Flat histogram diagrammatic Monte Carlo method: Motion of a hole in a columnar antiferromagnet

We will present a version of the diagrammatic Monte Carlo (Diag-MC) method in which we incorporate the flat histogram principle and we term the improved version ``Flat Histogram Diagrammatic Monte Carlo'' method. We demonstrate the superiority of the method over the standard Diag-MC in extracting the long-imaginary-time behavior of the Green's function, without incorporating any {\it a priori} knowledge about this function, by applying the technique to the polaron problem. We have also applied the technique to the motion of a hole inside a $J_1$-$J_2$ quantum antiferromagnet with columnar order.