We propose a new Monte Carlo algorithm for the free energy calculation based on configuration space sampling. Upward or downward temperature scan can be used to produce F (T ). We implement this algorithm for Ising model on square lattice and on triangular lattice. Comparison with the exact free energy shows an excellent agreement. We analyse the properties of this algorithm and compare it with Wang-Landau algorithm which samples in energy space. This method is applicable to general classical statistical models. The possibility of extending it to quantum systems is discussed.

We implement the Lanczos algorithm on natural orbital basis to solve the zero-temperature Green's function of Anderson impurity models, following the work of Y. Lu, M. Höppner, O.Gunnarsson, and M. W. Haverkort, Phys. Rev. B 90 (2014) 085102. We present the technical details, generalize the algorithm to the cases of particle-hole asymmetry, with local magnetic field, and of two impurities. The results are benchmarked with conventional Lanczos, quantum Monte Carlo, and numerical renormalization group methods, demonstrating its potential as a powerful impurity solver for the dynamical mean-field theory.Keywords: Lanczos, natural orbital, Anderson impurity model, quantum impurity solver IntroductionThe Anderson impurity model (AIM) [1] is one of the basic models in condensed matter physics. It describes the physics of a local electron orbital with on-site Coulomb repulsion embedded in a conduction electron band and is widely used to describe the dilute magnetic impurities in metals [2], Kondo effect [3], as well as impurity quantum phase transitions [4]. In the past two decades, stimulated by the development and application of the dynamical mean-field theory (DMFT) [5,6], the study of AIM receives revived attention because in DMFT, a lattice Hamiltonian for the correlated electrons is mapped into an AIM with self-consistently determined electron bath. The core calculation of DMFT is the iterative solution of the self-energy *