Lecture notes on Diagrammatic Monte Carlo for the Fr\"ohlich polaron

TL;DR

This paper provides a comprehensive guide on implementing the diagrammatic Monte Carlo method for the Fr"ohlich polaron, including sampling schemes, updates, data structures, and extensions, with practical code examples.

Contribution

It offers detailed implementation instructions, sample codes, and discusses extensions of the diagrammatic Monte Carlo method for the Fr"ohlich polaron problem.

Findings

Effective sampling schemes for Green functions and self-energy

Demonstration of control over parameters in the bold scheme

Availability of sample codes and documentation

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 Fr\"ohlich 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.