5. Tutorial for calculations of dynamical properties

5.1. Dynamical spin structure factor

Let’s solve the following Hubbard model on the square lattice.

(5.1)\[H = -t \sum_{\langle i,j\rangle , \sigma}(c_{i\sigma}^{\dagger}c_{j\sigma}+{\rm H.c.})+U \sum_{i} n_{i\uparrow}n_{i\downarrow}\]

The input files (stan1.in and stan2.in) for 8-site Hubbard model are as follows

stan1.in

a0W = 2
a0L = 2
a1W = -2
a1L = 2
model = "hubbard"
method = "CG"
lattice = "square"
t = 1.0
t' = 0.5
U = 4.0
2Sz = 0
nelec = 8
EigenvecIO = "out"
stan2.in

a0W = 2
a0L = 2
a1W = -2
a1L = 2
model = "hubbard"
method = "CG"
lattice = "square"
t = 1.0
t' = 0.5
U = 4.0
2Sz = 0
nelec = 8
LanczosEPS = 8
CalcSpec = "Normal"
SpectrumType = "SzSz"
SpectrumQW = 0.5
SpectrumQL = 0.5
OmegaMin = -10.0
OmegaMax = 20.0
OmegaIM = 0.2
OmegaOrg = 10.0

You can execute HPhi as follows

HPhi -s stan1.in
HPhi -s stan2.in

After finishing calculations, the spectrum \(G_{S_z S_z}({\bf Q} \equiv (\pi, \pi), \omega) = \langle S_z(-{\bf Q}) \left[H-\omega-\omega_0 + i\eta\right]^{-1}S_z({\bf Q})\rangle\) is outputted in output/zvo_DynamicalGreen.dat. Here, \(S_z({\bf Q})= \sum_{i}e^{i {\bf Q} \cdot {\bf r}_i} S_z^i\) and the frequency \(\omega\) moves from \(-10\) to \(10\), \(\omega_0 = 10\), and \(\eta\) is set as \(0.2\). You can check the result by executing the following command on gnuplot:

gnuplot
gnuplot> set xlabel "Energy"
gnuplot> set ylabel "G_{SzSz}(E)"
gnuplot> set xzeroaxis
gnuplot> plot "output/zvo_DynamicalGreen.dat" u 1:3 w l tit "Real", \
> "output/zvo_DynamicalGreen.dat" u 1:4 w l tit "Imaginary"

You can see the following output image.

_images/spectrum.png

5.2. Hubbard chain (optical conductivity)

Here, we calculate the optical conductivity for the one-dimensional Hubbard model.

The optical conductivity \(\sigma(\omega)\) can be calculated from the current-current correlation \(I(\omega,\eta)\), which is defined as

(5.2)\[ \begin{align}\begin{aligned}j_{x}={i}\sum_{i,\sigma}(c_{{\bf r}_{i}+{\bf e}_{x},\sigma}^{\dagger}c_{{\bf r}_{i},\sigma}-c_{{\bf r}_{i},\sigma}^{\dagger}c_{{\bf r}_{i}+{\bf e}_{x},\sigma}),\\I(\omega,\eta)={\rm Im}\Big[\langle 0|j_{x}[H-(\omega-E_{0}-{i}\eta)I]^{-1}j_{x}|0\rangle\Big],\end{aligned}\end{align} \]

where \({\bf e}_{x}\) is the unit translational vector in the x direction. From this the regular part of the optical conductivity is defined as

(5.3)\[\sigma_{\rm reg}(\omega)=\frac{I(\omega,\eta)+I(-\omega,-\eta)}{\omega N_{s}},\]

where \(N_{\rm s}\) is the number of sites.

An input file (stan.in) for 6-site Hubbard model is as follows:

model = "Hubbard"
method = "CG"
lattice = "chain"
L = 6
t = 1
U = 10
2Sz = 0
nelec = 6
exct = 1
EigenVecIO  = "out"

Scripts for calculating the optical conductivity are available at HPhi/tool/ForOpticalConductivity.

By performing the all-in-one script (All.sh),

sh ./All.sh

you can obtain optical.dat.

A way for plotting optical.dat is as follows

plot "optical.dat" u 1:(-($4+$8)/$1) w l