The BravaisLattice and TightBinding classes: definitions and example
The following example is aimed at demonstrating the use of TRIQS Lattice tools.
BravaisLattice
A BravaisLattice is constructed as
BravaisLattice(units, orbital_positions ) where
unitsis the list the coordinates (given as triplets) of the basis vectors \(\lbrace \mathbf{e}_i \rbrace _{i=1\dots d}\) (\(d\) is the dimension)orbital_positionsis a dictionary of the atoms forming the basis of the Bravais Lattice: the key is the name of the atom/orbital, the value is the triplet of its coordinates.
TightBinding
A tight-binding lattice is defined by the relation:
where \(\mathbf{t}_i\) is the matrix of the hoppings from a documentation/manual/triqs unit cell (\(\mathbf{R}=O\)) to a unit cell indexed by \(\mathbf{R}\). \((\mathbf{t}_\mathbf{R})_{n,m}\) is the tight-binding integral between atom \(n\) of site \(O\) and atom \(m\) of site \(\mathbf{R}\), ie
where \(\phi_n(\mathbf{r}-\mathbf{R})\) is the Wannier orbital of
atom \(n\) centered at site \(\mathbf{R}\). The corresponding
class in Lattice Tools is the TightBinding class. Its instances
are constructed as follows:
TightBinding ( bravais_lattice, hopping_dictionary) where
bravais_latticeis an instance ofBravaisLatticehopping_dictionaryis a dictionary of the hoppings \(\mathbf{t}_\mathbf{R}\), where the keys correspond to the coordinates of \(\mathbf{R}\) in the unitary basis \(\lbrace \mathbf{e}_i \rbrace _{i=1\dots d}\), and the values to the corresponding matrix: \((\mathbf{t}_\mathbf{R})_{n,m}\)
energies_on_bz_path
The function energies_on_bz_path (TB, start, end, n_pts) returns a
\(n_{at} \times n_{pts}\) matrix \(E\) such that
E[n,k]\(= \epsilon_n(\mathbf{k})\)
where k indexes the n_pts \(\mathbf{k}\)-points of the
line joining start and end, and \(\epsilon_n(k)\) is the
\(n\)th eigenvector of \(t_\mathbf{k}\).
Example
The following example illustrates the usage of the above tools for the
case of a two-dimensional, square lattice with various unit cells. We
successively construct three Bravais lattices BL_1, BL_2 and
BL_4 with, respectively, 1, 2 and 4 atoms per unit cell, as well as
three tight-binding models with hopping dictionaries hop_1,
hop_2 and hop_4
from numpy import array, zeros
import math
from triqs.lattice.tight_binding import *
# Define the Bravais Lattice : a square lattice in 2d
BL_1 = BravaisLattice(units = [(1,0,0), (0,1,0) ], orbital_positions= [(0,0,0)] )
BL_2 = BravaisLattice(units = [(1,1,0) , (-1,1,0) ], orbital_positions= [ (0,0,0),(.5,.5,0)] )
BL_4 = BravaisLattice(units = [(2,0,0) , (0,2,0) ], orbital_positions= [(0,0,0),(0,.5,0), (.5,0,0), (.5,.5,0)] )
# Hopping dictionaries
t = .25; tp = -.1;
hop_1= { (1,0) : [[ t]], (-1,0) : [[ t]], (0,1) : [[ t]], (0,-1) : [[ t]],
(1,1) : [[ tp]], (-1,-1): [[ tp]], (1,-1) : [[ tp]], (-1,1) : [[ tp]]
}
hop_2= { (0,0) :[[0.,t],
[t,0.]],
(1,0) : [[ tp, 0],
[ t ,tp]],
(-1,0) : [[ tp, t],
[ 0 ,tp]],
(0,1) :[[ tp, 0],
[ t, tp]],
(0,-1) :[[ tp, t],
[ 0 ,tp]],
(1,1) : [[ 0, 0],
[ t,0]],
(-1,-1) :[[ 0, t],
[ 0,0]],
(-1,1) : [[ 0, 0],
[ 0,0]],
(1,-1) : [[ 0, 0],
[ 0,0]]
}
hop_4= { (0,0) :[[0.,t, tp,t],
[t,0., t,tp],
[tp,t,0,t],
[t,tp,t,0]],
(1,0) : [[0.,0, 0,0],
[t,0.,0,tp],
[tp,0,0,t],
[0,0,0,0]],
(-1,0) : [[0.,t, tp,0],
[0,0.,0,0],
[0,0,0,0],
[0,tp,t,0]],
(0,1) : [[0.,0, 0,0],
[0,0.,0,0],
[tp,t,0,0],
[t,tp,0,0]],
(0,-1) :[[0.,0, tp,t],
[0,0.,t,tp],
[0,0,0,0],
[0,0,0,0]],
(1,1) : [[0.,0, 0,0],
[0,0.,0,0],
[tp,0,0,0],
[0,0,0,0]],
(-1,-1) : [[0.,0, tp,0],
[0,0.,0,0],
[0,0,0,0],
[0,0,0,0]],
(-1,1) : [[0.,0, 0,0],
[0,0.,0,0],
[0,0,0,0],
[0,tp,0,0]],
(1,-1) :[[0.,0, 0,0],
[0,0.,0,tp],
[0,0,0,0],
[0,0,0,0]],
}
TB_1 = TightBinding(BL_1, hop_1)
TB_2 = TightBinding(BL_2, hop_2)
TB_4 = TightBinding(BL_4, hop_4)
# High-symmetry points
Gamma = array([0. ,0. ]);
PiPi = array([math.pi ,math.pi ])*1/(2*math.pi);
Pi0 = array([math.pi ,0 ])*1/(2*math.pi);
PihPih = array([math.pi/2 ,math.pi/2])*1/(2*math.pi)
TwoPi0 = array([2*math.pi ,0 ])*1/(2*math.pi);
TwoPiTwoPi= array([math.pi*2 ,math.pi*2])*1/(2*math.pi)
n_pts=50
# Paths along high-symmetry directions
path_1=[Gamma,Pi0,PiPi,Gamma]
path_2=[Gamma,PiPi,TwoPi0,Gamma] #equivalent to path_1 in coordinates of 2at/ucell basis
path_4=[Gamma,TwoPi0,TwoPiTwoPi,Gamma] #equivalent to path_1 in coordinates of 4at/ucell basis
def energies_on_path(path, TB, n_pts, n_orb=1):
E=zeros((n_orb,n_pts*(len(path)-1)))
for i in range(len(path)-1,0,-1):
energies = energies_on_bz_path (TB, path[i-1], path[i], n_pts)
for orb in range(n_orb): E[orb,(i-1)*n_pts:(i)*n_pts]=energies[orb,:]
print("index of point #"+str(i-1)+" = "+str((i-1)*n_pts))
return E
E_1= energies_on_path(path_1,TB_1,n_pts,1)
E_2= energies_on_path(path_2,TB_2,n_pts,2)
E_4= energies_on_path(path_4,TB_4,n_pts,4)
from matplotlib import pylab as plt
plt.plot(E_1[0], '--k', linewidth=4, label = "1 at/unit cell")
plt.plot(E_2[0],'-.g', linewidth=4, label = "2 ats/unit cell")
plt.plot(E_2[1],'-.g', linewidth=4)
plt.plot(E_4[0],'-r', label = "4 ats/unit cell")
plt.plot(E_4[1],'-r')
plt.plot(E_4[2],'-r')
plt.plot(E_4[3],'-r')
plt.grid()
plt.legend()
plt.axes().set_xticks([0,50,100,150])
plt.axes().set_xticklabels([r'$\Gamma_1$',r'$M_1$',r'$X_1$',r'$\Gamma_1$'])
plt.ylabel(r"$\epsilon$")