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Rimu.jl Package Guide

Random Integrators for many-body quantum systems

The grand aim is to develop a toolbox for many-body quantum systems that can be represented by a Hamiltonian in second quantisation language. Currently supported features include:

Interacting with quantum many-body models

  • Full configuration interaction quantum Monte Carlo (FCIQMC), a flavour of projector quantum Monte Carlo for stochastically solving the time-independent Schrödinger equation. See References.
  • Matrix-free exact diagonalisation of quantum Hamiltonians (with external package KrylovKit.jl).
  • Sparse matrix representation of quantum Hamiltonians for exact diagonalisation with sparse linear algebra package of your choice (fastest for small systems).

Representing quantum many-body models

  • A composable and efficient type system for representing single- and multi-component Fock states of bosons, fermions, and mixtures thereof, to be used as a basis for representing Hamiltonians.
  • An interface for defining many-body Hamiltonians.
  • Pre-defined models include:
    • Hubbard model in real space for bosons and fermions and mixtures in 1, 2, and 3 spatial dimensions.
    • Hubbard and related lattice models in momentum space for bosons and fermions in one spatial dimension.
    • Transcorrelated Hamiltonian for contact interactions in one dimension for fermions, as described in Jeszenski et al. arXiv:1806.11268.

Statistical analysis of Monte Carlo data

  • Blocking analysis following Flyvberg & Peterson JCP (1989), and automated with hypothesis testing by Jonsson

PRE (2018).

  • Unbiased estimators for the ground state energy by re-reweighting following Nightingale & Blöte PRB (1986) and Umrigar et al. JCP (1993).

The code supports parallelisation with MPI (harnessing MPI.jl) as well as native Julia threading (experimental). In the future, we may add tools to solve the time-dependent Schrödinger equation and Master equations for open system time evolution.

Concept: Joachim Brand and Elke Pahl.

Contributors: Joachim Brand, Elke Pahl, Mingrui Yang, Matija Čufar, Chris Bradly.

Discussions, help, and additional contributions are acknowledged by Ali Alavi, Didier Adrien, Chris Scott (NeSI), Alexander Pletzer (NeSI).

Installation

Installing Rimu for usage

Rimu is a registered package and can be installed with the package manager. Hit the ] key at the Julia REPL to get into Pkg mode and type

pkg> add Rimu

Alternatively, use

julia> using Pkg; Pkg.add(name="Rimu")

in order to install Rimu from a script.

Installing Rimu for development

In order to be able to edit the source code, push changes, change and make new git branches, etc., clone the git repository with git clone to a convenient location, e.g. ~/mygitpackagefolder/. Then hit the ] key at the Julia REPL to get into Pkg mode and type

pkg> develop ~/mygitpackagefolder/rimu.jl

where the file path has to be adjusted to the location of the cloned git repository.

Usage

The package is now installed and can be imported with

julia> using Rimu

When planning to edit the code of the package it is advisable to use the Revise package by issuing

julia> using Revise

before using Rimu. This will track any changes made to the source code of Rimu and the changed package will be available after saving the source code (hopefully, and most of the time, without restarting the Julia REPL).

Rimu offers a number of tools for representing Hamiltonians (see Hamiltonians) and state vectors / wave functions (see DictVectors) as well as algorithms to find the ground state, e.g. lomc!.

Scripts

Rimu is written as a Julia package to be imported with using Rimu as described above. It supplies useful functions and types. Performing actual calculations and analysing the results is done with scripts. The folder scripts/ contains a collections of scripts that are either examples for use of the Rimu package or useful scripts for data analysis. In particular:

  • scripts/BHM-example.jl is an example script that runs fciqmc on the 1D Bose-Hubbard model. A data frame with results is written to the file fciqmcdata.arrow.
  • scripts/BHM-example-mpi.jl demonstrates basic usage of Rimu with MPI.

MPI

The Rimu package can run in parallel on different processes or node and distribute work by making use of MPI, or "message passing interface". For example, running

> julia scripts/BHM-example.jl

will run on one processor with the main lomc!() computation (i.e. after package loading and compilation) completing in 2.69 seconds.

Running

> mpirun -np 4 julia scripts/BHM-example-mpi.jl

on the same hardware makes use of 4 cores and the main part completes in 1.04 seconds, a speedup factor of 2.6. This seems reasonable, given that extra work needs to be done for communicating between different processes.

Using MPI parallelism with Rimu is easy. Enabling MPI for use in lomc!() is enabled automatically if PDVec is used to store a vector. In that case, data will be stored in a distributed fashion among the MPI ranks and only communicated between ranks when necessary. Additional MPI-related functionality is provided by the module RMPI.

References

The code implements the FCIQMC algorithm originally described in

  • "Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space", G. H. Booth, A. J. W. Thom, A. Alavi, J. Chem. Phys. 131, 054106 (2009).
  • "Communications: Survival of the fittest: accelerating convergence in full configuration-interaction quantum Monte Carlo.", D. Cleland, G. H. Booth, A. Alavi, J. Chem. Phys. 132, 041103 (2010).

Scientific papers describing additional features implemented in Rimu:

  • "Improved walker population control for full configuration interaction quantum Monte Carlo", M. Yang, E. Pahl, J. Brand, J. Chem. Phys. 153, 170143 (2020); arXiv:2008.01927.
  • "Stochastic differential equation approach to understanding the population control bias in full configuration interaction quantum Monte Carlo", J. Brand, M. Yang, E. Pahl, arXiv:2103.07800 (2021).

Papers discussing results obtained with Rimu:

  • "Polaron-Depleton Transition in the Yrast Excitations of a One-Dimensional Bose Gas with a Mobile Impurity", M. Yang, M. Čufar, E. Pahl, J. Brand, Condens. Matter 7, 15 (2022).
  • "Magnetic impurity in a one-dimensional few-fermion system", L. Rammelmüller, D. Huber, M. Čufar, J. Brand, A. Volosniev, arXiv:2204.01606 (2022).