Documentation

Introduction

This webpage contains the details of a ground state (i.e, zero-temperature) lattice worm algorithm path-integral quantum Monte Carlo (WA-PIMC) code actively developed in c++ since 2020 based on:

• N.V. Prokof'ev, B.V. Svistunov, I.S. Tupitsyn: Exact, Complete, and Universal Continuous-Time Worldline Monte Carlo Approach to the Statistics of Discrete Quantum Systems
• N. Prokof'ev Lattice Path-Integral Monte Carlo Lecture Notes

In its current version, it can be used to simulate indistinguishable bosons in the one,two, and three-dimensional hypercubic Bose-Hubbard Model. As written, it takes a large number of command line options and allows for the measurement of the system energy and the Rényi Entanglement Entropy between spatial bipartitions of the lattice.

If you have questions on code usage or bug reports, please contact me at ecasiano@vols.utk.edu.

The development and maintenance of this code base has been supported in part by the National Science Foundation under Award No. DMR-2041995 and Award No. DMR-1553991

Installation

This program has been successfully compiled and runs on Intel systems using clang and g++. Before installing, one needs to ensure that all dependencies are met. We recommend that the required libraries (boost) are installed in a local folder inside your home directory: $HOME/local.

To clone the repository:

git clone https://github.com/DelMaestroGroup/pigsfli.git

Dependencies

The code is written in c++ and makes use of boost libraries and, for handling of command-line arguments, the cxxopts header-only library. For generation of random numbers and the ability to save the state of an RNG for simulation restarts, we use RNG_CLASS. Both cxxopts and RNG_CLASS are included in the repository when cloning.

Compilation

After dependencies are satisfied, you are now ready to compile the main pigsfli program on your system. pigsfli 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:

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

The following are some CMake options that can be edited in before compilation, where xxx should be replaced with the appropriate values:

• -D NDIM=1|2|3 the number of spatial dimensions
• -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 pigsfli to a non-standard location
• -D BOOST_ROOT=xxx to add non-standard location for Boost install
• -D STATIC=1 to enable a static build
• -D CMAKE_BUILD_TYPE=Debug to build pigsfli in debug mode
• -E env CXXFLAGS="xxx" add additional compiler flags
• -E env LDFLAGS="xxx" add additional linker flags

Executables will be installed to the CMAKE_INSTALL_PREFIX location or if the install step is skipped, they will be located in build/pigsfli. The executable produced will be pigsfli.e. Or pigsflid.e for CMAKE_BUILD_TYPE=Release|Debug.

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:

pigsfli.e --help

Quick Start

If you want to perform a quick test-run for a small one-dimensional Bose-Hubbard lattice you could try something like:

/pigsfli.e -D 1 -L 4 -N 4 -l 2 -U 3.3578 --mu 1.998 --sweeps 100000 --seed 2001 --measurement-frequency 25 --rng boost_mt19937 --bin-size 10000 --bins-wanted 100 --num-replicas 1 --beta 1.0 --canonical

Code Option	Description
D	Dimension of hypercubic lattice
L	Linear size of hypercube
N	Total number of particles
U	Interaction potential
l	Linear size of hypercubic subregion
sweeps	Number of sweeps before attempting measurements
beta	Set length of imaginary time
mu	Chemical potential
t	Tunneling parameter
canonical	set to false for grand canonical simulation
seed	Random seed value
sweeps-pre	Number sweeps for each pre-equilibration step
bin-size	Number of measurements per bin
bins-wanted	Number of bins desired in data file
subgeometry	Shape of subregion: square OR strip
num-replicas	Number of replicas
measurement-frequency	Measurements will be performed every other this amount
rng	Random Number Generator type
restart	continue simulation from a loaded rng state
no-accessible	do not calculate accessible entanglement entropies

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:

        _            __ _ _
       (_)          / _| (_)
  _ __  _  __ _ ___| |_| |_
 | '_ \| |/ _` / __|  _| | |
 | |_) | | (_| \__ \ | | | |
 | .__/|_|\__, |___/_| |_|_|
 | |       __/ |
 |_|      |___/

 Path-Integral Ground State (Monte Carlo) For Lattice Implementations


Stage (1/3): Determining mu and eta...

mu: 1.998 eta: 0.5 Z-frac: 1.285%
N     P(N)
3     ***
4     **************************************************************************************************
<N>: 3.97285

mu: -0.984349 eta: 0.190673 Z-frac: 27.9514%
N     P(N)
3     **********************************************
4     *******************************************************
<N>: 3.54578

mu: -1.13738 eta: 0.208524 Z-frac: 21.5446%
N     P(N)
4     *******************************************************************
5     **********************************
<N>: 4.33464

mu: -0.564649 eta: 0.205204 Z-frac: 21.4041%
N     P(N)
3     **********************************
4     *******************************************************************
<N>: 3.66301

mu: -1.12861 eta: 0.204957 Z-frac: 24.5636%
N     P(N)
3     ***********************
4     ***********************************************************
5     ********************
<N>: 3.97396

Fine tuning eta... (Want: 10% < Z-frac < 15%)

mu: -1.12861 eta: 0.206175 Z-frac: 24.4551%
N     P(N)
3     **************************************
4     ******************************************************
5     **********
<N>: 3.72151

mu: -1.12861 eta: 0.103088 Z-frac: 40.3038%
N     P(N)
3     *******
4     **********************************************************************
5     ************************
<N>: 4.17121

Stage (2/3): Equilibrating...

Stage (3/3): Main Monte Carlo loop...

-------- Detailed Balance --------

Insert Worm: 3721/163694
Delete Worm: 3438/3994

Insert Anti: 3525/123099
Delete Anti: 3232/3604

InsertZero Worm: 38086/378100
DeleteZero Worm: 38496/45527

InsertZero Anti: 40086/265865
DeleteZero Anti: 40496/52085

InsertBeta Worm: 37640/375699
DeleteBeta Worm: 37806/45155

InsertBeta Anti: 39576/269448
DeleteBeta Anti: 39742/50686

Advance Head: 96559/97385
Recede  Head: 96107/98277

Advance Tail: 98285/100513
Recede  Tail: 98454/99213

IKBH: 87069/234971
DKBH: 87130/95065

IKAH: 53877/234475
DKAH: 53999/61909

IKBT: 54638/165410
DKBT: 54503/62090

IKAT: 88504/238488
DKAT: 88439/96651

SWAP: 347255/861002
UNSWAP: 347253/934478

SWAP Advance Head: 28600/29543
SWAP Recede Head: 29080/31833

SWAP Advance Tail: 28552/31357
SWAP Recede Tail: 28457/29492

Elapsed time: 1.74691 seconds

Output explanation

In Stage (1/3), histograms of the total particle number distribution are shown, where each of the asterisks (*) represents a normalized count. For canonical ensemble simulations, like the one shown above, the only particle numbers visited are $N-1$, $N$, and $N+1$, where $N$ is the target number of particles. A grand canonical simulation will show histograms with more particle numbers than these. Once the peak of the distribution is at $N$, and it's at least 33% larger than the next largest sector, we proceed to the "fine tuning eta" stage. Here, eta is either shrunk or augmented until we reach the desired percentage of configurations with no worms present (currently, we set this window between 40 and 45%).

Stage (2/3), the code is ran without taking any measurement as a en equilibration step. The number of equilibration steps are currently determined by the sweeps parameter from the command line.

Finally, Stage (3/3) is where measurements are performed and samples collected. Once the desired number of samples are collected, the simulation stops. Near the bottom of the terminal output above, the number of times that each of the updates is accepted and proposed are shown as a fraction. The number of times that the update is accepted is shown in the numerator, whereas the times that it was proposed is shown in the denominator.

The total run time of equilibration and Main Monte Carlo loops is shown at the bottom of the output, in seconds.

After this has been completed, you can analyze the results of your run using the scripts in the https://github.com/DelMaestroGroup/papers-code-pigsfli.