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—documentation:language_reference:functions:meanfieldoperator [2018/07/23 17:08] – Created Simon Heinze
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 +====== MeanFieldOperator ======
 +
 +###
 +//MeanFieldOperator($O$, $\rho$)// creates the mean-field version of operator $O$ with the corresponding density matrix $\rho$.
 +$rho$ stores the expectation values of $a^{\dagger}_{\tau}a^{\phantom{\dagger}}_{\tau'}$, a table of dimensions $NFermion$ by $NFermion$.
 +
 +Any two particle parts of the operator will be replaced in mean-field, using the Hartree-Fock approximation by:
 +\begin{eqnarray}
 +a^{\dagger}_{i}a^{\dagger}_{j}a^{\phantom{\dagger}}_{k}a^{\phantom{\dagger}}_{l} &\to&\\
 +\nonumber &-& a^{\dagger}_{i}a^{\phantom{\dagger}}_{k} \langle a^{\dagger}_{j}a^{\phantom{\dagger}}_{l} \rangle \\
 +\nonumber &+& a^{\dagger}_{i}a^{\phantom{\dagger}}_{l} \langle a^{\dagger}_{j}a^{\phantom{\dagger}}_{k} \rangle \\
 +\nonumber &+& a^{\dagger}_{j}a^{\phantom{\dagger}}_{k} \langle a^{\dagger}_{i}a^{\phantom{\dagger}}_{l} \rangle \\
 +\nonumber &-& a^{\dagger}_{j}a^{\phantom{\dagger}}_{l} \langle a^{\dagger}_{i}a^{\phantom{\dagger}}_{k} \rangle \\
 +\nonumber &-& \langle a^{\dagger}_{i}a^{\phantom{\dagger}}_{l} \rangle \langle a^{\dagger}_{j}a^{\phantom{\dagger}}_{k} \rangle \\
 +\nonumber &+& \langle a^{\dagger}_{i}a^{\phantom{\dagger}}_{k} \rangle \langle a^{\dagger}_{j}a^{\phantom{\dagger}}_{l} \rangle 
 +\end{eqnarray}
 +
 +If the option AddDFTSelfInteraction was set to true more terms are added to the Mean-Field Operator, namely
 +\begin{equation}
 +\sum_{m} U \langle a^\dagger_m a^{\phantom{\dagger}}_m \rangle a^\dagger_m a^{\phantom{\dagger}}_m
 +\end{equation}
 +where
 +\begin{equation}
 +U
 +=
 +\left(
 +\frac{N_{Fermion} (N_{Fermion}-1)}{2}
 +\right)^{-1}
 +\sum_m
 +\left(
 +U_{m\,n\,n\,m}
 +-
 +U_{m\,n\,m\,n}
 +\right)
 +\end{equation}
 +is the average interaction energy electrons have with one another.
 +###
 +
 +===== Input =====
 +
 +  * $O$ : Operator
 +  * $rho$ : Matrix (Table of Table of length $O.NF$) of doubles
 +  * Possible options are:
 +    * "AddDFTSelfInteraction" bool defining if the electron self-interaction is to be included. (Standard false)
 +
 +===== Output =====
 +
 +  * $O_{MF}$ The mean-field approximated operator
 +
 +===== Example =====
 +
 +
 +
 +==== Input ====
 +<code Quanty Example.Quanty>
 +NF = 4
 +op = NewOperator("Number",NF,{1},{1},{0.1+I}) + NewOperator("U",NF,{0},{1},{5}) + 3
 +rho = {{0.7,0.3+I,0,0},{0.3-I,0.4,0,0},{0,0,0,0},{0,0,0,0}}
 +
 +print("Full Operator:")
 +print(op)
 +print("\nDensity:")
 +print(rho)
 +print("\nMeanFieldOperator:")
 +print( MeanFieldOperator(op, rho) )
 +print("\nMeanFieldOperator with electron self-interaction:")
 +print( MeanFieldOperator(op, rho, {{"AddDFTSelfInteraction",true}}) )
 +</code>
 +
 +==== Result ====
 +<file Quanty_Output>
 +Full Operator:
 +
 +Operator: CrAn
 +QComplex         =          2 (Real==0 or Complex==1 or Mixed==2)
 +MaxLength        =          4 (largest number of product of lader operators)
 +NFermionic modes =          4 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis)
 +NBosonic modes   =          0 (Number of bosonic modes (phonon modes, ...) in the one particle basis)
 +
 +Operator of Length   0
 +QComplex      =          0 (Real==0 or Complex==1)
 +N             =          1 (number of operators of length   0)
 +|  3.000000000000000E+00
 +
 +Operator of Length   2
 +QComplex      =          1 (Real==0 or Complex==1)
 +N             =          1 (number of operators of length   2)
 +C  1 A  1 |  1.000000000000000E-01  1.000000000000000E+00
 +
 +Operator of Length   4
 +QComplex      =          0 (Real==0 or Complex==1)
 +N             =          1 (number of operators of length   4)
 +C  1 C  0 A  1 A  0 | -5.000000000000000E+00
 +
 +
 +
 +Density:
 +{ { 0.7 , (0.3 + 1 I) , 0 , 0 } , 
 +  { (0.3 - 1 I) , 0.4 , 0 , 0 } , 
 +  { 0 , 0 , 0 , 0 } , 
 +  { 0 , 0 , 0 , 0 } }
 +
 +MeanFieldOperator:
 +
 +Operator: 
 +QComplex         =          0 (Real==0 or Complex==1 or Mixed==2)
 +MaxLength        =          4 (largest number of product of lader operators)
 +NFermionic modes =          4 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis)
 +NBosonic modes   =          0 (Number of bosonic modes (phonon modes, ...) in the one particle basis)
 +
 +Operator of Length   0
 +QComplex      =          0 (Real==0 or Complex==1)
 +N             =          1 (number of operators of length   0)
 +|  1.255000000000000E+01
 +
 +Operator of Length   2
 +QComplex      =          1 (Real==0 or Complex==1)
 +N             =          4 (number of operators of length   2)
 +C  1 A  1 | -3.400000000000000E+00  1.000000000000000E+00
 +C  1 A  0 |  1.500000000000000E+00  5.000000000000000E+00
 +C  0 A  1 |  1.500000000000000E+00 -5.000000000000000E+00
 +C  0 A  0 | -2.000000000000000E+00  0.000000000000000E+00
 +
 +
 +
 +MeanFieldOperator with electron self-interaction:
 +
 +Operator: 
 +QComplex         =          0 (Real==0 or Complex==1 or Mixed==2)
 +MaxLength        =          4 (largest number of product of lader operators)
 +NFermionic modes =          4 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis)
 +NBosonic modes   =          0 (Number of bosonic modes (phonon modes, ...) in the one particle basis)
 +
 +Operator of Length   0
 +QComplex      =          0 (Real==0 or Complex==1)
 +N             =          1 (number of operators of length   0)
 +|  1.255000000000000E+01
 +
 +Operator of Length   2
 +QComplex      =          1 (Real==0 or Complex==1)
 +N             =          4 (number of operators of length   2)
 +C  1 A  1 | -3.066666666666666E+00  1.000000000000000E+00
 +C  1 A  0 |  1.500000000000000E+00  5.000000000000000E+00
 +C  0 A  1 |  1.500000000000000E+00 -5.000000000000000E+00
 +C  0 A  0 | -1.416666666666667E+00  0.000000000000000E+00
 +</file>
 +
 +===== Table of contents =====
 +{{indexmenu>.#1}}
  

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