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In this lecture we discuss various properties of the phase factor of the fermion determinant for QCD at nonzero chemical potential. Its effect on physical observables is elucidated by comparing the phase diagram of QCD and phase quenched QCD and by illustrating the failure of the Banks-Casher formula with the example of one-dimensional QCD. The average phase factor and the distribution of the phase are calculated to one-loop order in chiral perturbation theory. In quantitative agreement with lattice QCD results, we find that the distribution is Gaussian with a width $sim mu T sqrt V$ (for $m_pi ll T ll Lambda_{rm QCD}$). Finally, we introduce, so-called teflon plated observables which can be calculated accurately by Monte Carlo even though the sign problem is severe.
We investigate the properties of QCD at finite isospin chemical potential at zero and non-zero temperatures. This theory is not affected by the sign problem and can be simulated using Monte-Carlo techniques. With increasing isospin chemical potential
We present results for the phase diagram of QCD with two massless quark flavours as obtained from a first-principles functional renormalisation group approach. In particular we compute order parameters for chiral symmetry breaking and quark confineme
We provide the most accurate results for the QCD transition line so far. We optimize the definition of the crossover temperature $T_c$, allowing for its very precise determination, and extrapolate from imaginary chemical potential up to real $mu_B ap
We study genuine finite density effects in QCD-like theories with three-dimensional Polyakov-loop effective theories for heavy quarks. These are derived from the full QCD-like theories by combined strong-coupling and hopping expansions. In particular
We investigate the real and imaginary chemical-potential dependence of pion and $rho$-meson screening masses in both the confinement and the deconfinement region by using two-flavor lattice QCD. The spatial meson correlators are calculated in the ima