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documentation:language_reference:objects:responsefunction:functions:new [2024/12/17 16:20] – created Maurits W. Haverkortdocumentation:language_reference:objects:responsefunction:functions:new [2024/12/30 09:46] (current) – Maurits W. Haverkort
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 ### ###
-alligned paragraph text+ 
 +ResponseFunction.New(Table) creates a new response function object according to the values in Table. Response functions can be of 4 different types (ListOfPoles, Tri, And, or Nat) and single-valued or matrix-valued. Below 8 examples for creating each of these response functions by hand at some arbitrary values. 
 ### ###
  
-===== Example =====+### 
 + 
 +The input table contains the elements 
 +  * "type" a string equal to "ListOfPoles", "Tri", "And", or "Nat" defining the type used for the internal storage of a response functoin 
 +  * "name" an arbitrary string used to recognise the response function 
 +  * "mu" a double giving the chemical potential. We highly recommend to shift the energy scale such that $\mu=0$. Documentation for the behaviour of several functions at finite chemical potential, $\mu$ is still missing. You can always shift the chemical potential to zero by shifting the onsite energy of all orbitals by $-\mu$. 
 +  * Depending on the type lists of doubles, matrices, or complex matrices $A$ and $B$. Each section below starts by defining $G(\omega,\Gamma)$ in terms of $A$'s and $B$'s, the input below uses these variables. We use capital font for matrices and small font for numbers. Note that the matrices $A$ and $B$ must fulfil several conditions for the response function to be physical. Only physical response functions can be transformed between all types as unitary transformations.
  
 ### ###
-description text+ 
 +===== ListOfPoles representation ===== 
 ### ###
 +Response functions stored as list of poles are defined via 
 +$$ G(\omega,\Gamma) = A_0 + \sum_{i=1}^{n}  \frac{B_{i-1}}{\omega + \mathrm{i}\Gamma/2 - a_i} $$
 +###
 +
 +==== Single valued functions ====
 +<code Quanty Example.Quanty>
 +a = {10, -1,-0.5, 0,   0.5,  1,  1.5}
 +b = {  0.1, 0.1, 0.1, 0.1, 0.2, 0.3}
 +G = ResponseFunction.New( {a,b,mu=0,type="ListOfPoles", name="A"} )
 +print("The resposne function definition is")
 +print(G)
 +omega = 1.1
 +Gamma = 0.001
 +print()
 +print("Evaluated at omega =",omega," and Gamma =",Gamma," yields ",G(omega,Gamma))
 +</code>
 +Generates the output
 +<file Quanty_Output>
 +The resposne function definition is
 +{ { 10 , -1 , -0.5 , 0 , 0.5 , 1 , 1.5 } , 
 +  { 0.1 , 0.1 , 0.1 , 0.1 , 0.2 , 0.3 } ,
 +  name = A ,
 +  type = ListOfPoles ,
 +  mu = 0 }
 +
 +Evaluated at omega = 1.1 and Gamma = 0.001 yields (11.617645834991 - 0.011148328755289 I)
 +</file>
 +
 +
 +==== Matrix valued functions ====
 +<code Quanty Example.Quanty>
 +A0 = Matrix.New( {{0,0,0},{0,0,0},{0,0,0}} )
 +a1 = -1
 +a2 = 1/2
 +a3 = 1
 +B1s = Matrix.New( {{1,1,3},{1,5,6},{3,6,9}} )
 +B1 = B1s * B1s
 +B2s = Matrix.New( {{2,0,3},{0,5,6},{3,6,9}} )
 +B2 = B2s * B2s
 +B3s = Matrix.New( {{3,0,3},{0,5,6},{3,6,9}} )
 +B3 = B3s * B3s
 +G = ResponseFunction.New( { {A0,a1,a2,a3}, {B1,B2,B3}, mu=0, type="ListOfPoles", name="ML"} )
 +print("The resposne function definition is")
 +print(G)
 +omega = 1.1
 +Gamma = 0.001
 +print()
 +print("Evaluated at omega =",omega," and Gamma =",Gamma," yields ")
 +print(G(omega,Gamma))
 +</code>
 +
 +Generates the output
 +<file Quanty_Output>
 +{ { { { 0 , 0 , 0 } , 
 +  { 0 , 0 , 0 } , 
 +  { 0 , 0 , 0 } } , -1 , 0.5 , 1 } , 
 +  { { { 11 , 24 , 36 } , 
 +  { 24 , 62 , 87 } , 
 +  { 36 , 87 , 126 } } , 
 +  { { 13 , 18 , 33 } , 
 +  { 18 , 61 , 84 } , 
 +  { 33 , 84 , 126 } } , 
 +  { { 18 , 18 , 36 } , 
 +  { 18 , 61 , 84 } , 
 +  { 36 , 84 , 126 } } } ,
 +  type = ListOfPoles ,
 +  name = ML ,
 +  mu = 0 }
 +
 +Evaluated at omega = 1.1 and Gamma = 0.001 yields 
 +{ { (206.90024667403 - 0.91928020904165 I) , (221.42405005987 - 0.9276985714825 I) , (432.13381820162 - 1.8498699350513 I) } , 
 +  { (221.42405005987 - 0.9276985714825 I) , (741.17515429623 - 3.1416753933531 I) , (1021.4074723828 - 4.3264255332922 I) } , 
 +  { (432.13381820162 - 1.8498699350513 I) , (1021.4074723828 - 4.3264255332922 I) , (1529.9683515529 - 6.4891280958856 I) } }
 +</file>
 +
 +
 +
 +===== Tridiagonal representation =====
 +
 +###
 +Response functions stored in tridiagonal format are defined via 
 +$$ G(\omega,\Gamma) = A_0 + B_0^* \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_1 - B_{1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_2 - B_{2}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_2 - B_{3}^{\phantom{\dagger}} \frac{...}{\omega + \mathrm{i}\Gamma/2 - A_{n-1} - B_{n-1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_n } B_{n-1}^{\dagger}} B_{3}^{\dagger} } B_{2}^{\dagger} } B_{1}^{\dagger} } B_0^T $$
 +###
 +
 +==== Single valued functions ====
 +<code Quanty Example.Quanty>
 +a = {0, 1,   1,   1,   1,   1,  1}
 +b = {   1, 0.5, 0.5, 0.5, 0.5, 0.5}
 +G = ResponseFunction.New( {a,b,mu=0,type="Tri", name="GT"} )
 +print("The resposne function definition is")
 +print(G)
 +omega = 1.1
 +Gamma = 0.001
 +print()
 +print("Evaluated at omega =",omega," and Gamma =",Gamma," yields ")
 +print(G(omega,Gamma))
 +</code>
 +Generates the output
 +<file Quanty_Output>
 +{ { 0 , 1 , 1 , 1 , 1 , 1 , 1 } , 
 +  { 1 , 0.5 , 0.5 , 0.5 , 0.5 , 0.5 } ,
 +  name = GT ,
 +  type = Tri ,
 +  mu = 0 }
 +
 +Evaluated at omega = 1.1 and Gamma = 0.001 yields 
 +(-1.4800882525182 - 0.010904814637879 I)
 +</file>
 +
 +
 +==== Matrix valued functions ====
 +<code Quanty Example.Quanty>
 +A0 = Matrix.New( {{0,0,0},{0,0,0},{0,0,0}} )
 +A1 = Matrix.New( {{1,2,3},{2,5,6},{3,6,9}} )
 +A2 = Matrix.New( {{2,2,3},{2,5,6},{3,6,9}} )
 +A3 = Matrix.New( {{3,2,3},{2,5,6},{3,6,9}} )
 +B0s = Matrix.New( {{1,0,0},{0,1,0},{0,0,1}} )
 +B0 = B0s * B0s
 +B1s = Matrix.New( {{1,1,3},{1,5,6},{3,6,9}} )
 +B1 = B1s * B1s
 +B2s = Matrix.New( {{2,0,3},{0,5,6},{3,6,9}} )
 +B2 = B2s * B2s
 +B3s = Matrix.New( {{3,0,3},{0,5,6},{3,6,9}} )
 +B3 = B3s * B3s
 +G = ResponseFunction.New( { {A0,A1,A2,A3}, {B0,B1,B2}, mu=0, type="Tri", name="MT"} )
 +print("The resposne function definition is")
 +print(G)
 +omega = 1.1
 +Gamma = 0.001
 +print()
 +print("Evaluated at omega =",omega," and Gamma =",Gamma," yields ")
 +print(G(omega,Gamma))
 +</code>
 +
 +Generates the output
 +<file Quanty_Output>
 +{ { { { 0 , 0 , 0 } , 
 +  { 0 , 0 , 0 } , 
 +  { 0 , 0 , 0 } } , 
 +  { { 1 , 2 , 3 } , 
 +  { 2 , 5 , 6 } , 
 +  { 3 , 6 , 9 } } , 
 +  { { 2 , 2 , 3 } , 
 +  { 2 , 5 , 6 } , 
 +  { 3 , 6 , 9 } } , 
 +  { { 3 , 2 , 3 } , 
 +  { 2 , 5 , 6 } , 
 +  { 3 , 6 , 9 } } } , 
 +  { { { 1 , 0 , 0 } , 
 +  { 0 , 1 , 0 } , 
 +  { 0 , 0 , 1 } } , 
 +  { { 11 , 24 , 36 } , 
 +  { 24 , 62 , 87 } , 
 +  { 36 , 87 , 126 } } , 
 +  { { 13 , 18 , 33 } , 
 +  { 18 , 61 , 84 } , 
 +  { 33 , 84 , 126 } } } ,
 +  type = Tri ,
 +  mu = 0 ,
 +  BlockSize = { 3 , 3 , 3 , 3 } ,
 +  name = MT }
 +
 +Evaluated at omega = 1.1 and Gamma = 0.001 yields 
 +{ { (0.82041346528466 - 0.0005287731551253 I) , (-0.10773626514221 + 0.00047135055668369 I) , (-0.18782055050369 - 0.00021908813997981 I) } , 
 +  { (-0.10773626514222 + 0.00047135055668371 I) , (0.9906660606812 - 0.0018715007457328 I) , (-0.67958409204703 + 0.0013049040629232 I) } , 
 +  { (-0.18782055050369 - 0.00021908813997982 I) , (-0.67958409204703 + 0.0013049040629232 I) , (0.51775011277794 - 0.00095267059250443 I) } }
 +</file>
 +
 +
 +===== Anderson representation =====
 +
 +###
 +Response functions stored in Anderson format are defined via 
 +$$ G(\omega,\Gamma) = A_0 + B_0^* \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_1 - \sum_{i=2}^{n} B_{i-1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_{i} } B_{i-1}^{\dagger} } B_0^T $$
 +###
 +
 +==== Single valued functions ====
 +<code Quanty Example.Quanty>
 +a = {0, 1, 1.5,   2, 2.5,   3, 3.5}
 +b = {   1, 0.5, 0.5, 0.5, 0.5, 0.5}
 +G = ResponseFunction.New( {a,b,mu=0,type="And", name="GA"} )
 +print("The resposne function definition is")
 +print(G)
 +omega = 1.1
 +Gamma = 0.001
 +print()
 +print("Evaluated at omega =",omega," and Gamma =",Gamma," yields ")
 +print(G(omega,Gamma))
 +</code>
 +Generates the output
 +<file Quanty_Output>
 +The resposne function definition is
 +{ { 0 , 1 , 1.5 , 2 , 2.5 , 3 , 3.5 } , 
 +  { 1 , 0.5 , 0.5 , 0.5 , 0.5 , 0.5 } ,
 +  type = And ,
 +  name = GA ,
 +  mu = 0 }
 +
 +Evaluated at omega = 1.1 and Gamma = 0.001 yields 
 +(0.70566877797716 - 0.00077467678957667 I)
 +</file>
 +
 +
 +==== Matrix valued functions ====
 +<code Quanty Example.Quanty>
 +A0 = Matrix.New( {{0,0,0},{0,0,0},{0,0,0}} )
 +A1 = Matrix.New( {{1,2,3},{2,5,6},{3,6,9}} )
 +A2 = Matrix.New( {{2,2,3},{2,5,6},{3,6,9}} )
 +A3 = Matrix.New( {{3,2,3},{2,5,6},{3,6,9}} )
 +B0s = Matrix.New( {{1,0,0},{0,1,0},{0,0,1}} )
 +B0 = B0s * B0s
 +B1s = Matrix.New( {{1,1,3},{1,5,6},{3,6,9}} )
 +B1 = B1s * B1s
 +B2s = Matrix.New( {{2,0,3},{0,5,6},{3,6,9}} )
 +B2 = B2s * B2s
 +B3s = Matrix.New( {{3,0,3},{0,5,6},{3,6,9}} )
 +B3 = B3s * B3s
 +G = ResponseFunction.New( { {A0,A1,A2,A3}, {B0,B1,B2}, mu=0, type="And", name="MA"} )
 +print("The resposne function definition is")
 +print(G)
 +omega = 1.1
 +Gamma = 0.001
 +print()
 +print("Evaluated at omega =",omega," and Gamma =",Gamma," yields ")
 +print(G(omega,Gamma))
 +</code>
 +
 +Generates the output
 +<file Quanty_Output>
 +{ { { { 0 , 0 , 0 } , 
 +  { 0 , 0 , 0 } , 
 +  { 0 , 0 , 0 } } , 
 +  { { 1 , 2 , 3 } , 
 +  { 2 , 5 , 6 } , 
 +  { 3 , 6 , 9 } } , 
 +  { { 2 , 2 , 3 } , 
 +  { 2 , 5 , 6 } , 
 +  { 3 , 6 , 9 } } , 
 +  { { 3 , 2 , 3 } , 
 +  { 2 , 5 , 6 } , 
 +  { 3 , 6 , 9 } } } , 
 +  { { { 1 , 0 , 0 } , 
 +  { 0 , 1 , 0 } , 
 +  { 0 , 0 , 1 } } , 
 +  { { 11 , 24 , 36 } , 
 +  { 24 , 62 , 87 } , 
 +  { 36 , 87 , 126 } } , 
 +  { { 13 , 18 , 33 } , 
 +  { 18 , 61 , 84 } , 
 +  { 33 , 84 , 126 } } } ,
 +  mu = 0 ,
 +  name = MA ,
 +  type = And }
 +
 +Evaluated at omega = 1.1 and Gamma = 0.001 yields 
 +{ { (0.079202023515427 - 0.00018271816123949 I) , (0.019672301598063 - 0.00021050428128743 I) , (-0.033329362936266 + 0.00019529208541705 I) } , 
 +  { (0.019672301598062 - 0.00021050428128743 I) , (-0.028178571653589 - 0.00029307470840254 I) , (0.014801870346139 + 0.00026254621032145 I) } , 
 +  { (-0.033329362936266 + 0.00019529208541705 I) , (0.014801870346139 + 0.00026254621032145 I) , (-0.0017775335507766 - 0.00023672956892774 I) } }
 +</file>
 +
 +===== Natural impurity orbital representation =====
 +
 +###
 +Response functions stored in Natural impurity format are defined via 
 +$$ G(\omega,\Gamma) = A_0 + B_0^* \left( G_{val}(\omega,\Gamma) + G_{con}(\omega,\Gamma) \right) B_0^T$$, with $G_{val}(\omega,\Gamma)$ and $G_{con}(\omega,\Gamma)$ as response functions with poles either at positive energy ($G_{con}(\omega,\Gamma)$) or poles at negative energy ($G_{val}(\omega,\Gamma)$).
 +###
 +
 +==== Single valued functions ====
 +<code Quanty Example.Quanty>
 +acon = {0,         1,   1,   1,   1,   1,  1}
 +bcon = {   sqrt(1/2), 0.5, 0.5, 0.5, 0.5, 0.5}
 +Gcon = ResponseFunction.New(  {acon,bcon,mu=0,type="Tri"} )
 +aval = {0,        -1,  -1,  -2,  -1,  -1, -1}
 +bval = {   sqrt(1/2), 0.5, 0.5, 0.5, 0.5, 0.5}
 +Gval = ResponseFunction.New( {aval,bval,mu=0,type="Tri"} )
 +a0=0
 +b0=1
 +G = ResponseFunction.New( {{a0,b0},val=Gval,con=Gcon,mu=0,type="Nat", name="GD"} )
 +print("The resposne function definition is")
 +print(G)
 +omega = 1.1
 +Gamma = 0.001
 +print()
 +print("Evaluated at omega =",omega," and Gamma =",Gamma," yields ")
 +print(G(omega,Gamma))
 +</code>
 +Generates the output
 +<file Quanty_Output>
 +The resposne function definition is
 +{ { 0 , 1 } ,
 +  type = NaturalImpurityOrbital ,
 +  name = GD ,
 +  mu = 0 ,
 +  con = { { 0 , 1 , 1 , 1 , 1 , 1 , 1 } , 
 +  { 0.70710678118655 , 0.5 , 0.5 , 0.5 , 0.5 , 0.5 } ,
 +  mu = 0 ,
 +  type = Tri ,
 +  name = Matrix } ,
 +  epsilon = 0 ,
 +  val = { { 0 , -1 , -1 , -2 , -1 , -1 , -1 } , 
 +  { 0.70710678118655 , 0.5 , 0.5 , 0.5 , 0.5 , 0.5 } ,
 +  mu = 0 ,
 +  type = Tri ,
 +  name = Matrix } }
 +
 +Evaluated at omega = 1.1 and Gamma = 0.001 yields 
 +(-0.48700605787262 - 0.0055204934115643 I)
 +</file>
 +
  
-==== Input ====+==== Matrix valued functions ====
 <code Quanty Example.Quanty> <code Quanty Example.Quanty>
 -- some example code -- some example code
 </code> </code>
  
-==== Result ====+Generates the output
 <file Quanty_Output> <file Quanty_Output>
 text produced as output text produced as output

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