tightbinder.models.AgarwalaChern

class AgarwalaChern(side=1, t=1, m=1, t2=0, lda=0, r=1.1)

Bases: System

Implementation of the Agarwala model of amorphous Chern insulators in 2D, which is a generalization of the BHZ model for arbitrary atomic positions, with additional terms to break particle-hole and inversion symmetry.

Parameters:

• side – First-neighbour distance in the crystalline system. Defaults to 1.

• t – Hopping parameter. Defaults to 1.

• m – Mass term. Defaults to 1.

• t2 – Particle-hole symmetry breaking parameter. Defaults to 0.

• lda – Inversion symmetry breaking parameters. Defaults to 0

• r – Cutoff distance for neighbour interaction. Defaults to side*1.1.

Methods

add_atom	Method to add one atom from a numbered species into a specific position.
add_atoms	Method to add a list of atoms of specified species at some positions.
add_bond	Method to add a hopping between two atoms of the motif.
add_bonds	Same method as add_hopping but we input a list of hoppings at once.
atom_coordination_number	Method to find the coordination number for a specific atom in the crystal.
brillouin_zone_mesh	Routine to compute a mesh of the first Brillouin zone using the Monkhorst-Pack algorithm.
compute_first_neighbour_distance	Routine to compute the first neighbour distance.
compute_neighbour_distances	Method to compute neighbour distance up to neighbours nni.
compute_unit_cell_area	Method to compute the area of the unit cell.
coordination_number	Method to find the coordination number of the solid.
crystal_center	Method to compute the geometric center of the crystal from the positions of the atoms of the motif.
crystallographic_group	Determines the crystallographic group associated to the material from the configuration file.
determine_high_symmetry_points	Routine to compute high symmetry points depending on the dimension of the system.
extend_copy	Method to extend the system by attaching a copy of the motif displaced by some input vector.
find_disconnected_atoms	Method to find which atoms do not have neighbours, i.e. they are disconnected from the lattice.
find_first_neighbours	Method to find the first neighbours of the atoms of the motif.
find_lowest_coordination_atoms	Routine to determine the atoms of the solid with the lowest coordination atoms.
find_neighbours	From a list of atoms (motif), it sets the list of bonds within the solid.
hamiltonian_k	Routine to evaluate the Bloch Hamiltonian at a given k point.
high_symmetry_path	Routine to generate a path in reciprocal space along the high symmetry points (HSPs) indicated.
identify_boundary	Method to obtain the indices of the outermost atoms of the motif, taking into account boundary conditions.
identify_convex_hull	Deprecated in favour of identify_motif_edges.
identify_motif_edges	Method to obtain the indices of the outermost atoms of the motif.
initialize_hamiltonian	Routine to initialize the matrices that compose the Bloch Hamiltonian
plot_2d_crystal	Method to visualize the crystalline structure (Bravais lattice + motif) in two dimensions.
plot_crystal	Method to visualize the crystalline structure (Bravais lattice + motif).
plot_wireframe	Method to plot a wireframe of the system according to the detected bonds.
reciprocal_lattice	Routine to compute the reciprocal lattice basis vectors from the Bravais lattice basis.
reduce	Routine to reduce the dimensionality of the System object along the specified directions, by repeating unit cells along those directions until a given size (number of unit cells) is reached.
remove_atom	Method to remove the atom at the specified index from the motif.
remove_atoms	Method to remove the atoms specified in a list with all their indices.
remove_disconnected_atoms	Method to remove disconnected atoms from the motif.
restrict_lattice2rectangle	TO BE IMPLEMENTED YET (is it really necessary?).
ribbon	Routine to generate a ribbon for a 2D crystal.
rotate	Method to rotate the whole system in the xy plane by some angle.
solve	Diagonalize the Hamiltonian to obtain the band structure and the eigenstates.
supercell	Routine to generate a supercell for a system using the number of cells along each Bravais vectors specified in ncells.
update	Routine to initialize or update the intrinsic attributes of the Crystal class whenever the Bravais lattice is changed.
visualize	Method to render a 3d visualization of the crystal using the self.vpython library.

Attributes

basisdim
boundary
bravais_lattice
filling
matrix_type
motif
ndim
norbitals
unit_cell_list

add_atom(position, species=0)

Method to add one atom from a numbered species into a specific position.

Parameters:

• position (Union[list, ndarray]) – Array with atom coordinaes. Must be three-dimensional.

• species (int) – Index of corresponding chemical species. Defaults to 0.

Return type:

None

add_atoms(atoms, species=None)

Method to add a list of atoms of specified species at some positions. Built on top of method add_atom.

Parameters:

• atoms (Union[list, ndarray]) – List of length n, containing the positions of each atom.

• species (list) – List with the indices of the chemical species. If None, all default to zero.

Return type:

None

add_bond(initial, final, cell=(0.0, 0.0, 0.0), neigh='1')

Method to add a hopping between two atoms of the motif. NB: The hopping has a specified direction, from initial to final. The HSince the final Hamiltonian is computed taking into account hermiticity, it is not necessary to specify the hopping in the other direction. (TODO: Change this, no longer working like this).

Parameters:

• initial (int) – Index of initial atom of the bond.

• final (int) – Index of final atom of the bond.

• cell (Union[list, tuple, ndarray]) – Array with cell containing the final atom. Defaults to [0., 0., 0.].

• neigh (str) – String to denote if the bond corresponds to first neighbour, second, etc. Defaults to ‘1’.

Return type:

None

add_bonds(initial, final, cells=None, neigh=None)

Same method as add_hopping but we input a list of hoppings at once.

Parameters:

• initial (List[int]) – List of indices of initial atom of the bond.

• final (List[int]) – List of indices of final atom of the bond.

• cell – List of cell vectors containing the final atom. Defaults to None, which corresponds to the origin cell.

• neigh (List[str]) – List of strings to denote if the bonds corresponds to first neighbour, second, etc respectively. Defaults to None, which corresponds to all being ‘1’.

Return type:

None

static atom_coordination_number(index, bonds)

Method to find the coordination number for a specific atom in the crystal.

Parameters:

• index (int) – Index of atom whose coordination number we want to compute.

• bonds (list) – List of indices within the solid.

Return type:

int

Returns:

Number of neighbours.

brillouin_zone_mesh(mesh)

Routine to compute a mesh of the first Brillouin zone using the Monkhorst-Pack algorithm.

Parameters:

mesh (list) – List with the number of kpoints along each axis.

Return type:

ndarray

Returns:

Array with all kpoints, nk x 3

compute_first_neighbour_distance(near_cells=None)

Routine to compute the first neighbour distance. DEPRECATED since there is compute_neighbour_distances which works up to n neighbour distances.

Parameters:

near_cells (Union[list, ndarray]) – List of unit cells to loop over. Defaults to None.

Return type:

float

Returns:

First neighbour distance.

compute_neighbour_distances(nni)

Method to compute neighbour distance up to neighbours nni.

Parameters:

nni (int) – Maximum neighbour distance to consider. nni=5 mean that it will find first, second up to fifth neighbours distances.

Return type:

List[float]

Returns:

List of neighbour distances.

compute_unit_cell_area()

Method to compute the area of the unit cell. TODO: Adapt for 1D and 3D

Return type:

float

Returns:

Value of unit cell area.

coordination_number()

Method to find the coordination number of the solid. Returns coordination number of solid, and a list with the different individual coordinations numbers found within the solid.

Return type:

Tuple[float, List[int]]

Returns:

Coordination number, and list of all individual coordination numbers found.

crystal_center()

Method to compute the geometric center of the crystal from the positions of the atoms of the motif. Calculated as the average of all atomic positions.

Return type:

ndarray

Returns:

Numpy array with the cartesian coordinates of the center

crystallographic_group()

Determines the crystallographic group associated to the material from the configuration file. NOTE: Only the group associated to the Bravais lattice, reduced symmetry due to presence of motif is not taken into account. Its purpose is to provide a path in k-space to plot the band structure. Workflow is the following: First determine length and angles between basis vectors. Then we separate by dimensionality: first we check angles, and then length to determine the crystal group.

Return type:

None

determine_high_symmetry_points()

Routine to compute high symmetry points depending on the dimension of the system. These symmetry points will be used to plot the band structure along the main reciprocal paths of the system (irreducible BZ). Returns a dictionary with pairs {letter denoting high symmetry point: its coordinates}

Return type:

None

extend_copy(vector, n=1)

Method to extend the system by attaching a copy of the motif displaced by some input vector.

Parameters:

• vector – Vector by which we displace the copy of the system.

• n – Number of copies to attach. The n-th copy will be displaced by n*vector. Defaults to 1.

Returns:

Extended system.

find_disconnected_atoms()

Method to find which atoms do not have neighbours, i.e. they are disconnected from the lattice.

Return type:

List[int]

Returns:

List of indices of disconnected atoms.

find_first_neighbours()

Method to find the first neighbours of the atoms of the motif. :rtype: list :return: List of the corresponding bonds.

find_lowest_coordination_atoms(include_first_neighbours=False)

Routine to determine the atoms of the solid with the lowest coordination atoms. TODO: Define parameter to specify storing atoms with coordinations lower than one specified.

Parameters:

include_first_neighbours (bool) – Includes in the list of atoms, also the first neighbours of those found originally. Defaults to False.

Return type:

List[int]

Returns:

List of indices of atoms.

find_neighbours(mode='minimal', nn=1, r=None)

From a list of atoms (motif), it sets the list of bonds within the solid. I.e.: Atom list -> List of neighbours/atom. By default it will look for the minimal distance between atoms to determine first neighbours. For amorphous systems the option radius is available to determine neighbours within a given radius R. Boundary conditions can also be set, either PBC (default) or OBC.

Parameters:

• mode (str) – Search mode, can be either ‘minimal’ or ‘radius’. Defaults to ‘minimal’.

• nn (int) – Next neighbour, used to specify up to which neighbour there are hoppings if mode=’minimal’

• r (float) – Value for radius sphere to detect neighbours if mode=’radius’

Return type:

None

hamiltonian_k(k)

Routine to evaluate the Bloch Hamiltonian at a given k point. It adds the k dependency of the Bloch Hamiltonian through the complex exponentials.

Parameters:

• k (Union[list, ndarray]) – k vector (Array 1x3)

• conditions – defaults to PBC. Can be either PBC or OBC

Return type:

ndarray

Returns:

Bloch Hamiltonian matrix

high_symmetry_path(nk, points)

Routine to generate a path in reciprocal space along the high symmetry points (HSPs) indicated. The HSPs must be within those corresponding to the associated symmetry group, otherwise the method will throw an error.

Parameters:

• nk (int) – Number of kpoints in the path. They are evenly distributed between HSPs.

• points (List[str]) – List of labels of desired HSPs.

Return type:

list

Returns:

List of kpoints following the path specified with the labels.

identify_boundary(alpha=0.4, ndim=2, verbose=False, corner_atoms=False)

Method to obtain the indices of the outermost atoms of the motif, taking into account boundary conditions. NB: Currently works only in 2D

Parameters:

• alpha (float) – This parameters tunes the shape of the boundary. For alpha=0, the boundary is a convex hull; as alpha increases the boundary becomes more concave. Default value is 0.4

• ndim (int) – Dimension of the expected boundary. Defaults to 2.

• verbose (bool) – Boolean parameter to print indices of boundary atoms. Defaults to False.

• cornes_atoms – Boolean to toggle detection of atoms that might be shared by the periodic boundary and the open boundary.

Return type:

ndarray

Returns:

Indices of atoms in the boundary.

Raises:

AssertionError – Crystal dimension different from two raises error. Also raises if bonds are not computed.

identify_convex_hull(loop=6)

Deprecated in favour of identify_motif_edges. Method to obtain the atoms that correspond to the edges of the system. This method uses directly the ConvexHull method from Scipy to identify the outmost points of the motif. Compute the ConvexHull three times, each time removing the points detected in the previous iteration.

NB: If given 3d points, they must not be coplanar for the algorithm to work. If they are, then the points must be given as 2d

Parameters:

loop (int) – Number of times the ConvexHull is applied, to increase number of points in the edge.

Return type:

ndarray

Returns:

Indices of atoms belonging to edges of motif.

identify_motif_edges(alpha=0.4, ndim=2)

Method to obtain the indices of the outermost atoms of the motif. Note that this method returns exclusively the outermost atoms, independently of the boundary conditions. To take into account boundary conditions, one should call identify_boundary() from System. NB: Currently works only in 2D

Parameters:

• alpha (float) – This parameters tunes the shape of the boundary. For alpha=0, the boundary is a convex hull; as alpha increases the boundary becomes more concave. Default value is 0.4

• ndim (int) – Dimension of the boundary. If the system lies on a plane, the boundary is two-dimensional, so ndim = 2 (default value).

Return type:

ndarray

Returns:

Indices of atoms in the boundary.

Raises:

• AssertionError – Crystal dimension different from two raises error.

• ValueError – Crystal motif must have at least three atoms. Raised by Delaunay class.

initialize_hamiltonian(find_bonds=True)

Routine to initialize the matrices that compose the Bloch Hamiltonian

Parameters:

find_bonds (bool) – Toggle to find bonds in the system before initializating the Hamiltonian.

Return type:

None

plot_2d_crystal(ax=None, fontsize=10, size=10)

Method to visualize the crystalline structure (Bravais lattice + motif) in two dimensions. If the crystal is 3D, it will plot only the x, y coordinates.

Parameters:

• ax – Axis to plot the projection of the crystal. If None, creates one.

• fontsize (int) – Size of labels. Defaults to 10.

• size (int) – Size of markers. Defaults to 10.

Return type:

None

plot_crystal(cell_number=1, crystal_name='')

Method to visualize the crystalline structure (Bravais lattice + motif).

Parameters:

• cell_number (int) – Number of cells appended in each direction with respect to the origin. Defaults to 1.

• crystal_name (str) – Name of the crystal. Defaults to the empty string.

Return type:

None

plot_wireframe(ax=None, linewidth=1, alpha=0.5, color='black')

Method to plot a wireframe of the system according to the detected bonds.

Parameters:

• ax (Axes) – Matplotlib axes object. If None, a new Axis object is created.

• linewidth (float) – Linewidth of the wireframe. Defaults to 1.

• alpha (float) – Sets wireframe transparency. Defaults to 0.5

• color – Wireframe color. Defaults to black.

Return type:

None

reciprocal_lattice()

Routine to compute the reciprocal lattice basis vectors from the Bravais lattice basis. The algorithm is based on the fact that a_idot b_j=2PIdelta_ij, which can be written as a linear system of equations to solve for b_j. Resulting vectors have 3 components independently of the dimension of the vector space they span.

Return type:

None

reduce(**ncells)

Routine to reduce the dimensionality of the System object along the specified directions, by repeating unit cells along those directions until a given size (number of unit cells) is reached. Thus we make the original system finite along those directions.

Parameters:

ncells (dict) – Number of cells of direction(s) along which we want to reduce the dimensionality of the system, n1, n2 and/or n3.

Return type:

System

Returns:

Modified System object.

remove_atom(index)

Method to remove the atom at the specified index from the motif. Note that this method does not remove existing bonds corresponding to the removed atom.

Parameters:

index (int) – Index of atom to be removed.

Return type:

None

remove_atoms(indices)

Method to remove the atoms specified in a list with all their indices. Built upon the method remove_atom.

Parameters:

indices (list) – List with the indices of the atoms to be removed.

Return type:

None

remove_disconnected_atoms()

Method to remove disconnected atoms from the motif. Based on the method remove_atoms.

restrict_lattice2rectangle()

TO BE IMPLEMENTED YET (is it really necessary?).

Return type:

None

ribbon(width, orientation='horizontal', periodic=False)

Routine to generate a ribbon for a 2D crystal. It is designed to create rectangular ribbons, based on a rectangular lattice. Therefore there must exist a combination of basis vectors such that we can obtain a rectangular supercell. Otherwise the method returns an error.

Parameters:

• width (int) – Width of the ribbon (number of cells).

• orientation (str) – ‘horizontal’ or ‘vertical’. Defaults to ‘horizontal’.

• periodic (bool) – Boolean to keep the ribbon periodic along the direction it is built. Defauts to False. Turning this one would correspond to a nanotube instead of nanoribbon (finite).

Return type:

System

Returns:

Modified System object.

rotate(angle=90)

Method to rotate the whole system in the xy plane by some angle.

Parameters:

angle (float) – Value of rotation angle in degrees. Defaults to 90 degrees.

Return type:

System

Returns:

Rotated system.

solve(kpoints=None, neigval=None)

Diagonalize the Hamiltonian to obtain the band structure and the eigenstates.

Parameters:

• kpoints (list) – List of kpoints where we want to diagonalize the Bloch Hamiltonian. Defaults to None, which corresponds to [[0., 0., 0.]] for both PBC and OBC.

• neigval (int) – Number of eigenvalues to compute if using sparse matrices. Defaults to None, which in turn is set to the filling.

Returns:

Spectrum object.

supercell(update=True, **ncells)

Routine to generate a supercell for a system using the number of cells along each Bravais vectors specified in ncells.

Parameters:

• update (bool) – Boolean parameter to update all of Crystal class atributes. Defaults to True, only set to false when used inside other routines (reduce)

• dict – Specify number of cells in some direction: n1, n2 and/or n3.

Return type:

System

Returns:

Modified System object.

update()

Routine to initialize or update the intrinsic attributes of the Crystal class whenever the Bravais lattice is changed.

Return type:

None

visualize()

Method to render a 3d visualization of the crystal using the self.vpython library.

Return type:

None