For finite baryon chemical potential, conventional lattice descriptions of quantum chromodynamics (QCD) have a sign problem which prevents straightforward simulations based on importance sampling. In this thesis we investigate heavy dense QCD by representing lattice QCD with Wilson fermions at finite temperature and density in terms of Polyakov loops. We discuss the derivation of $3$-dimensional effective Polyakov loop theories from lattice QCD based on a combined strong coupling and hopping parameter expansion, which is valid for heavy quarks. The finite density sign problem is milder in these theories and they are also amenable to analytic evaluations. The analytic evaluation of Polyakov loop theories via series expansion techniques is illustrated by using them to evaluate the $\SU{3}$ spin model. We compute the free energy density to $14$th order in the nearest neighbor coupling and find that predictions for the equation of state agree with simulations to $\mathcal{O}(1\%)$ in the phase were the (approximate) $Z(3)$ center symmetry is intact. The critical end point is also determined but with less accuracy and our results agree with numerical results to $\mathcal{O}(10\%)$. While the accuracy for the endpoint is limited for the current length of the series, analytic tools provide valuable insight and are more flexible. Furthermore they can be generalized to Polyakov-loop-theories with $n$-point interactions. We also take a detailed look at the hopping expansion for the derivation of the effective theory. The exponentiation of the action is discussed by using a polymer expansion and we also explain how to obtain logarithmic resummations for all contributions, which will be achieved by employing the finite cluster method know from condensed matter physics. The finite cluster method can also be used to evaluate the effective theory and comparisons of the evaluation of the effective action and a direction evaluation of the partition function are made. We observe that terms in the evaluation of the effective theory correspond to partial contractions in the application of Wick's theorem for the evaluation of Grassmann-valued integrals. Potential problems arising from this fact are explored. Next to next to leading order results from the hopping expansion are used to analyze and compare the onset transition both for baryon and isospin chemical potential. Lattice QCD with an isospin chemical potential does not have a sign problem and can serve as a valuable cross-check. Since we are restricted by the relatively short length of our series, we content ourselves with observing some qualitative phenomenological properties arising in the effective theory which are relevant for the onset transition. Finally, we generalize our results to arbitrary number of colors $N_c$. We investigate the transition from a hadron gas to baryon condensation and find that for any finite lattice spacing the transition becomes stronger when $N_c$ is increased and to be first order in the limit of infinite $N_c$. Beyond the onset, the pressure is shown to scale as $p \sim N_c$ through all available orders in the hopping expansion, which is characteristic for a phase termed quarkyonic matter in the literature. Some care has to be taken when approaching the continuum, as we find that the continuum limit has to be taken before the large $N_c$ limit. Although we currently are unable to take the limits in this order, our results are stable in the controlled range of lattice spacings when the limits are approached in this order.
A study of the sign problem for lattice QCD with chemical potential
