Generating fermionic pseudo-snapshots

This repository contains subroutines for nested componentwise direct sampling (NCDS) of occupation number states from pseudo free-fermion density matrices as they arise naturally in finite-temperature determinantal QMC (DQMC) simulations.

The code can be used in two forms, either

1. as a stand-alone code, which uses as input the Green's functions which have been output and written on disk during a previous DQMC run or
2. directly called from a DQMC code so that fermionic snapshots are output "on the fly".

The two use cases are descibed in the following.

Reading Green's function from a previous DQMC run:

Compile the code sample_pseudo_DM.f90 with MPI support by calling

make

This requires that you have mpif90 installed. The resulting executable is called sample_pseudo_DM. Then, copy the provided test data for a 12x12 system at U/t=4, inverse temperature = 4 / t and half filling

cp _resources/U4.0_mu0.0_L12_beta4.0/Green_ncpu00000_*.dat .

The equal-time Green's function in real space for spin up and spin down is assumed to be stored in the files Green_ncpuXXXXX_up.dat and Green_ncpuXXXXX_dn.dat. The two files contain a stream of Green's function matrices for spin up and spin down in a synchronized fashion, i.e. the n-th Green's function in the "up-file" must be combined with the n-th Green's function in the "down-file". Successive Green's functions are separated by two empty lines. The five-digit number code XXXXX labels DQMC output from different CPUs, parallelized via MPI, and the code for sampling of fermionic pseudosnapshots can be run with the same number of CPUs by calling

mpiexec.openmpi -np 1 ./sample_pseudo_DM

(We have only one set of Green's functions as test data, so we use only one CPU here.)

The number of snapshots to be generated can be set in the input file simparams.in.

filename = 'list_of_sitearrays.txt'   ! Here, a subset of sites for sampling can be selected (currently not used)
Nsites = 144                          ! Total number of sites, 12x12=144
max_HS_samples = 12                   ! Read a maximum number of Green's functions from the files Green_ncpuXXXXX_up(dn).dat  
Nsamples_per_HS = 10                  ! Generate `Nsamples_per_HS` number of occupation number snapshots per Green's function                   
skip = 0                              ! Discard the first `skip` number of Green's function 

With the above settings 120 snapshots will be generated per CPU. The snapshots will be written line-by-line into two synchronized files for spin up and spin down called Fock_samples_ncpuXXXXX_up.dat and Fock_samples_ncpuXXXXX_dn.dat. The first two entries in each line are the sign and the reweighting factor. Note that the ordering of sites in a line is the same as the ordering used to write the Green's function matrix in real space.

The meaning of a line in the output files is as follows

1.0       1.0        0.0        1.253               1 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 ...
BSS_sign  Re(phase)  Im(phase)  reweighting_factor  occupation_vector(1:N_sites)

BSS_sign = sign(det(GF_up) * det(GF_dn)) is the sign of the current Green's function from which the sample was drawn. If the Hamiltionian is sign-problem free, it is equal to one. If there is a sign problem, you need to calculate this one (the code does not do it right now). Re(phase) and Im(phase) give the phase of the pseudo-snapshot (note that for sampling in momentum space there is a phase problem). When calculating any quantity from the pseudo-snapshots they need to be weighted by reweighting_factor.

Visualizing the generated pseudo-snapshots:

After completing the above steps, we can visualize the pseudo-snapshots. The python script illustrates how to combine pseudo-snapshots for spin up and down.

cp _resources/visualization/show_movie.py .
python show_movie.py

Interfacing with the QUantum Electron Simulation Toolbox (QUEST) DQMC code:

All relevant subroutines are in NCDS_for_quest.F90.

From the open-source QUEST determinantal QMC code (-> http://quest.ucdavis.edu/index.html) the provided driver subroutine could be called like so in the subroutine DQMC_Phy0_Meas:

call run_sample_pseudo_DM(G_up=G_up, G_dn=G_dn, BSS_sign_up=sgnup, BSS_sign_dn=sgndn,  sp_basis="real_space", &
       Nsamples_per_HS=20, outfile_basename="Fock_samples", &
       MPI_rank=qmc_sim%rank )

Then it will write the pseudo-napshots for spin-up and spin-down in two "synchronized" files together with the sign (or phase factor) and reweighting factor.

If you use this code, please cite:

@article{arXiv:2009.07377,
      title={Numerically exact quantum gas microscopy for interacting lattice fermions}, 
      author={Stephan Humeniuk and Yuan Wan},
      year={2020},
      eprint={2009.07377},
      archivePrefix={arXiv},
      primaryClass={cond-mat.quant-gas}
}