To reduce the rapidly growing computational cost of the dual-fermion lattice calculation with increasing system size, we introduce two embedding schemes. One is the real fermion embedding, and the other is the dual-fermion embedding. Our numerical tests show that the real fermion and dual-fermion embedding approaches converge to essentially the same result. The application on the Anderson disorder and Hubbard models shows that these embedding algorithms converge more quickly with system size as compared to the conventional dual-fermion method, for the calculation of both single- and two-particle quantities.
To reduce the rapidly growing computational cost of the dual-fermion lattice calculation with increasing system size, we introduce two embedding schemes. One is the real fermion embedding, and the other is the dual-fermion embedding. Our numerical tests show that the real fermion and dual-fermion embedding approaches converge to essentially the same result. The application on the Anderson disorder and Hubbard models shows that these embedding algorithms converge more quickly with system size as compared to the conventional dual-fermion method, for the calculation of both single- and two-particle quantities.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. \nIf electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, \ntreating both interactions and disorder on equal footing. \nThe approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. \nTerletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. \nWe extended the approach from Terletska et al. to treat disordered interacting systems. \n \nDual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field \ntheory. \nThe perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, \nbut experience shows that oftentimes good results are obtained for this region. \n \nAn advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. \nAdditionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. \nThus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. \nCluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. \n \nThe new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. \nFor both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. \nThe Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. \n \nFor the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are \nqualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. \nWe found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result \nshows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. \nThis behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. \nWe found this behavior for all values of the disorder that we considered.
We apply the recently developed dual-fermion algorithm for disordered interacting systems to the Anderson-Hubbard model. This algorithm is compared with dynamical cluster approximation calculations for a one-dimensional system to establish the quality of the approximation in comparison with an established cluster method. We continue with a three-dimensional (3D) system and look at the antiferromagnetic, Mott, and Anderson localization transitions. The dual-fermion approach leads to quantitative as well as qualitative improvement of the dynamical mean-field results, and it allows one to calculate the hysteresis in the double occupancy in 3D, taking into account nonlocal correlations.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. If electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, treating both interactions and disorder on equal footing. The approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. Terletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. We extended the approach from Terletska et al. to treat disordered interacting systems. Dual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field theory. The perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, but experience shows that oftentimes good results are obtained for this region. An advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. Additionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. Thus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. Cluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. The new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. For both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. The Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. For the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are qualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. We found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result shows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. This behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. We found this behavior for all values of the disorder that we considered.
The dual fermion approach provides a formally exact prescription for calculating properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). Most practical implementations, however, neglect higher-order interaction vertices beyond two-particle scattering in the dual effective action and further truncate the diagrammatic expansion in the two-particle scattering vertex to a leading-order or ladder-type approximation. In this work, we compute the dual fermion expansion for the two-dimensional Hubbard model including all diagram topologies with two-particle interactions to high orders by means of a stochastic diagrammatic Monte Carlo algorithm. We benchmark the obtained self-energy against numerically exact diagrammatic determinant Monte Carlo simulations to systematically assess convergence of the dual fermion series and the validity of these approximations. We observe that, from high temperatures down to the vicinity of the DMFT N\'eel transition, the dual fermion series converges very quickly to the exact solution in the whole range of Hubbard interactions considered ($4\ensuremath{\le}U/t\ensuremath{\le}12$), implying that contributions from higher-order vertices are small. As the temperature is lowered further, we observe slower series convergence, convergence to incorrect solutions, and ultimately divergence. This happens in a regime where magnetic correlations become significant. We find, however, that the self-consistent particle-hole ladder approximation yields reasonable and often even highly accurate results in this regime.
We have designed a multiscale approach for strongly correlated systems by combining the dynamical cluster approximation (DCA) and the recently introduced dual fermion formalism. This approach employs an exact mapping from a real lattice to a DCA cluster of linear size Lc embedded in a dual fermion lattice. Short-length-scale physics is addressed by the DCA cluster calculation, while longer-length-scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as O(1/Lc), so perturbation theory on the dual lattice converges very quickly, e.g., the dual Fermion self-energy calculated with simple second-order perturbation theory is of order O(1/Lc3) with third-order and three-body corrections down by an additional factor of O(1/Lc).
We introduce a numerical algorithm to stochastically sample the dual fermion perturbation series around the dynamical mean field theory, generating all topologies of two-particle interaction vertices. We show results in the weak and strong coupling regime of the half-filled Hubbard model in two dimensions, illustrating that the method converges quickly where dynamical mean field theory is a good approximation, and show that corrections are large in the strong correlation regime at intermediate interaction. The fast convergence of dual corrections to dynamical mean field results illustrates the power of the approach and opens a practical avenue towards the systematic inclusion of nonlocal correlations in correlated materials simulations. An analysis of the frequency scale shows that only low-frequency propagators contribute substantially to the diagrams, putting the inclusion of higher order vertices within reach.