of Fig. \ref{['fig:actionCharge']}. The difference $E^2(x) - B^2(x)$ has both positive and negative values and is rendered in the same manner used for the topological charge in Fig. \ref{['fig:actionCharge']}. Thus, violet/blue indicates a negative value (i.e. $B^2(x) > E^2(x)$) and red/yellow indicates a positive value (i.e. $E^2(x) > B^2(x)$). An abundance of violet/blue in the bottom right plots reveals an impact of the chemical potential on the balance of $E^2(x)$ and $B^2(x)$ field strengths.
• Figure 3: Representative plots of one configuration with zero chemical potential illustrating the first derivative of ${\cal F}$ for the four gradient flow actions: Moran (top-left), Intermediate (bottom-left), Iwasaki (top-right) and DBW2 (bottom-right).
• Figure 4: Representative gradient flows for a coarser $16^3 \times 24$ lattice with $a\mu = 0.25$ and $a j = 0.2$. Both the $1^{st}$ derivative (black lines) and $2^{nd}$ derivative (blue lines) of ${\cal F}$ are illustrated for Moran (left) and Iwasaki (right) gradient flows.
• Figure 5: A comparison of the actions reconstructed from the improved field strength tensor for Moran and Iwasaki gradient flows after $n_s$ sweeps at $\rho = 0.005$. Using the same representative configuration illustrated in Fig. \ref{['fig:Lat24_smoC']}, the reconstructed action, $S_R$, is reported in terms of the single instanton action, $S_0$. The dashed lines highlight how the action remaining at 200 sweeps of Moran gradient flow cor responds to 154 sweeps of Iwasaki gradient flow.