Lattice field theory is a non-perturbative method of solving the equations of gauge theories in particle physics. It is a well-established method of exploring the phase diagram of quantum chromodynamics (QCD), the theory of the strong interaction. Numerical studies of lattice QCD on the computer are capable of yielding exact, non-perturbative results. However, regions of the QCD phase diagram pose great challenges for lattice QCD because of the difficulties which arise for numerical calculations. We present numerical studies directed towards eventually simulating in more difficult regions of the phase diagram.

In the first study, we demonstrate the effectiveness of a generalization of the parallel tempering algorithm, originally developed for spin systems by Marinari and Parisi \cite{Marinari:1992qd}, in mitigating the issue of critical and supercritical slowing-down in lattice simulations in the vicinity of second- and first-order phase transitions respectively. We do so by performing large-scale simulations to characterize the phase transition of pure ${\rm SU}(3)$ Yang-Mills theory, or quenched QCD. We compare the autocorrelation times of parallel tempering simulations with those of brute force calculations. We compute the transition temperature to be $w_0T_c = 0.25384(25)$---which is the first per-mill accurate result in lattice QCD---and by a finite-volume scaling, we show that the transition is first-order.

In the second study, we look further into the phase transition in pure ${\rm SU}(3)$ by studying its topological features. We consider the behavior of the kurtosis of the topological charge across the deconfinement transition, which is a quantity useful in determining the onset of the dilute instanton gas picture in the deconfined phase.

In the final study, we investigate the renormalization of so-called minimally doubled fermions. The Karsten-Wilczek action is a implementation of minimally doubled fermions on the lattice. It explicitly breaks hypercubic symmetry and introduces three counterterms with respective bare parameters. We present a tuning of the bare parameters of the Karsten-Wilczek action on stored gauge configurations that were computed with the staggered fermion action at the physical point. We also study the magnitude of the taste-splitting of several fermion channels as a function of the lattice spacing.