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documentation:language_reference:functions:meanfieldoperator [2018/06/21 15:21]
Simon Heinze created
documentation:language_reference:functions:meanfieldoperator [2018/09/17 17:09] (current)
Simon Heinze sum over m and n
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 ### ###
-alligned paragraph text+//​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} 
 +
 +
 +\left( 
 +\frac{N_{Fermion} (N_{Fermion}-1)}{2} 
 +\right)^{-1} 
 +\sum_{m,​n} 
 +\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 ===== ===== Input =====
  
-  * bla Integer +  * $O$ Operator 
-  * bla2 Real+  * $rho$ 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 ===== ===== Output =====
  
-  * bla : real+  * $O_{MF}$ The mean-field approximated operator
  
 ===== Example ===== ===== Example =====
  
-### +
-description text +
-###+
  
 ==== Input ==== ==== Input ====
 <code Quanty Example.Quanty>​ <code Quanty Example.Quanty>​
--- some example code+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>​ </​code>​
  
 ==== Result ==== ==== Result ====
 <file Quanty_Output>​ <file Quanty_Output>​
-text produced as 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>​ </​file>​
  
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