6. Adding twist angles

This example covers building a Hubbard model Hamiltonian on square Lattice with twist angles applied 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 input1.toml, for a Hubbard model on a 4x8 square lattice with periodic boundary conditions.

[hamiltonian]
U = 4

[lattice]
L1 = 4
L2 = 8
boundary1 = "PBC" # or "open" for open boundaries
boundary2 = "PBC" # or "open" for open boundaries
twist = ["0 Pi","7/8 Pi"] # 'Pi' is parsed and converted! So are fractions expressed as 'num/denom'

[misc_params]
nelec = [24,24]

We note that both the symbol ‘Pi’ and fractions of the form “numerator/denominator” are parsed and converted when included as strings. Explicit decimal inputs are, of course, also allowed

[hamiltonian]
U = 4

[lattice]
L1 = 4
L2 = 8
boundary1 = "PBC" # or "open" for open boundaries
boundary2 = "PBC" # or "open" for open boundaries
twist = [0.0,2.748893571891069] # 'Pi' is parsed and converted! So are fractions expressed as 'num/denom'

[misc_params]
nelec = [24,24]

afqmctools can be invoked within a Python script as

from afqmctools.wavefunction.free_electron import free_electron

infile = "input1.toml"

# Build and save a lattice model Hamiltonian
hamiltonian = HamiltonianDirector(source=infile).build()
input_params = io.read_input_params(infile)
nelec = input_params["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,
    twist=input_params["lattice"].get("twist",None),
    output="afqmc.h5"
)

The free_electron() function accepts a twist keyword argument for the sake of reproducibility for open-shell systems. Unlike many of the other examples, here we explicitly set the twist to that of the lattice. By default, i.e. if twist is None, a small incommensurate twist angle is used instead.

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=(31.609146118164062-6.798654794692993e-07j) with EK=(-40.39085388183594-2.384185791015625e-07j) EU=(72-4.414469003677368e-07j) EU1=0 EU2=0 EJ=0
Reference HF Energy = 31.609146118164062

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.

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

infile = "input_charge.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"
)

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=(-13.653738975524902+0j) with EK=(-13.951533317565918+0j) EU=(0.2977941334247589+0j) EU1=0 EU2=0 EJ=0
Reference HF Energy = -13.653738975524902

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