1. Rectangular Hubbard Model

This example covers building a Hubbard model Hamiltonian on square Lattice, and generating a free-electron trial wavefunction.

A lattice model Hamiltonian can be generated using afqmctools and a toml-based input file. Below is a sample input file, which we name input.toml, for a Hubbard model on a 3x2 square lattice with periodic boundary conditions.

[hamiltonian]
t = 1.0 # defaults to 1.0
U = 2.0 

[lattice]
L1 = 2
L2 = 3
boundary1 = "PBC" # or "open" for open boundaries
boundary2 = "PBC" # or "open" for open boundaries

[misc_params]
nelec = [2,2]

afqmctools can be invoked within a Python script as

from afqmctools.hamiltonian.model.director import HamiltonianDirector
import afqmctools.utils.io as io
from afqmctools.wavefunction.free_electron import free_electron

infile = "input.toml"

# Build and save a lattice model Hamiltonian
hamiltonian = HamiltonianDirector(infile).build()
nelec = io.read_input_params(infile)["misc_params"]["nelec"]
io.write_model_hamiltion(
    hamiltonian=hamiltonian,
    fname="afqmc.h5",
    nelec=nelec
)

# compute and save a free-electron trial wfn
free_electron(
    source=infile,
    nelec=nelec,
    output="afqmc.h5"
)

The free_electron() function accepts a twist keyword argument for the sake of reproducibility for open-shell systems. twist should be a 2-d iterable containing \(\vec{\theta}=(\theta_1,\theta_2)\) in radians even if one component is set to 0.0. By default, a small incommensurate twist angle is used.

If the sample Python script is run above with the sample input file, the following should be present at the end of the output.

Running in Slater Detemrinant Mode
energyCall: Etotal=(-8.5+0j) with EK=(-10+0j) EU=(1.500000238418579+0j) EU1=0 EU2=0 EJ=0
Reference HF Energy = -8.5

Internally, the AutoHF Hartree-Fock code is used to evaluate the energy of the free-electron trial wavefunction with respect to the interacting Hamiltonian. This allows the initial energy to be checked within the AFQMC code.

See the examples in Running SAFIRE for how to run AFQMC.