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https://github.com/LCPQ/DEHam
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35 KiB
35 KiB
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1. t-J Model¶
1. t-J Hamiltonian¶
a. Basis¶
The $t-J$ Hamiltonian with a S=1/2 spin-model is constructed in a basis of $\mathcal{R}^3$ given by the vector B as follows:
$$ B= \begin{bmatrix} 0 \\ \uparrow \\ \downarrow \end{bmatrix} $$b.) Operators¶
The Hamiltonian is constructed using two types of operators.
- The particle creation and annihilation operators $\hat{C}^\dagger,\hat{C}$
- The spin ladder operators $\hat{S}^+, \hat{S}^-, \hat{S}^z$
The matrix form of the five operators in the above basis is given as follows:
$$ \hat{C}^{\dagger}_{\uparrow}= \left( \hat{C}_{\uparrow} \right)^\dagger= \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$$$ \hat{C}^{\dagger}_{\downarrow}= \left( \hat{C}_{\downarrow} \right)^\dagger= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} $$$$ \hat{S}^{z}\ \ \ \ =\ \ \ \ \ \begin{bmatrix} 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & -\frac{1}{2} \end{bmatrix} $$$$ \hat{S}^{+}_{\downarrow}= \left( \hat{S}^- \right)^\dagger= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} $$c.) Hamiltonian¶
Nearest Neighbor coupling
The $t-J$ Hamiltonian is written in the above basis with the above five operators and the two parameters $t$ which governs the kinetic energy of the particles and the magnetic-coupling parameter $J$ which governs the coupling of the localized spins on neighboring sites.
$$
\hat{H_{t-J}} = \sum^{N-1}_i \left\{ t_{i,i+1} \sum_{\sigma} \left(\hat{C}^{\dagger}_{\sigma,i}\cdot\hat{C}_{\sigma,i+1} + h.c. \right) + J_{i,i+1} \left( \frac{\hat{S}^+_{i}\cdot\hat{S}^-_{i+1} + \hat{S}^-_{i}\cdot\hat{S}^+_{i+1}}{2} + \hat{S}^z_i\cdot\hat{S}^z_{i+1} \right)\right\}
$$
$h.c.$ stands for Hermitian conjugate.
2. 2D Lattice¶
1.) Labeling¶
Here a 4x4 lattice is shown with periodic boundary conditions where each site and bond is labelled.
In [150]:
import matplotlib as mplt
import networkx as nx
import numpy as np
import matplotlib.pyplot as plt
Lx = 4
Ly = 4
#fig = plt.figure()
#draw2DLattice(Lx,Ly,fig)
draw2DLattice(Lx,Ly)
#fig.savefig("/tmp/graph.png",dpi=1000)
- The DEHam program requires an input file which has the topology of the Hamiltonian and the various parameters as explained below in a sample inputfile:
8 # The number of orbitals (total)
1 # The largest number of non-zero elements per row (Multiple of Ndet)
1 # The total number of processors used in parallel (Multiple of Ndet)
1 # The number of holes
0 # The isz (ms-1/2) value
true # Restrict the hole to the 1'st (i.e. half of natom) Family of states. *false* for no restrictions
1,2,3,1,2,3,4,5,6,7 # (L1) The topology of the system is specified here
2,3,4,8,7,6,5,6,7,8 # (L2) first and second line contain the two sites linked
1,1,1,2,2,2,2,3,3,3 # (ltype) third line contains the type of link (1 for t or J, 2 for K and 3 for none)
.1430,-0.20,0.0000 # The three types of links this line gives Jz, Kz
.1430,-0.20,0.0000 # The three types of links this line gives Jx,y; Kx,y
-1.00,0.0,0.00 # This line gives t(i,i+1), t(i,i+2), t(i,i+3)
0.,0.,0.,0.,0.,0.,0.,0.,0. # Energy of each orbital + additional energy
2 # The total number of roots
1 # I The position of the first
1 # I SBox
1 # I
1 # I
1 # II The positions of the second
1 # II SBox
1 # II
1 # II
1 # III
1 # III The positions of the third
1 # III SBox
1 # III
1 # positio of the hole
0 # fix the position of the first hole during the CI
0 # fix the position of the second hole during the CI
0 # Print the wavefunction. It is stored in the FIL666 file after the run
a.) Script for $t-J$ input file¶
In [139]:
l1, l2, ltype = getInput(Lx,Ly)
print(l1)
print(l2)
print(ltype)
3.) Misc¶
1. Draw Lattice¶
In [149]:
"""
Function that draws a 2D Lattice
"""
def draw2DLattice(Lx,Ly):
n=Lx+2
m=Ly+2
G = nx.grid_2d_graph(n,m) # Lx, Ly grid
pos = dict( (n, n) for n in G.nodes() );
#labels = Dict( ((i, j), i + (Ly-j) * (Lx+1) ) for (i,j) in G.nodes() );
#edges=Dict()
#for (i,edge) in enumerate(G.edges)
# edges[edge]=i
# print(i,"\t",edge,"\n")
#end
for j in range(0,m-1):
G.remove_edge((0,j),(0,j+1))
G.remove_edge((n-1,j),(n-1,j+1))
for i in range(0,n-1):
G.remove_edge((i,0),(i+1,0))
G.remove_edge((i,m-1),(i+1,m-1))
edgescenter = dict()
countedge = 1
for j in range(1,m-1):
for i in range(1,n-1):
edgescenter[((i,j),(i+1,j))] = countedge
countedge += 1
for j in range(1,m-1):
for i in range(1,n-1):
edgescenter[((i,j),(i,j+1))] = countedge
countedge += 1
labelscenter = dict()
countlabels = 1
for j in range(1,m-1):
for i in range(1,n-1):
labelscenter[(i,j)]=countlabels
countlabels+=1
color_map=[]
countcolor=0
for i in range(0,n):
for j in range(0,m):
if i==0 or j==0 or i==n-1 or j==m-1:
color_map.append("white")
else:
if(countcolor%2 == 0):
color_map.append("blue")
else:
color_map.append("red")
countcolor+=1
countcolor+=1
# nx.draw(G, pos=pos, labels=labelscenter, node_color=color_map, font_color="white", ax=fig.add_subplot(111));
# nx.draw_networkx_edge_labels(G,pos,edge_labels=edgescenter,font_color="black", ax=fig.add_subplot(111));
nx.draw(G, pos=pos, labels=labelscenter, node_color=color_map, font_color="white");
nx.draw_networkx_edge_labels(G,pos,edge_labels=edgescenter,font_color="black");
2. Generate Input¶
In [132]:
"""
Function that generates the input
"""
def getInput(Lx,Ly):
n=Lx+2
m=Ly+2
labelscenter = dict()
countlabels = 1
for j in range(1,m-1):
for i in range(1,n-1):
labelscenter[(i,j)]=countlabels
countlabels+=1
edgescenter = dict()
countedge = 1
for j in range(1,m-1):
for i in range(1,n-1):
edgescenter[((i,j),(i+1,j))] = countedge
countedge += 1
for j in range(1,m-1):
for i in range(1,n-1):
edgescenter[((i,j),(i,j+1))] = countedge
countedge += 1
l1 = []
l2 = []
itype = [1 for x in range(0,len(edgescenter))]
for x in edgescenter:
if Lx + 1 not in x[1]:
l1.append(labelscenter[x[0]])
l2.append(labelscenter[x[1]])
else:
l1.append(labelscenter[x[0]])
if x[1][0] > Lx:
l2.append(labelscenter[(x[1][0]-Lx,x[1][1])])
elif x[1][1] > Ly:
l2.append(labelscenter[(x[1][0],x[1][1]-Ly)])
# print(l1,"\n")
# print(l2,"\n")
# print(itype,"\n")
return l1,l2,itype
In [133]:
getInput(3,3)
Out[133]:
header=r"""
"""
In [ ]: