Summary:
In this thesis we introduce a new method, the "Flat Histogram Diagrammatic
Monte Carlo" (FDHMC), which is the combination of the "Diagrammatic Monte
Carlo" (DMC) with Flat Histogram methods. By applying the FHDMC and other
versions of the DMC to a specific solvable problem, namely, the Froehlich
polaron problem for Feynman diagrams of specific form, we calculate the
imaginary-time Green function. A comparison of the results make clear that, the
FHDMC excels in terms of the quality of the results and the direct manner of
its application. We apply the FHDMC and other versions of the DMC to calculate
the imaginary-time Green function $G(\tau)$ in a solvable problem, which is the
$t-J$ model in the spin wave linear approximation with a limitation of the
calculations in the area of the Feynman diagrams that are determined by the non
crossing approximation (NCA). We invert the results of $G(\tau)$ with the
analytic continuation method "Stochastic Analytic Inference" (SAI) and we
determine the spectral function $A(\omega)$ for each case. By comparing the
results of this inversion with the exact solution for the spectral function, we
conclude the clear superiority of the FHDMC compared to other methods. Finally,
we apply the FHDMC in order to calculate the spectral function $A(\omega)$, the
dispersion relation and the spectral weight of the quasi particle, without
approximation for the movement of a hole for the $t-J$ model and the variation
of the $t-J_1-J_2$ in the spin wave linear approximation, in which the problem
of the fermion sign also appears. The superiority of the FHDMC in dealing with
the problem of the fermion sign is ascertained, and the accuracy of the NCA
approach is evaluated in any case.
Keywords:
Flat histogram diagrammatic Monte Carlo, Analytic continuation, Spectral function , T-J model, Quantum Monte Carlo