Quantum Monte Carlo Studies of Strongly Correlated Electron Systems

Electronic correlations are at the heart of modern solid state physics. The interest lies in emergent collective phenomena which appear at low energy scales and which often originate from competing interactions. In this article, we summarize three research subjects where the effects of correlations dominate and can be elucidated with the combined use of supercomputers and state-of-the-art stochastic algorithms. i) We show that the semimetallic state of the two-dimensional honeycomb lattice with a point-like Fermi surface is unstable towards a canted antiferromagnetic insulator upon application of an in-plane magnetic field. The magnetic field shifts the up- and the down-spin cones in opposite directions thereby generating a finite density of states at the Fermi surface which triggers a nesting instability leading to antiferromagnetic insulating state. Our conclusions, based on mean-field arguments, are confirmed by large scale auxiliary field projective quantum Monte Carlo methods. ii) Unbiased weak-coupling continuous time quantum Monte Carlo (CTQMC) is used to study the transition between the singlet and doublet (local moment) states of a single magnetic impurity coupled to s-wave superconducting leads, focusing on the Josephson current with 0 to π phase shift and the crossing of the Andreev bound states in the single particle spectral function. Extended to dynamical mean-field theory (DMFT), this impurity problem provides a link to the periodic Anderson model with superconducting conduction electrons (BCS-PAM). We compute the spectral functions which signal the transition from a coherent superposition of Andreev bound states to incoherent quasiparticle excitations. iii) Dynamical quantum-cluster approaches, such as different cluster extensions of the DMFT (cluster DMFT) or the variational cluster approximation (VCA), combined with efficient cluster solvers, such as CTQMC provide controlled approximations of the single-particle Green’s function for lattice models of strongly correlated electrons. To access the thermodynamics, however, a thermodynamical potential is needed. We compute the numerically exact cluster grand potential within VCA using CTQMC in combination with a quantum Wang-Landau technique to reweight the coefficients in the expansion of the partition function of the two-dimensional Hubbard model at finite temperatures.