Lattice Model Tools for AFQMC#
Overview#
A lattice model Hamiltonian consists of a series of Hamiltonian terms (i.e. a hopping term, hubbard on-site interaction, etc.) which are defined in terms of a generic lattice. In practical calculations, a specific lattice must be used and some set of Hamiltonian terms are constructed on the specific lattice. In the AFQMC lattice model tools, the creation of a Lattice class instance and of a Hamiltonian class instance are separated. First, the Lattice instance is generated The lattice is an object which is aware of geometrical properties such as the positions of lattice sites, neighbors, boundary conditions, etc.
Lattice Class#
Hamiltonian Builder#
Model Hamiltonian Builder#
TODO: explain the combined (lattice,band) indices
The afqmctools Python module includes tooling to build general, multi-band Hubbard-Kanamori Hamiltonians of the type:
where \(i\),\(j\) are combined lattice and band indices, \(\hat{c}^\dagger_{i\sigma}\), \(\hat{c}_{j\sigma'}\) create/annihilate and electron on the site (and band) corresponding to \(i\)/\(j\) with spin \(\sigma\)/\(\sigma'\), \(\hat{n}_{i\sigma}\) is the number operator, \(t^{\sigma \sigma'}_{ij}\) includes all one-body terms (\(n^{th}\)-order neighbor hopping, spin-orbit coupling, etc.), \(U\) is the traditional on-site hubbard interaction, \(U^1\) is a density-density interaction, \(U^2\) is a spin-spin interaction, and \(J\) is a Hund’s coupling term. \(U^1\), \(U^2\), and \(J\) are typically only non-zero between bands on the same lattice site, but are by no means limited in this way.
Currently, SAFIRE can use any form of this Hamiltonian. For convenience, we supply tooling to build the most common forms of the Hamiltonian described above. Only on-site, but inter-band, \(U\),\(U^1\),\(U^2\), and \(J\) are implemented; however, a motivated user can build a custom Hamiltonian and save it in the format described below in the section, “ input file format”
Building a Model Hamiltonian#
This section explains how to build a model Hamiltonian using the included Python tooling. Input format/conventions are described in more detail in the section Model Hamiltonian Builder Input Conventions
A Hamiltonian may be built from a Python dict as follows:
from afqmctools.hamiltonian.director import HamiltonianDirector
params = {
'hamiltonian' : {
't' : 1.0,
'U' : 2.0,
'U1' : 1.5,
'U2' : 1.0,
'J' : 0.5
},
'lattice' : {
'L1' : 6,
'L2' : 1,
'boundary1' : 'PBC',
}
}
hamiltonian = HamiltonianDirector(
source = params
).build()
...
the Hamiltonian can then be saved in the SAFIRE format with:
...
from afqmctools.utils.io import write_model_hamiltion
write_model_hamiltion(
hamiltonian=hamiltonian,
fname='afqmc.h5',
nelec=(6,6)
)
Alternatively, the cli includes a tool to automatically build and save a
Hamiltonian from parameters saved in an input file (in toml format).
If afqmctools was installed as described in
Installation, a model Hamiltonian can be built
using:
$ make_model_ham -i model_ham_params.toml -o afqmc.h5
For example, the following input file will make the same Hamiltonian as above.
In TOML:
[hamiltonian]
nbands = 2
t = 1.0
U = 2.0
U1 = 1.5
U2 = 1.0
J = 0.5
[lattice]
L1 = 6
L2 = 1
boundary1 = "PBC"
[cli_params]
nelec = [6,6]
Model Hamiltonian Builder Input Conventions#
Model Hamiltonian parameters can be specified either directly as a Python dict, or via an input file in toml format. both cases share the following conventions.
The model Hamiltonian builder will reference two different input blocks;
The hamiltonian block - where
Hamiltonian parameters are specified - and a lattice block - where
the lattice that underlies the Hamiltonian is defined.
Any supported Hamiltonian can be constructed on any supported
lattice.
hamiltonian block#
The hamiltonian block consists of the following fields. In the
following, \(i,j\) are combined indices including both lattice
indices (\(\mu\),\(\nu\)), and band indices
(\(m\),\(m'\))
TODO: This is a little too busy. Try making each parameter a sub-sub-section (whatever level is appropriate), and space out each case.
t: specifies hopping terms, \(\sum_{ij,\sigma \sigma'}t^{\sigma \sigma'}_{ij}\hat{c}^\dagger_{i\sigma}\hat{c}_{j\sigma'}\).tmay be any of:None : no hopping : \(t^{\sigma \sigma'}_{ij} = 0\)
a single number : interpreted as the nearest-neighbor hopping. Assumes no hopping between band : \(t^{\sigma \sigma'}_{ij} = t\delta_{mm'}\delta_{\nu,\mu\pm1}\delta_{\sigma\sigma'}\)
a 1-D array of length \(n-1\) : interpreted as multiple hopping terms where
t[n-1]is the hopping strength between \(n^{th}\)-order neighbors. Assumes no hopping between bands : \(t^{\sigma \sigma'}_{ij} = \sum_{n=1} t_n \delta_{mm'}\delta_{\nu,\mu\pm n}\delta_{\sigma\sigma'}\)a 2-D array with shape \(N_{bands}\) x \(N_{bands}\) : defines hopping between bands and/or allows different bands to use different hopping strengths. In this case, the same
tarray is used for all lattice sites. \(t^{\sigma \sigma'}_{ij} = t_{mm'} \delta_{\nu,\nu\pm 1}\delta_{\sigma\sigma'}\)
U: specifies the on-site Hubbard interaction. May be any of:None : no hubbard U : \(U_i = 0\)
a single number : interpreted as a constant U across lattice sites and bands \(U_i = U \delta_{mm'} \delta_{\mu\nu}\)
a 1-D array of length \(N_{bands}\) : interpreted as \(U_m\) where \(m\) is the band index, but otherwise constant in lattice index. \(U_i = U_m \delta_{\mu\nu}\)
a 1-D array of length \(N_{sites}\) : interpreted as \(U_\mu\) where \(\mu\) is the lattice site index, but otherwise constant in band index. \(U_i = U_{\mu} \delta_{mm'}\)
TODO: update the description of U1, U2, and J to include the intersite case.
U1: specifies the on-site, density-density Hubbard interaction. \(\sum_{i<j} U_{ij}^1 (\hat{n}_{i\uparrow} \hat{n}_{j\downarrow} + \hat{n}_{i\downarrow} \hat{n}_{j\uparrow} )\) May be any of:None : no density-density Hubbard interaction \(U_{ij}^1 = 0\)
a single number : interpreted as a constant U1 for all valid interband interactions, on the same site. \(U_{ij}^1 = U^1 \delta_{\mu\nu} \forall m,m'\)
a 2-D array with shape \(N_{bands}\) x \(N_{bands}\) : interpreted as the onsite, interband \(U^1\) interaction which is the same for all sites. The diagonal is ignored since it should be specified via by the parameter
U. \(U_{ij}^1 = U^1_{mm'} \delta_{\mu\nu}\)
U2: specifies the interband, on-site spin-spin Hubbard interaction. It follows similar input conventions asU1, but is understood to describe interactions between electrons of the same spin. SeeU1for more. \(\sum_{i<j} U_{ij}^2 (\hat{n}_{i\uparrow} \hat{n}_{j\uparrow} + \hat{n}_{i\downarrow} \hat{n}_{j\downarrow} )\)J: specifies the on-site Hund’s interaction. It follows similar input conventions asU1, but is understood to describe interactions between electrons of the same spin. SeeU1for more. \(\sum_{i<j} J_{ij} (\hat{c}^\dagger_{i\uparrow}\hat{c}^\dagger_{j\downarrow}\hat{c}_{i\downarrow}\hat{c}_{j\uparrow} +\hat{c}^\dagger_{i\uparrow}\hat{c}^\dagger_{i\downarrow}\hat{c}_{j\downarrow}\hat{c}_{j\uparrow} +\hat{c}^\dagger_{j\uparrow}\hat{c}^\dagger_{i\downarrow}\hat{c}_{j\downarrow}\hat{c}_{i\uparrow} +\hat{c}^\dagger_{j\uparrow}\hat{c}^\dagger_{j\downarrow}\hat{c}_{i\downarrow}\hat{c}_{i\uparrow})\)
lattice block#
The lattice block consists of the following fields. In the
following, \(\hat{a}_1\) and \(\hat{a}_2\) are the lattice
vectors.
L1: the number of lattice sites along \(\hat{a}_1\).L2: the number of lattice sites along \(\hat{a}_2\).(optional)
type: eithersquareortriangular: use either a square lattice or a triangular lattice.(optional)
boundary1: either'open'or'pbc'. the type of boundary to use in the direction perpendicular to \(\hat{a}_2\).openis the default.(optional)
boundary2: either'open'or'pbc'. the type of boundary to use in the direction perpendicular to \(\hat{a}_1\).openis the default.