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