spinless hamiltonian (zero hopping)

[rp092]

Good evening, this one is the spinless hamiltonian I would like to run on HPhi

H = Δ Σ_ j (-1)^j n_j + V Σ_ j n_ j n_ j+1

with Δ = 1 and V = 1. Here is a sketch of the excitations of the system.

In the case L=6 nelec = 3 I'm expecting two kinds of excitations:
the first one with energy E = 2Δ + 2V, the second one with energy E= 2Δ + V.
So in the spectrum there would be two peaks, the first at E = 4 and the second E = 3 with double spectral weight respect to the first. I insert the staggered potential trought the trans.def file like this:

`

NTransfer 48

========i_j_s_tijs======

1     0     0     0         0.000000000000000         0.000000000000000
0     0     1     0         0.000000000000000        -0.000000000000000
1     1     0     1         0.000000000000000         0.000000000000000
0     1     1     1         0.000000000000000        -0.000000000000000
2     0     1     0         0.000000000000000         0.000000000000000
1     0     2     0         0.000000000000000        -0.000000000000000
2     1     1     1         0.000000000000000         0.000000000000000
1     1     2     1         0.000000000000000        -0.000000000000000
3     0     2     0         0.000000000000000         0.000000000000000
2     0     3     0         0.000000000000000        -0.000000000000000
3     1     2     1         0.000000000000000         0.000000000000000
2     1     3     1         0.000000000000000        -0.000000000000000
4     0     3     0         0.000000000000000         0.000000000000000
3     0     4     0         0.000000000000000        -0.000000000000000
4     1     3     1         0.000000000000000         0.000000000000000
3     1     4     1         0.000000000000000        -0.000000000000000
5     0     4     0         0.000000000000000         0.000000000000000
4     0     5     0         0.000000000000000        -0.000000000000000
5     1     4     1         0.000000000000000         0.000000000000000
4     1     5     1         0.000000000000000        -0.000000000000000
0     0     5     0         0.000000000000000         0.000000000000000
5     0     0     0         0.000000000000000        -0.000000000000000
0     1     5     1         0.000000000000000         0.000000000000000
5     1     0     1         0.000000000000000        -0.000000000000000
0     0     0     0         1.000000000000000         0.000000000000000
0     1     0     0         1.000000000000000         0.000000000000000
0     0     0     1         1.000000000000000         0.000000000000000
0     1     0     1         1.000000000000000         0.000000000000000
1     0     1     0        -1.000000000000000         0.000000000000000
1     1     1     0        -1.000000000000000         0.000000000000000
1     0     1     1        -1.000000000000000         0.000000000000000
1     1     1     1        -1.000000000000000         0.000000000000000
2     0     2     0         1.000000000000000         0.000000000000000
2     1     2     0         1.000000000000000         0.000000000000000
2     0     2     1         1.000000000000000         0.000000000000000
2     1     2     1         1.000000000000000         0.000000000000000
3     0     3     0        -1.000000000000000         0.000000000000000
3     1     3     0        -1.000000000000000         0.000000000000000
3     0     3     1        -1.000000000000000         0.000000000000000
3     1     3     1        -1.000000000000000         0.000000000000000
4     0     4     0         1.000000000000000         0.000000000000000
4     1     4     0         1.000000000000000         0.000000000000000
4     0     4     1         1.000000000000000         0.000000000000000
4     1     4     1         1.000000000000000         0.000000000000000
5     0     5     0        -1.000000000000000         0.000000000000000
5     1     5     0        -1.000000000000000         0.000000000000000
5     0     5     1        -1.000000000000000         0.000000000000000
5     1     5     1        -1.000000000000000         0.000000000000000`

and I calculate the density-density dynamical GF using that BASH code:

#!/bin/bash
export LC_NUMERIC="en_US.UTF-8"

Q_start=0
Q_end=0.5
Dq=0.01

mkdir GF
for q in seq $Q_start $Dq $Q_end
do
cat scenario_gf.in | sed 's/SpectrumQL = .*/SpectrumQL = '$q'/g' > scenario_$q.in
echo "Running scenario $q..."
HPhi -sdry scenario_$q.in > tmp
rm trans.def
cp Delta1/trans.def trans.def

HPhi -e namelist.def
cp output/zvo_DynamicalGreen.dat GF/dyn_gf_$q.dat
done

given that the output of the calculation is that spectrum

(it is in logartimic scale + an ε to avoid any divergence)

If you look at q=π (0.5 in fractional coordinates) and q=0 there are no excitations apart from the "elastic" terms of the GF at ω = 0. Are there any errors among my inputs or am I misunderstanding something in the calculation of the GF with HPhi?

Thanks a lot.
R.

[mitsuaki1987]

Dear rp092

Would you please provide us the scenario_gf.in ?

Best regards,
Mitsuaki Kawamura
ISSP, University of Tokyo

[mitsuaki1987]

Also,
How about the single-particle dynamical GF ?

[rp092]

Sure. I' write down here the new bash code that I'm using. If you put Q_start = Q_end and Delta_start = Delta_end the code runs only one time at fixed Q and Delta. In my formalism I use Q as the wave vector of the excitation. Next I'm using a python script to plot the GF.

#################################################################################
BASH code
#################################################################################

#!/bin/bash
export LC_NUMERIC="en_US.UTF-8"

if [ $# -ne 7 ]
then
echo "Pass me the scenario.in as first argument."
echo "Pass me Delta_start as second argument."
echo "Pass me Delta_end as third argument."
echo "Pass me Delta_Delta as fourth argument."
echo "Pass me Q_star as fifth argument."
echo "Pass me Q_end as sixth argument."
echo "Pass me Dq as seventh argument."
exit
fi

input=$1
input_gf=$input.gf

Delta_Start=$2
Delta_End=$3
Delta_Delta=$4

Q_start=$5
Q_end=$6
Dq=$7

cat $input | grep -v Eigenvec > $input_gf
cat >> $input_gf <<EOF
CalcSpec = "Normal"
SpectrumType = "Density"

SpectrumQW = 0
SpectrumQL = XXX

OmegaIM = -0.1
NOmega = 5000
EOF

L=cat $input | grep "L" | awk -F "=" '{name=$1; value=$2; gsub(" ", "", name); gsub(" ", "", value); if (name == "L") print value}'

mkdir -p Results
cp $input Results

cp $input_gf Results
cd Results

for delta in seq $Delta_Start $Delta_Delta $Delta_End
do
rm -f trans.def
HPhi -sdry $input > output_$delta
cat > generate_trans.py <<EOF

for i in range($L):
sign = (-1)*i/2.
for s1 in range(2):
for s2 in range(2):
line = "%5d %5d %5d %5d%26.15f%26.15f" % (i, s1, i, s2, sign$delta, 0)
print line
EOF
python generate_trans.py >> trans.def
cat trans.def | sed 's/NTransfer./NTransfer '$(($L8))'/g' > trans_new
mv trans_new trans.def

cp trans.def trans.def.bak

HPhi -e namelist.def > output_e_$delta

for q in seq $Q_start $Dq $Q_end
do
cat $input_gf | sed 's/XXX/'$q'/g' > scenario_$q.in

echo "Running scenario $delta | $q..."
HPhi -sdry scenario_$q.in > tmp
cp trans.def.bak trans.def

HPhi -e namelist.def > GF_calc_output_$delta_$q
outfile=printf "dyn_delta_%.1f_Q_%.2f.dat" $delta $q
cp output/zvo_DynamicalGreen.dat $outfile

energy=cat output/zvo_energy.dat | grep Energy | awk '{print $2}'

cat > shift_gf.py <<EOF

from numpy import *
GF = loadtxt("dyn_delta_${delta}Q$q.dat")
w = GF[:,0]
GF_Im = GF[:,3]

new_data = zeros( ( len(w), 2))
new_data[:,0] = w - $energy
new_data[:,1] = GF_Im

savetxt("green_function_D${delta}_Q$q.dat", new_data)
EOF
python shift_gf.py
done
done

#################################################################################
python code
#################################################################################
from numpy import *
from matplotlib.pyplot import *
import sys

if len(sys.argv) != 5:
print "Pass me the GS energy (first row in zvo_energy.dat) as first argument."
print "Pass me Delta (staggered potential) as second argument."
print "Pass me Q_start as third argument."
print "Pass me Q_end as fourth argument."
exit()

d_ENERGY = float(sys.argv[1])
Delta = float(sys.argv[2])
Q_start = float(sys.argv[3])
Q_end = float(sys.argv[4])

NQ = 101
q = linspace(Q_start, Q_end, NQ)

datas = []
for q_val in q:
data = loadtxt("Results/dyn_delta_%.1f_Q_%.2f.dat" % (Delta,q_val))
datas.append(data)

w = datas[0][:,0]
GF_im = zeros((len(w), NQ))

for i, data in enumerate(datas):
GF_im[:, i] = data[:,3]

savetxt("GF_im.dat", GF_im)
GF_im[abs(GF_im) > 40] = 40
w = w - d_ENERGY
imshow(abs(GF_im), aspect="auto", origin= "lower", extent = [ Q_start, Q_end, min(w), max(w)])
ylabel(r"$\omega$")
xlabel("q")
title("Imaginary part of the Dynamical Green function")
colorbar()

show()

[mitsuaki1987]

Dear rp092

Thank you.
Where can I see your input file for HPhi.

Also, did you try SpectrumType = "up" (or "down") ?

Best regards,
Mitsuaki Kawamura

[rp092]

Here is my input file scenario.in:

L = 6
model = "Fermion Hubbard"
method = "Lanczos"
lattice = "Chain Lattice"

U = 0
mu = 0
t = 0
V = 1.0
nelec = 3
2Sz = 3

EigenvecIO = "out"

I didn't try any other spectrum type beyond the density one.

[mitsuaki1987]

Why do you focus on the density-density correlation function ?
I think all eigenstates of this system are also eigenstates of every number operator.

[rp092]

Because I would like to study the electron-hole pair excitations spectrum on varying the hopping term t and the Coulomb repulsion V>0.

[mitsuaki1987]

Is it correct ?

[rp092]

What do you mean by n_{R0}?
So what you're saying to me is that I'm looking to the wrong GF?

[mitsuaki1987]

Sorry, I wrote too indirectly, and I modified the above formula.
What I wanted to say is

• Your dynamical correlation function can be derived analytically.
• The numerical result with HPhi agrees this analytical result.
• You are misunderstanding; there is no excitation in the density-density correlation function.
• If you compute the one-body correlation function, you can see peaks at E = 2Δ + 2V, 2Δ + V, and so on.

[rp092]

Thanks a lot.

R.