Large-$N$ $SU(N)$ Yang-Mills theories with milder topological freezing

We simulate $4d$ $SU(N)$ pure-gauge theories at large $N$ using a parallel tempering scheme that combines simulations with open and periodic boundary conditions, implementing the algorithm originally proposed by Martin Hasenbusch for $2d$ $C{P}^{N-1}$ models. That allows to dramatically suppress the topological freezing suffered from standard local algorithms, reducing the autocorrelation time of $Q^2$ up to two orders of magnitude. Using this algorithm in combination with simulations at non-zero imaginary $\theta$ we are able to refine state-of-the-art results for the large-$N$ behavior of the quartic coefficient of the $\theta$-dependence of the vacuum energy $b_2$, reaching an accuracy comparable with that of the large-$N$ limit of the topological susceptibility.