Abstract
These notes are intended as a detailed discussion on how to implement the diagrammatic Monte Carlo method for a physical system which is technically simple and where it works extremely well, namely the Fröhlich polaron problem. Sampling schemes for the Green function as well as the self-energy in the bare and skeleton (bold) expansion are disclosed in full detail. We discuss the Monte Carlo updates, possible implementations in terms of common data structures, as well as techniques on how to perform the Fourier transforms for functions with discontinuities. Control over the variety of parameters, especially in the bold scheme, is demonstrated. Sample codes are made available online along with extensive documentation. Towards the end, we discuss various extensions of the method and their applications. After working through these notes, the reader will be well equipped to explore the richness of the diagrammatic Monte Carlo method for quantum many-body systems.
Highlights
When physicists use the term diagrammatic Monte Carlo (DiagMC) in the sense of the expression “sampling over all Feynman diagrams” it implies a number of differences compared to the previous paragraph: The thermodynamic limit is taken from the start, the partition function is usually not used for normalization (instead, the lowest order diagram is often chosen), nor does the sampling necessarily take place in the space of the partition function diagrams: The method relies on the cancellation of disconnected diagrams when computing correlation functions as can be found in standard textbooks [4,5,6,7,8]
We spend about 4% of the time in the zeroth order diagram, and close to 40% of the time in fourth order, the code has occasionally gone to 16th order
One is typically interested in the transport of quantum dot like systems coupled to external leads, and attempts to monitor the time evolution for a long enough period of time such that a steady state sets in. The purpose of these notes is to provide a pedagogical overview of the technical aspects of diagrammatic Monte Carlo simulations, lowering the barrier for newcomers, and giving a flavor of its power to experienced researchers acquainted with other numerical techniques
Summary
Our goal is to show the similarities between these systems from the algorithmic point of view rather than a full discussion of the physics of these models, which is beyond the scope of these lecture notes
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