Drude-Lorentz Spectral Density

\[J(\omega) = 2 \pi \sum_k |g_k|^2 \delta(\omega-\omega_k) = \frac{4\Delta W\omega}{\omega^2+W^2}\]

Here, $\Delta$ represents the coupling strength between system and the bosonic environment with band-width $W$.

Matsubara Expansion

With Matsubara Expansion, the correlation function can be analytically solved and expressed as follows:

\[C(t_1, t_2)=\sum_{l=1}^{\infty} \eta_l \exp(-\gamma_l (t_1-t_2))\]

with

\[\begin{aligned} \gamma_{1} &= W,\\ \eta_{1} &= \Delta W\left[-i+\cot\left(\frac{W}{2 k_B T}\right)\right],\\ \gamma_{l\neq 1} &= 2\pi l k_B T,\\ \eta_{l\neq 1} &= -2 k_B T \cdot \frac{2\Delta W \cdot \gamma_l}{-\gamma_l^2 + W^2}. \end{aligned}\]

This can be constructed by the built-in function Boson_DrudeLorentz_Matsubara:

Vs # coupling operator
Δ  # coupling strength
W  # band-width  of the environment
kT # the product of the Boltzmann constant k and the absolute temperature T
N  # Number of exponential terms
bath = Boson_DrudeLorentz_Matsubara(Vs, Δ, W, kT, N - 1)

Padé Expansion

With Padé Expansion, the correlation function can be analytically solved and expressed as the following exponential terms:

\[C(t_1, t_2)=\sum_{l=1}^{\infty} \eta_l \exp(-\gamma_l (t_1-t_2))\]

with

\[\begin{aligned} \gamma_{1} &= W,\\ \eta_{1} &= \Delta W\left[-i+\cot\left(\frac{W}{2 k_B T}\right)\right],\\ \gamma_{l\neq 1} &= \zeta_l k_B T,\\ \eta_{l\neq 1} &= -2 \kappa_l k_B T \cdot \frac{2\Delta W \cdot \zeta_l k_B T}{-(\zeta_l k_B T)^2 + W^2}, \end{aligned}\]

where the parameters $\kappa_l$ and $\zeta_l$ are described in J. Chem. Phys. 134, 244106 (2011). This can be constructed by the built-in function Boson_DrudeLorentz_Pade:

Vs # coupling operator
Δ  # coupling strength
W  # band-width  of the environment
kT # the product of the Boltzmann constant k and the absolute temperature T
N  # Number of exponential terms
bath = Boson_DrudeLorentz_Pade(Vs, Δ, W, kT, N - 1)