Residual entropy from temperature incremental Monte Carlo method

Abstract: Residual entropy, which reflects the degrees of freedom in a system at absolute zero temperature, is crucial for understanding quantum and classical ground states. Despite its key role in explaining low-temperature phenomena and ground state degeneracy, accurately measuring residual entropy remains a difficult task owing to computational limitations. In this Letter, we introduce the temperature incremental Monte Carlo (TIMC) method, our approach to overcoming these challenges. The TIMC method incrementally calculates the partition function ratio of neighboring temperatures within Monte Carlo simulations, enabling precise entropy calculations and revealing other temperature-dependent properties in a single computational sweep of temperatures. We have rogorously tested TIMC on several complex systems, including the frustrated antiferromagnetic Ising model on both C60 and 2D triangular lattices, the Newman-Moore glassy model, and a 2D quantum transverse field Ising model. Notably, our method overcomes the difficulties encountered in partition function measurements when mapping $d$-dimensional quantum models to $d+1$-dimensional classical counterparts. These challenges arise from singular interactions that emerge in the small $\Delta_\tau$ limit during the quantum-to-classical mapping procedure. The TIMC method enables precise entropy calculations across the entire temperature range, as demonstrated in our studies of frustrated spin models, glassy phases, and phases exhibiting spontaneous symmetry breaking. This method's capability to calculate residual entropy could provide insights when applied to systems lacking analytical solutions.