# SciPost Thesis Link

Title: | w2dynamics: continuous time quantum Monte Carlo calculations of one- and two-particle propagators | |

Author: | Markus Wallerberger | |

As Contributor: | Markus Wallerberger | |

Type: | Ph.D. | |

Discipline: | Physics | |

Subject area: | Condensed Matter Physics - Computational | |

Approaches: | Theoretical, Computational | |

URL: | http://repositum.tuwien.ac.at/urn:nbn:at:at-ubtuw:1-3537 | |

Degree granting institution: | TU Wien - Vienna University of Technology | |

Supervisor(s): | Karsten Held | |

Defense date: | 2016-06-01 |

### Abstract:

The single-impurity Anderson model (SIAM), comprised of a single impurity with a local Hubbard interaction immersed in a bath of non-interacting electrons, is a fundamental model of electronic correlation. It has been used to study the Kondo effect and transport through quantum junctions. The SIAM in its multi-orbital generalisation also lies at the computational core of dynamical mean field theory (DMFT) and its combination with density functional theory (DFT). DMFT improves on the rough treatment of correlations in DFT by including all local correlations in a non-perturbative manner, corresponding to the assumption of a local self-energy, and was successfully used to study systems with partially filled 3d and 4f shells (Georges et al., 1996). While for the study of the SIAM and also in DMFT one typically focuses on the one-particle impurity propagators, recently the local two-particle irreducible vertex Gamma has come into focus. From a physical point of view, Gamma enters as vertex corrections into the charge and magnetic susceptibilities. Moreover, it was recently discovered that a set of divergencies of Gamma surround the Mott transition, marking a bifurcation of the self-energy that proves problematic for methods relying on an expansion in Gamma (SchÃ¤fer et al., 2013). Finally, the impurity vertex is the central ingredient for post-DMFT methods like the dynamical vertex approximation (DGA). DGA augments DMFT with a diagrammatic treatment of non-local correlations on all length scales, which in turn are dominant for low-dimensional systems or systems close to a second-order phase transition (Toschi et al., 2007). A state-of-the-art method for solving the SIAM at finite temperatures is continuous time quantum Monte Carlo in the hybridisation expansion (CT-HYB). CT-HYB expands the partition function Z with respect to the hybridisation with the bath and stochastically sums up the resulting series of strong-coupling Feynman diagrams. CT-HYB can treat continuous baths, multiple orbitals, different types of local interaction and is free of systematic bias and thus numerically exact (Gull et al., 2011). The many-body propagators are usually obtained as a 'by-product' of partition function sampling, as this allows for an easy implementation. The multi-orbital vertex however is a large object, which is challenge for CT-HYB from a computational and memory point of view. I will show how by using decompositions and non-equidistant fast Fourier transforms, one can overcome the problem. I will also analyse the symmetries, conserved quantities and asymptotics of the vertex. Furthermore, I will show that the estimator for Gamma has severe ergodicity problems for strong insulators and fails to yield spin-flip and pair-hopping terms of the vertex in high-symmetry cases. Worm sampling avoids above complications by directly sampling the many-body propagators. I will show that its use in CT-HYB significantly improves the quality and statistical uncertainties of the propagators. I will also demonstrate how by using worm sampling for the impurity vertex, one can calculate frequency boxes of arbitrary sizes. I will then focus on the application of DMFT to models and real systems. I will study the oxide heterostructure formed by a thin layer of SrVO3 grown on a SrTiO3 substrate. I will show that local correlation is responsible for pushing the system close to a metal-insulator transition by enhancing the crystal field splitting. Thus, a small perturbation of a system by an electric field or a compressive strain may be used to form a 'Mott transistor'. I will show that this effect is stable with respect to choosing a d-only or dp basis. CT-HYB also allows us to study the divergency lines of Gamma at lower temperatures than other methods. I will show that at low temperatures, the divergencies start to bend away from the Mott transition and towards the non-interacting limit.