Consistent regularization and renormalization in models with inhomogeneous phases

Abstract

In many models in condensed matter and high-energy physics, one finds inhomogeneous phases at high density and low temperature. These phases are characterized by a spatially dependent condensate or order parameter. A proper calculation requires that one takes the vacuum fluctuations of the model into account. These fluctuations are ultraviolet divergent and must be regularized. We discuss different ways of consistently regularizing and renormalizing quantum fluctuations, focusing on momentum cutoff, symmetric energy cutoff, and dimensional regularization. We apply these techniques calculating the vacuum energy in the Nambu-Jona-Lasinio model in $1+1$ dimensions in the large-${N}_{c}$ limit and in the $3+1$ dimensional quark-meson model in the mean-field approximation both for a one-dimensional chiral-density wave.