5. Hubbard with Pinning Fields#

This example covers building a Hubbard model Hamiltonian on square Lattice with an antiferromagnetic pinning field and a charge pinning field, and generating a free-electron trial wavefunction.

Currently, pinning is applied only at the boundaries perpendicular to axis 1, and is applied to both boundaries or not at all. A more flexible pinning field interface is planned.

Currently the following spin pinning are available:

AFM pinning - alternating \(\pm1\) on each site.
- h_afm_pin - magnitude of pinning field
- afm_pin_type options:
  - “staggered” or “afm” - each edge has the opposite signs (default)
  - “same” or “fm” - each edge has the same pinning
FM pinning - a constant Sz field on each edge.
- h_fm_pin - magnitude of pinning field
- fm_pin_type options:
  - “staggered” - +1 on one edge and -1 on the other (default)
  - “same” or “fm” - +1 on both edges

TODO add a picture illustrating the pinning field location!

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_afm.toml, for a Hubbard model on a 8x4 square lattice with periodic boundary conditions.

[hamiltonian]
t = [1.0,0.5] # a 't' array of length n give nth-order neighbor hoping
U = 2.0 
h_afm_pin = 0.5 # NOTE: syntax likely to change in the near future
afm_pin_type = "staggered" # "afm"; alternative is "fm"

[lattice]
L1 = 8
L2 = 4
boundary1 = "PBC" # or "open" for open boundaries
boundary2 = "PBC" # or "open" for open boundaries

[misc_params]
nelec = [16,16]

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_afm.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=(-14.568331718444824+0j) with EK=(-14.848626136779785+0j) EU=(0.2802944779396057+0j) EU1=0 EU2=0 EJ=0
Reference HF Energy = -14.568331718444824

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.

Charge Pinning

Charge pinning is applying a constant charge field \(h \hat{n}_i\) applied along the edge. More flexible pinning is planned for the future. The charge pinning case can be set up similarly using

[hamiltonian]
t = [1.0,0.5] # a 't' array of length n give nth-order neighbor hoping
U = 2.0
h_charge_pin = 0.5 # NOTE: syntax likely to change in the near future

[lattice]
L1 = 8
L2 = 4
boundary1 = "PBC" # or "open" for open boundaries
boundary2 = "PBC" # or "open" for open boundaries

[misc_params]
nelec = [16,16]

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.