lattpy 0.6.4

Python package for modeling Bravais lattices in solid state physics.

LattPy

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LattPy is a simple and efficient Python package for modeling Bravais lattices and constructing (finite) lattice structures in d dimensions.

:warning: WARNING: This project is still in development and might change significantly in the future!

Installation

Install via pip

pip install lattpy

or Github

pip install git+https://github.com/dylanljones/lattpy.git@VERSION

or download/clone the package, navigate to the root directory and install via

python setup.py install

Usage

Before accessing the attributes of the Lattice-model the lattice has to be configured

Configuration

A new instance of a lattice model is initialized using the unit-vectors of the Bravais lattice. After the initialization the atoms of the unit-cell need to be added. To finish the configuration the connections between the atoms in the lattice have to be set. This can either be done for each atom-pair individually by calling add_connection or for all possible pairs at once by callling add_connections. The argument is the number of unique distances of neighbors. Setting a value of 1 will compute only the nearest neighbors of the atom.

import numpy as np
from lattpy import Lattice

latt = Lattice(np.eye(2))                 # Construct a Bravais lattice with square unit-vectors
latt.add_atom(pos=[0.0, 0.0])             # Add an Atom to the unit cell of the lattice
latt.add_connections(1)                   # Set the maximum number of distances between all atoms

latt = Lattice(np.eye(2))                 # Construct a Bravais lattice with square unit-vectors
latt.add_atom(pos=[0.0, 0.0], atom="A")   # Add an Atom to the unit cell of the lattice
latt.add_atom(pos=[0.5, 0.5], atom="B")   # Add an Atom to the unit cell of the lattice
latt.add_connection("A", "A", 1)          # Set the max number of distances between A and A
latt.add_connection("A", "B", 1)          # Set the max number of distances between A and B
latt.add_connection("B", "B", 1)          # Set the max number of distances between B and B
latt.analyze()

Configuring all connections using the add_connections-method will call the analyze-method directly. Otherwise this has to be called at the end of the lattice setup or by using analyze=True in the last call of add_connection. This will compute the number of neighbors, their distances and their positions for each atom in the unitcell.

To speed up the configuration prefabs of common lattices are included. The previous lattice can also be created with

from lattpy import simple_square

latt = simple_square(a=1.0, neighbors=1)  # Initializes a square lattice with one atom in the unit-cell

So far only the lattice structure has been configured. To actually construct a (finite) model of the lattice the model has to be built:

latt.build(shape=(5, 3))

This will compute the indices and neighbors of all sites in the given shape and store the data.

After building the lattice periodic boundary conditions can be set along one or multiple axes:

latt.set_periodic(axis=0)

To view the built lattice the plot-method can be used:

from lattpy import simple_square

latt = simple_square(a=1.0, neighbors=1)
latt.build((5, 3), periodic=0)
latt.plot()

General lattice attributes

After configuring the lattice the attributes are available. Even without building a (finite) lattice structure all attributes can be computed on the fly for a given lattice vector, consisting of the translation vector n and the atom index alpha. For computing the (translated) atom positions the get_position method is used. Also, the neighbors and the vectors to these neighbors can be calculated. The dist_idx-parameter specifies the distance of the neighbors (0 for nearest neighbors, 1 for next nearest neighbors, ...):

from lattpy import simple_square

latt = simple_square()

# Get position of atom alpha=0 in the translated unit-cell
positions = latt.get_position(n=[0, 0], alpha=0)

# Get lattice-indices of the nearest neighbors of atom alpha=0 in the translated unit-cell
neighbor_indices = latt.get_neighbors(n=[0, 0], alpha=0, distidx=0)

# Get vectors to the nearest neighbors of atom alpha=0 in the translated unit-cell
neighbor_vectors = latt.get_neighbor_vectors(alpha=0, distidx=0)

Also, the reciprocal lattice vectors can be computed

rvecs = latt.reciprocal_vectors()

or used to construct the reciprocal lattice:

rlatt = latt.reciprocal_lattice()

The 1. Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice:

bz = rlatt.wigner_seitz_cell()

The 1.BZ can also be obtained by calling the explicit method of the direct lattice:

bz = latt.brillouin_zone()

Finite lattice data

If the lattice has been built the needed data is cached. The lattice sites of the structure then can be accessed by a simple index i. The syntax is the same as before, just without the get_ prefix:

from lattpy import simple_square

latt = simple_square()
latt.build((5, 2))
idx = 2

# Get position of the atom with index i=2
positions = latt.position(idx)

# Get the atom indices of the nearest neighbors of the atom with index i=2
neighbor_indices = latt.neighbors(idx, distidx=0)

# the nearest neighbors can also be found by calling (equivalent to dist_idx=0)
neighbor_indices = latt.nearest_neighbors(idx)

Performance

Even though lattpy is written in pure python, it achieves high performance and a low memory footprint by making heavy use of numpy's vectorized operations. As an example the build-times of a square lattice for different number of sites are shown in the following plot:

Note that the overhead of the multi-thread neighbor search results in a slight increase of the build time for small systems. By using num_jobs=1 in the build-method this overhead can be eliminated for small systems. By passing num_jobs=-1 all cores of the system is used.

Examples

Using the (built) lattice model it is easy to construct the (tight-binding) Hamiltonian of a non-interacting model:

import numpy as np
from lattpy import simple_chain

# Initializes a 1D lattice chain with a length of 5 atoms.
latt = simple_chain(a=1.0)
latt.build(shape=4)
n = latt.num_sites

# Construct the non-interacting (kinetic) Hamiltonian-matrix
eps, t = 0., 1.
ham = np.zeros((n, n))
for i in range(n):
    ham[i, i] = eps
    for j in latt.nearest_neighbors(i):
        ham[i, j] = t

Since we loop over all sites of the lattice the construction of the hamiltonian is slow. An alternative way of mapping the lattice data to the hamiltonian is using the DataMap object returned by the map() method of the lattice data. This stores the atom-types, neighbor-pairs and corresponding distances of the lattice sites. Using the built-in masks the construction of the hamiltonian-data can be vectorized:

import numpy as np
from scipy import sparse
from lattpy import simple_chain

# Initializes a 1D lattice chain with a length of 5 atoms.
latt = simple_chain(a=1.0)
latt.build(shape=4)

# Vectorized construction of the hamiltonian
eps, t = 0., 1.
dmap = latt.data.map()               # Build datamap
values = np.zeros(dmap.size)         # Initialize array for data of H
values[dmap.onsite(alpha=0)] = eps   # Map onsite-energies to array
values[dmap.hopping(distidx=0)] = t  # Map hopping-energies to array

# The indices and data array can be used to construct a sparse matrix
ham_s = sparse.csr_matrix((values, dmap.indices))
ham = ham_s.toarray()

Both construction methods will create the following Hamiltonian-matrix:

[[0. 1. 0. 0. 0.]
 [1. 0. 1. 0. 0.]
 [0. 1. 0. 1. 0.]
 [0. 0. 1. 0. 1.]
 [0. 0. 0. 1. 0.]]

If periodic boundary conditions are set (along axis 0) the output is:

[[0. 1. 0. 0. 1.]
 [1. 0. 1. 0. 0.]
 [0. 1. 0. 1. 0.]
 [0. 0. 1. 0. 1.]
 [1. 0. 0. 1. 0.]]