New

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.

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

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))

Generates the 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)

Matrix valued functions

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))

Generates the 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) } }

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

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))

Generates the 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)

Matrix valued functions

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))

Generates the 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) } }

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

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))

Generates the 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)

Matrix valued functions

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))

Generates the 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) } }

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

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))

Generates the 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)

Matrix valued functions

Example.Quanty

-- some example code

Generates the output

text produced as output

Table of Contents

New

ListOfPoles representation

Single valued functions

Matrix valued functions

Tridiagonal representation

Single valued functions

Matrix valued functions

Anderson representation

Single valued functions

Matrix valued functions

Natural impurity orbital representation

Single valued functions

Matrix valued functions

Table of contents