Path Integral Quantum Monte Carlo
Documentation

# Introduction

This webpage contains the details of a worm algorithm path integral quantum Monte Carlo (WA-PIMC) code actively developed in c++ since 2008 in the Del Maestro group based on:

It can be used to simulate indistinguishable bosons with various types of realistic interactions in one, two and three spatial dimensions. As written, it takes a large number of command line options and allows for the measurement of essentially any physical observable of interest.

The design philosophy included the goal of abstracting the actual implementation of the WA-PIMC method to a kernel that will never need to be touched by the end user. The code can be easily extended to study a wide variety of situations by including new types of containers, potentials estimators and communicators.

If you have questions, bug reports or plan to use this code for scientific research, please contact me at Adria.nosp@m.n.De.nosp@m.lMaes.nosp@m.tro@.nosp@m.utk.e.nosp@m.du.

The development and maintenance of this code base has been supported in part by the National Science Foundation under Award Nos. DMR-1553991 and DMR-1808440.

This program has been successfully compiled and run on both Intel and AMD systems using clang, g++, pathscale and icpc. Before installing, one needs to ensure that all dependencies are met. We recommend that the required libraries (boost and blitz) are installed in a local folder inside your home directory: $HOME/local. # Dependencies The code is written in c++ and makes use of both the blitz++ and boost libraries. You should be able to grab blitz from github and compile from source via the instructions below. We use many of the boost header-only libraries, but two libraries will need to be compiled: boost_program_options and boost_serialization libraries. Let us assume that you will be installing both blitz and boost in the folder $HOME/local using the GNU C++ compiler. For icpc or clang, the changes should be obvious, and in particular for the Intel compiler you will need to use intel-linux as the toolset while for clang you will use darwin.

If you don't have a $HOME/local you should create this directory now via mkdir$HOME/local

## Blitz

Unless you need to use the blitz++'s internal debug functionality initiated through #define BZ_DEBUG which is set by including debug=1 when compiling, blitz can be used as a 'header only' library and does not need to be compiled. This is the most common use case. However, as it doesn't take very long to compile one can proceed as follows:

1. Move into your source directory (create if necessary).
cd $HOME/local/src 2. Clone the latest version of blitz++ from github into $HOME/local/src
3. Move into the blitz source directory
4. Read the instructions in the INSTALL file to determine if there is anything special you need to do on your system.
5. Execute
mkdir build; cd build
cmake -DCMAKE_INSTALL_PREFIX=PREFIX ..
make lib
make install

where PREFIX is the location you want to install the libraries, we suggest $HOME/local where $HOME is your expanded home directory.

## Boost

For detailed instructions on installing boost with compiled libraries please see Section 5.2 of the official Boost documentation.

1. Download and decompress boost into $HOME/local/src/ 2. Change to the boost source directory 3. Execute bootstrap.sh If you want to compile for a specific toolset you could add --with-toolset=gcc. Now you are ready to install. Execute bash ./b2 install --prefix=PREFIX --with-program_options --with-serialization cxxflags="-std=c++17" linkflags="-std=c++17"  or if you are using the clang compiler on mac os bash ./b2 install --prefix=PREFIX --toolset=darwin --with-program_options --with-serialization cxxflags="-std=c++17 -stdlib=libc++" linkflags="-std=c++17 -stdlib=libc++"   4. If you want to have multiple versions of the library compiled with different compilers you can use the --layout=versioned flag above, or you could add option.set layout : versioned ; to your project-config.jam. Note: you may have to rename the $HOME/include/blitz_VER directory to remove the version number.
5. You should now have a PREFIX/include directory containing the header files for blitz, boost and random and your PREFIX/lib directory will contain the following files (the .dylib files will only appear on Mac OS X) bash libblitz.a libboost_program_options.a libblitz.la libboost_program_options.dylib 
6. Update the LD_LIBRARY_PATH (or DYLD_LIBRARY_PATH on mac os) variable inside your .bahsrc or .bash_profile to include PREFIX/lib e.g.

bash export LD_LIBRARY_PATH=\$LD_LIBRARY_PATH:PREFIX/lib 

7. Source your .bashrc or .bash_profile.

bash source ~/.bashrc 

# Path Integral Monte Carlo

After successfully installing blitz and boost you are now ready to compile the main pimc program on your system. PIMC uses CMake for build, test and installation automation. For details on using CMake consult https://cmake.org/documentation/. In short, the following steps should work on UNIX-like systems:

mkdir build
cd build
cmake ../
make
sudo make install

On Windows try:

md build
cd build
cmake ../
cmake --build . --config Release
cmake --build . --target install

As above, and with further details below, but you should consider using the following CMake options with the appropriate value instead of xxx :

• -D NDIM=1|2|3 the number of spatial dimensions
• -D GPU_BLOCK_SIZE=xxx equal to the maximum threadblock size and enables GPU acceleration (using AMD's HIP language)
• -D MAX_GPU_STREAMS=xxx equal to maximum number of concurrent streams on GPU device
• -D CMAKE_C_COMPILER=xxx equal to the name of the C99 Compiler you wish to use (or the environment variable CC)
• -D CMAKE_CXX_COMPILER=xxx equal to the name of the C++17 compiler you wish to use (or the environment variable CXX)
• -D CMAKE_PREFIX_PATH=xxx to add a non-standard location for CMake to search for libraries, headers or programs
• -D CMAKE_INSTALL_PREFIX=xxx to install pimc to a non-standard location
• -D BOOST_ROOT=xxx to add non-standard location for Boost install
• -D BLITZ_ROOT=xxx to add non-standard location for Blitz++ install
• -D STATIC=1 to enable a static build
• -D CMAKE_BUILD_TYPE=Debug to build pimc in debug mode
• -D CMAKE_BUILD_TYPE=PIGS to build pigs
• -D CMAKE_BUILD_TYPE=PIGSDebug to build pigs in debug mode
• -D DEFAULT_CXX_FLAGS="xxx" to overwrite default compiler flags for Release and PIGS builds (default -Wall -fno-math-errno -O3)
• -D CMAKE_CXX_FLAGS="xxx" to set additional compiler flags for Release and PIGS builds
• -D CMAKE_CXX_FLAGS_DEBUG="xxx" to set additional compiler flags for PIMCDebug and PIGSDebug builds
• -E env CXXFLAGS="xxx" add additional compiler flags
• -E env LDFLAGS="xxx" add additional linker flags

Executables will be installed to CMAKE_INSTALL_PREFIX location or if the install is skiped will be located in build/pimc. Executables produced are pimc.e, pimcd.e, pigs.e, and pigsd.e for CMAKE_BUILD_TYPE=Release|Debug|PIGS|PIGSDebug respectively.

If you run into problems, failures with linking etc., common errors may include not properly setting your LD_LIBRARY_PATH or not starting from a clean build directory (issue make clean or rm -rf ./* inside the build directory).

# Usage

In order to get a quick idea of the options which the code accepts type:

pimc.e --help

The code requires various combinations of these options to run, and the help message should give you an idea about which ones are mandatory.

## Quick Start

If you want to perform a quick test-run for bulk helium you could try something like:

./pimc.e -T 5 -N 16 -n 0.02198 -t 0.01 -M 8 -C 1.0 -I aziz -X free -E 10000 -S 20 -l 7 -u 0.02 --relax

In order for this to work, you will need a folder named OUTPUT in the directory where you type the command as it will produce output files in OUTPUT that contain all the results of the code. Each run of the code is associated with a unique identifying integer: the PIMCID. The options used in this demo include a subset of all the possible options:

Code Option Description
T temperature in kelvin
N number of particles
n density in Å-ndim (ndim=spatial dimension)
t the imaginary time step tau
M number of time slices involved in a bisection move
C worm prefactor constant
I interaction potential
X external potential
E number of equilibration steps
S number of production bins to output
l potential cutoff length in Å
u chemical potential in kelvin
relax adjust the worm constant to ensure we are in the diagonal ensemble ~75% of the simulation
o the number of configurations to be stored to disk
p process or cpu number
R restart the simulation with a PIMCID
W the wall clock run limit in hours
s supply a gce-state-* file to start the simulation from
P number of imaginary time slices
D size of the center of mass move in Å
d size of the single slice displace move in Å
m mass of the particles in AMU
b the type of simulation cell
L linear system size in Å
a scattering length in Å
c strength of the integrated delta function interaction
Lx linear system size in the x-direction Å
Ly linear system size in the y-direction in Å
Lz linear system size in the z-direction in Å
action the type of effective action used in the simulation
canonical restrict to the canonical ensemble
window the particle number window for restricting number fluctuations in the canonical ensemble
imaginary_time_length the imaginary time extent in K-1
wavefunction the type of trial wavefunction
dimension output the spatial dimension that the code was compiled with
pigs perform a simulation at T = 0 K
max_wind The maximum winding sector to be sampled. Default=1
staging Use staging instead of bisection for diagonal updates.

All options, including lists of possible values and default values can be seen by using the --help flag.

The output of the above command should yield something like:

  _____    _____   __  __    _____
|  __ \  |_   _| |  \/  |  / ____|
| |__) |   | |   | \  / | | |
|  ___/    | |   | |\/| | | |
| |       _| |_  | |  | | | |____
|_|      |_____| |_|  |_|  \_____|

[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Pre-Equilibration Stage.
[▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇]  . Diagonal Pre-Equilibration.
[▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇]  . Off-Diagonal Pre-Equilibration.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Relax Worm Constant.

Converged on C0 =  1.00000

[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Equilibration Stage.
[▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇]
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Measurement Stage.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    1 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    2 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    3 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    4 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    5 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    6 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    7 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    8 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #    9 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   10 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   11 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   12 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   13 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   14 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   15 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   16 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   17 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   18 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   19 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Bin #   20 stored to disk.
[PIMCID: 395a20fc-f1aa-4169-81e0-97a77e251bc4] - Measurement complete.


during the relaxation process where the string following PIMCID: is a uuid that uniquely tags your run, and 20 measurements will be output to disk. To analyze the results the code, you will need to obtain and install the pimcscripts package via:

pip install --upgrade git+git://github.com/DelMaestroGroup/pimcscripts.git#egg=pimcscripts


Which will add the pimcscripts library and a number of useful python analysis programs to your path.

After this has been completed, you can analyze the results of your run via

pimcave.py OUTPUT/gce-estimator-05.000-008.996-+000.020-0.01000-395a20fc-f1aa-4169-81e0-97a77e251bc4.dat

where 395a20fc-f1aa-4169-81e0-97a77e251bc4 needs to be replaced with the unique identifier generated on your machine. The results should yield something like:

# PIMCID 395a20fc-f1aa-4169-81e0-97a77e251bc4
# Number Samples 20
K 315.54530 12.70789 4.03
V -509.32392 11.22932 2.20
V_ext 0.00000 0.00000 0.00
V_int -507.58296 10.92684 2.15
E -193.77862 10.05201 5.19
E_mu -194.11968 10.05299 5.18
K/N 18.40800 0.62058 3.37
V/N -29.71109 0.35392 1.19
E/N -11.30309 0.56667 5.01
N 17.05300 0.17947 1.05
N^2 292.21000 6.07779 2.08
density 0.02343 0.00025 1.05
us 1290.97997 22.17522 1.72
mcsteps 157.20000 3.86251 2.46
diagonal 0.64350 0.01594 2.48

The basic idea of running the program is that one needs to setup the simulation cell, by defining either its specific geometry via the size (L) flag, or by a combination of density (n) and number of particles (N). At present, two types of simulation cells are possible, a hypercube in 1,2 or 3 dimensions with periodic boundary conditions and a cylinder in 3 dimensions, that is obtained by defining a radius (r). One then needs to setup the details of the simulation, including the temperature (T), chemical potential (u), interaction (I) and external (X) potential. The simulation details are then set via the imaginary time step (t), worm parameter (C) and number of equilibration (E) steps and production output bins (S). A more detailed grasp of all possible program options can be obtained by reading the main driver file pdrive.cpp.

## Output

The results of running the code are a number of data, state and log files that reside in the OUTPUT directory. If the code is run for the cylinder geometry, there will be an additional copy of the files in OUTPUT/CYLINDER which contain measurements that have been restricted to some cutoff radius indicated by including the w flag when running. The generic output files are:

Output File Description
gce-estimator-T-L-u-t-PIMCID.dat The main estimator file. Includes binned averages of various non-vector estimators like the energy and density of particles.
gce-log-T-L-u-t-PIMCID.dat The log file, which includes all the details of the simulation (including the command needed to restart it) and details on acceptance and output.
gce-number-T-L-u-t-PIMCID.dat The number probability distribution
gce-obdm-T-L-u-t-PIMCID.dat The one body density matrix
gce-pair-T-L-u-t-PIMCID.dat The pair correlation function
gce-pcycle-T-L-u-t-PIMCID.dat The permutation cycle distribution
gce-radial-T-L-u-t-PIMCID.dat The radial density
gce-state-T-L-u-t-PIMCID.dat The state file (used to restart the simulation)
gce-super-T-L-u-t-PIMCID.dat Contains all superfluid estimators

Each line in either the scalar or vector estimator files contains a bin which is the average of some measurement over a certain number of Monte Carlo steps. By averaging bins, one can get the final result along with its uncertainty via the variance.

# General Description

A full understanding of this path integral Monte Carlo code requires an understanding of the WA-PIMC algorithm alluded to in the introduction. In this section, we describe the large building blocks of the code. The actual specific details of the implementation can be understood by reading the doxygen documentation included here as well as reading the actual source code.

Any Monte Carlo simulation whether quantum or classical shares a number of features in common. Some type of simulation cell is created with a set of parameters that describe its physical environment. The initial state of the system is guessed, and a series of Moves are performed on the constituents of the system in such a way that detailed balance is maintained. After some suitable equilibration period, measurements are made and their results are stored to disk.

As discussed above, the driver file for this PIMC program is called pdrive.cpp. It takes a series of command line options, which are used by the Setup class to initialize ConstantParameters, Container, LookupTable and Communicator objects. Next, a Potential object is created which describes the potential environment (any walls etc.) and the interactions between bosons. A Path object is then instantiated which holds all the details of the actual world lines of the quantum particles. An Action object is created based on the Potential which holds an approximation of the action to be discretized in the path integral decomposition of the partition function. Finally, the main operating object of the program, of type PathIntegralMonteCarlo is created, which requires both the Path and the Action. This object performs the actual simulation via a series of Moves, all of which generate trial world line configurations that exactly sample the kinetic part of the density matrix. All measurements are made via specific Estimators with the results being output to disk.

The main kernel of this program should remain relatively untouched, as it has been extensively tested and optimized. Generality can come from modifying just a few things. For example, in order to implement a new type of measurement, one would need to write a derived Estimator class along with modifying the Communicator class to define an output path. New types of particles and external environments can be added by adding new Potential then updating Setup to allow for their specification at the command line. Finally, radically different systems can be studied by modifying the Container class.