Fermion_Lorentz.md

[Lorentz Spectral Density](@id doc-Fermion-Lorentz)

$$J(\omega)=\frac{\Gamma W^2}{(\omega-\mu)^2+W^2}$$

Here, \Gamma represents the coupling strength between system and the fermionic environment with chemical potential \mu and band-width W.

Matsubara Expansion

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

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

with

$$\begin{aligned} \gamma_{1}^{\nu} &= W-\nu i \mu,\\\ \eta_{1}^{\nu} &= \frac{\Gamma W}{2} f\left(\frac{iW}{k_B T}\right),\\\ \gamma_{l\neq 1}^{\nu} &= \zeta_l k_B T - \nu i \mu,\\\ \eta_{l\neq 1}^{\nu} &= -i k_B T \cdot \frac{\Gamma W^2}{-(\zeta_l k_B T)^2+W^2},\\\ f(x) &= \{\exp(x) + 1\}^{-1}, \end{aligned}$$

where \zeta_l=(2 l - 1)\pi. This can be constructed by the built-in function Fermion_Lorentz_Matsubara:

ds # coupling operator
Γ  # coupling strength
μ  # chemical potential of the environment
W  # band-width  of the environment
kT # the product of the Boltzmann constant k and the absolute temperature T
N  # Number of exponential terms for each correlation functions (C^{+} and C^{-})
bath = Fermion_Lorentz_Matsubara(ds, Γ, μ, W, kT, N - 1)

Padé Expansion

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

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

with

$$\begin{aligned} \gamma_{1}^{\nu} &= W-\nu i \mu,\\\ \eta_{1}^{\nu} &= \frac{\Gamma W}{2} f\left(\frac{iW}{k_B T}\right),\\\ \gamma_{l\neq 1}^{\nu} &= \zeta_l k_B T - \nu i \mu,\\\ \eta_{l\neq 1}^{\nu} &= -i \kappa_l k_B T \cdot \frac{\Gamma W^2}{-(\zeta_l k_B T)^2+W^2},\\\ f(x) &= \frac{1}{2}-\sum_{l=2}^{N} \frac{2\kappa_l x}{x^2+\zeta_l^2}, \end{aligned}$$

where the parameters \kappa_l and \zeta_l are described in J. Chem. Phys. 134, 244106 (2011) and N represents the number of exponential terms for C^{\nu=\pm}. This can be constructed by the built-in function Fermion_Lorentz_Pade:

ds # coupling operator
Γ  # coupling strength
μ  # chemical potential of the environment
W  # band-width  of the environment
kT # the product of the Boltzmann constant k and the absolute temperature T
N  # Number of exponential terms for each correlation functions (C^{+} and C^{-})
bath = Fermion_Lorentz_Pade(ds, Γ, μ, W, kT, N - 1)