Topological sampling through windings

We propose a modification of the Hybrid Monte Carlo (HMC) algorithm that overcomes topological freezing two-dimensional $U(1)$ gauge theory with and without fermion content. This includes reversible jumps between sectors$-$winding steps$-$combined standard HMC steps. The full is referred to as winding (wHMC), it shows an improved behaviour autocorrelation time towards continuum limit. find excellent agreement wHMC estimates plaquette susceptibility analytical predictions in pure theory, which are known even at finite $\beta$. also study expectation values fixed sectors using both wHMC, fermions. Even when topology frozen HMC$-$leading significant deviations well non-topological quantities$-$the two algorithms agree on fixed-topology averages. Finally, we briefly compare results those obtained master-field simulations size $L\sim 8 \times 10^3$.