4.2. 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

(4.2)$$\begin{array}{r}\begin{array}{c}{j}_x=i\sum _{i,\sigma }(c_{{\mathbf{r}}_i+{\mathbf{e}}_x,\sigma }^{†}{c}_{{\mathbf{r}}_i,\sigma }-c_{{\mathbf{r}}_i,\sigma }^{†}{c}_{{\mathbf{r}}_i+{\mathbf{e}}_x,\sigma }),\\ I(\omega ,\eta )=\mathrm{I}\mathrm{m}[⟨0|j_x[H-(\omega -E_0-i\eta )I{]}^{-1}{j}_x|0⟩],\end{array}\end{array}$$

where ${\mathbf{e}}_x$ is the unit translational vector in the x direction. From this the regular part of the optical conductivity is defined as

(4.3)$${\sigma }_{\mathrm{r}\mathrm{e}\mathrm{g}}(\omega )=\frac{I(\omega ,\eta )+I(-\omega ,-\eta )}{\omega N_s},$$

where $N_{\mathrm{s}}$ is the number of sites.

An input file (samples/tutorial_4.2/stan1.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 samples/tutorial_4.2/.

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

sh ./All.sh

you can obtain optical.dat Note that samples/tutorial_4.2/OpticalSpectrum.py, samples/tutorial_4.1/lattice.py, samples/tutorial_4.2/lattice.py, and samples/tutorial_4.2/input.txt are necessary.

A way for plotting optical.dat is as follows

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