Lattice Green’s functions¶
Warning
This part of the library is largely experimental and subject to API breaks
This notebook shows a basic example of manipulation of \(G(\mathbf{k},\omega)\) defined on a 2D Brillouin zone.
from triqs.gf.multivar import *
from triqs.gf import *
from triqs.lattice import *
from triqs.lattice.tight_binding import *
from triqs.plot.mpl_interface import *
from matplotlib.pyplot import *
from numpy import cos, pi
Defining the meshes¶
Here we define a Bravais lattice object corresponding to a two-dimensional square lattice, as well as the corresponding Brillouin zone and a mesh defined on it.
BL = BravaisLattice(units = [(1,0,0) , (0,1,0) ]) #square lattice
nk=20
mk = MeshBrillouinZone(BrillouinZone(BL), nk)
miw=MeshImFreq(beta=1., S="Fermion", n_max=100) #not used (just demo)
mprod_iw = MeshBrillouinZoneImFreq(mk, miw) # not used (just demo)
mw=MeshReFreq(-5,5, 201)
mprod = MeshBrillouinZoneReFreq(mk, mw)
Definition of the Green’s function¶
Here we define the Green’s function as
\[G(\mathbf{k},i\omega) = \frac{1}{\omega+i\eta - \varepsilon(\mathbf{k}) - \Sigma(\omega)}\]
where \(\varepsilon(\mathbf{k}) = -2t\left(\cos(k_x)+\cos(k_y)\right)\) and \(\Sigma(\omega)\) is the atomic-limit self-energy:
\[\Sigma(\omega) = \frac{U^2}{4\omega}\]
#let us fill two G_k_w
G_w = GfReFreq(mesh=mw, shape=[1,1])
t=-0.25
U=5.0
eta=0.2
G_k_w = GfBrillouinZone_x_ReFreq(mprod, [1,1])
G_k_w_Mott = GfBrillouinZone_x_ReFreq(mprod, [1,1])
ik=0
for k in G_k_w.mesh.components[0]:
G_w << inverse(iOmega_n-2*t*(cos(k[0])+cos(k[1]))+eta*1j)
G_k_w.data[ik,:,0,0]=G_w.data[:,0,0]
G_w << inverse(iOmega_n-2*t*(cos(k[0])+cos(k[1]))+eta*1j - 0.25*U**2*inverse(iOmega_n+eta*1j))
G_k_w_Mott.data[ik,:,0,0]=G_w.data[:,0,0]
ik+=1
Various plots¶
We plot various slices of \(G(\mathbf{k},\omega)\) corresponding to \(U=0\) and \(U=4\).
figure()
gs=GridSpec(2,2)
subplot(gs[0])
oplot(G_k_w[0,0], path=[(0,0),(pi,pi),(pi,0),(0,0)], method="cubic", mode="I", cmap=cm.terrain)
colorbar()
#color plot of slice at constant omega in the Brillouin zone
#slice_at_const_w2 takes the linear frequency index as input
#method can be "nearest","linear", "cubic"
#mode can be "I" (imaginary) or "R" (real)
subplot(gs[1],aspect="equal")
oplot(G_k_w.slice_at_const_w2(len(G_k_w.mesh.components[1])/2), type="contourf", mode="I", method="cubic", cmap=cm.terrain)
colorbar()
#plot of slice at constant omega on a high-symmetry path
subplot(gs[2])
oplot(G_k_w.slice_at_const_w2(len(G_k_w.mesh.components[1])/2), path=[(0,0),(pi,pi),(pi,0),(0,0)], method="cubic",\
mode="I", label=r"$\mathrm{Im}G(\mathbf{k},\omega=0)$")
#plot of slice at constant k
#slice_at_const_w1 takes the integer coordinates of the k point
subplot(gs[3])
oplot(G_k_w.slice_at_const_w1([0,0,0]), label=r"$\mathrm{Im}G(\mathbf{k}=(0,0), \omega)$", mode="I")
oplot(G_k_w.slice_at_const_w1([nk/2,nk/2,0]), label=r"$\mathrm{Im}G(\mathbf{k}=(\pi,\pi), \omega)$", mode="I")
tight_layout()
figure()
gs=GridSpec(1,2)
subplot(gs[0])
oplot(G_k_w_Mott, path=[(0,0),(pi,pi),(pi,0),(0,0)], method="cubic", mode="I", cmap=cm.terrain)
colorbar()
subplot(gs[1])
oplot(G_k_w_Mott.slice_at_const_w1([0,0,0]), label=r"$\mathrm{Im}G(\mathbf{k}=(0,0), \omega)$", mode="I")
oplot(G_k_w_Mott.slice_at_const_w1([nk/2,nk/2,0]), label=r"$\mathrm{Im}G(\mathbf{k}=(\pi,\pi), \omega)$", mode="I")
tight_layout()
show()