On the growing length scale in a replica-coupled glassforming liquid

Computer simulations are used to study a three-dimensional polydisperse model glassformer in a replica-coupling setup where an attractive field $\propto - \varepsilon Q$ of strength $\varepsilon$ can adjust the similarity of the system to a fixed reference configuration with the overlap parameter $Q$. The polydispersity in the model enables the efficient use of swap Monte Carlo in combination with molecular-dynamics simulation from which we obtain fully equilibrated liquid configurations at very low temperature, i.e., far below the critical temperature of mode-coupling theory, $T_{\rm MCT}$. When the $\varepsilon$-field is switched on, the fast dynamics with swaps allow relaxation to the stationary state at temperatures below $T_{\rm MCT}$. In the stationary state, the overlap $Q$ has a finite value that increases with increasing $\varepsilon$. For a given temperature $T$, fluctuations of the overlap around the average value become maximal at a critical field strength $\varepsilon^\star(T)$. With decreasing $T$ along this $\varepsilon^\star(T)$-line, overlap fluctuations increase and a transition from a unimodal overlap distribution to a bimodal shape occurs. We give evidence that these bimodal distributions are not due to first-order phase transitions. However, they reflect finite-size effects due to a rapidly growing length scale with decreasing temperature. We discuss the significance of this length scale for the understanding of the glass transition.