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We carry out lattice simulations of two-color QCD and spectroscopy at finitedensity with two flavors of rooted-staggered quarks and a diquark source term.As in a previous four-flavor study, for small values of the inverse gaugecoupling we observe a Goldstone spectrum which reflects the symmetry-breakingpattern of a Gaussian symplectic chiral random-matrix ensemble (GSE) with Dysonindex $\beta_D=4$, which corresponds to any-color QCD with adjoint quarks inthe continuum instead of QC$_2$D wih fundamental quarks. We show that thisunphysical behavior occurs only inside of the bulk phase of $SU(2)$ gaugetheory, where the density of $Z_2$ monopoles is high. Using an improved gaugeaction and a somewhat larger inverse coupling to suppress these monopoles, wedemonstrate that the continuum Goldstone spectrum of two-color QCD,corresponding to a Gaussian orthogonal ensemble (GOE) with Dyson index$\beta_D=1$, is recovered also with rooted-staggered quarks once simulationsare performed away from the bulk phase. We further demonstrate how this changeof random-matrix ensemble is reflected in the distribution of eigenvalues ofthe Dirac operator. By computing the unfolded level spacings inside and outsideof the bulk phase, we demonstrate that, starting with the low-lying eigenmodeswhich determine the infrared physics, the distribution of eigenmodescontinuously changes from the GSE to the GOE one as monopoles are suppressed.

Continuum Goldstone spectrum of two-color QCD at finite density with staggered quarks