The Truncated Polynomial Expansion Monte Carlo Algorithm for Spin-fermion Models: Application to Diluted Magnetic Semiconductors and Manganites

A system of fermions coupled to classical fields is common to a wide range of strongly correlated electron problems where the fermionic operators appear in the Hamiltonian involving only quadratic terms. A conventional approach to solve these kinds of models is by diagonalizing the fermions in the one-electron sector at finite temperature for a given configuration of classical fields. However, this results in a high computational cost as the computational complexity grows with the 4-th power of the size of the system. The Truncated Polynomial Expansion Monte Carlo Algorithm (TPEM), developed by N. Furukawa and Y. Motome (J. Phys. Soc. Jpn. \textbf{73}, (2004) 1482), replaces the exact diagonalization of the one-electron sector in these models and has a complexity that is linear with the size of the system. In this talk, I will discuss the performance and reliability of the method as well as the parallelization of the algorithm. I will also show novel applications of the TPEM to disordered systems in the context of diluted magnetic semiconductors and to finite Hund coupling models for manganites (G. Alvarez \textit{et al.}, submitted to Computer Physics Communications). I will discuss how the TPEM can drastically improve the study of those systems.