CalculateHybridizationFunction

Responsefunction.CalculateHybridizationFunction(G0,Sigma) calculates the interacting impurity bath Green's function. Given a lattice with local Green's function $G_0(\omega)$ and a local self energy $\Sigma(\omega)$. The full Green's function then is $G(\omega) = G_0(\omega-\Sigma(\omega))$. If we want to add a self energy on all sites, except for the site we are looking at we get $$G_{Bath} = \frac{1}{G_0(\omega-\Sigma(\omega))^{-1} - \Sigma(\omega)}$$ This Green's function can be used to define the hybridisation function of an Anderson impurity model representing a lattice. This is useful for the DMFT approximation where we define a lattice model with local interactions on all lattice sites. We replace the interactions on all sites but one by a local self energy.

1. $G_0$ the one particle Green's function for the non-interacting lattice. Given in one of the available response function formats.
2. $\Sigma$ the local self energy. Given in one of the available response function formats.
3. A list of options. Available are

   • EnergyGrid - a table of discrete energies used for the possible values of the Bath energies for a representation of $G_{Bath}(\omega)$ in Anderson matrix format.

1. A response function representing $G_{Bath}(\omega)$.

The example below uses some arbitrary definition for $G_0$ and $\Sigma$. Examples where these functions are used can be found in the tutorials.

Example.Quanty

a = {0, 0, -2,0,2}
b = {   1,  1,1,1}
G0 = ResponseFunction.New( { a, b, mu=0, type="And", name="G0"} )
 
a = {0,0,0,0}
b = {0.1,0.1,0.1}
Sigma = ResponseFunction.New( { a, b, mu=0, type="Tri", name="Sigma"} )
 
GHyb = ResponseFunction.CalculateHybridizationFunction(G0,Sigma)
 
print("The non interacting Green's function")
print(G0)
print("The self energy")
print(Sigma)
print("The bath Green's function defining the hybridization function for the full interacting bath")
print(GHyb)

The non interacting Green's function
{ { 0 , 0 , -2 , 0 , 2 } , 
  { 1 , 1 , 1 , 1 } ,
  name = G0 ,
  mu = 0 ,
  type = And }
The self energy
{ { 0 , 0 , 0 , 0 } , 
  { 0.1 , 0.1 , 0.1 } ,
  name = Sigma ,
  mu = 0 ,
  type = Tri }
The bath Green's function defining the hybridization function for the full interacting bath
{ { 0 , 0 , -2.0049999999223 , -0.16180339887499 , -0.14261254104408 , -0.14010864678943 , -0.061803398874989 , -0.0024961056676115 , 0.0024961056676115 , 0.061803398874989 , 0.14010864678943 , 0.14261254104408 , 0.16180339887499 , 2.0049999999223 } , 
  { 1 , 0.99874921790792 , 0.37174803446018 , 0.0238150684481 , 0.026242558358305 , 0.60150095500755 , 0.03527279934937 , 0.03527279934937 , 0.60150095500755 , 0.026242558358305 , 0.0238150684481 , 0.37174803446018 , 0.99874921790792 } ,
  name = GBath ,
  mu = 0 ,
  type = And }