# Operators

The second example creates several operators using creation and annihilation operators

- Operators.Quanty
-- A basis consists of: -- a number of Fermionic modes or spin-orbitals NF=6 -- a number of Bosonic modes (phonon modes, ...) NB=0 -- Once we have a basis we can define operators. -- Operators are defined in second quantization. -- We thus need creation and annihilation operators. -- Operators are created with the function -- "NewOperator". This function takes as a first -- argument the type of operator to create and -- further arguments are type specific. -- The creation operator on site 0 is created by -- the following command OppC0=NewOperator("Cr",NF,0) -- The annihilation operator on site 0 is created -- by the following command OppA0=NewOperator("An",NF,0) -- you can print operators print(OppC0) -- the important part of this printout is given by -- the last line "C 0 | 1.000000000000000E+00" -- this tells you that the operator creates at -- position 0 with prefactor 1 -- you are allowed to do mathematics with -- operators, i.e. the number operator acting -- on site 0 is given as: OppCA0 = OppC0 * OppA0 print(OppCA0) -- the program knows about commutation relations, -- so a^+_0 a_0 + a_0 a^+_0 = 1 Oppdelta = OppC0 * OppA0 + OppA0 * OppC0 print(Oppdelta) -- You see that this operator indeed has the -- correct form: "| 1.000000000000000E+00" -- but it also has one element of length 2: -- "C 0 A 0 | 0.000000000000000E+00" -- the total operator is the sum over these two terms. -- The term of length two has prefactor 0, but it is still -- in the list of operators and thus takes memory. -- You can remove small elements with the command -- Chop Oppdelta.Chop() print(Oppdelta) -- the number operator can also be created at once, OppN0 = NewOperator("Number",NF,0,0) print(OppN0) -- you can create sum's of operators at once, -- evaluate to see what it does Opp2=NewOperator("Cr", NF, {0,1,2}, {0.1,0.2,0.3}) print(Opp2) Opp3=NewOperator("Number", NF, {0,1,2}, {0,1,5}, {0.1,0.2,0.3}) print(Opp3) -- Operators have a name which is shown in error -- messages and when you print the operator -- sometimes it can be good to set the name Opp4 = 1+((1+I) * Opp2*Opp3 + 2.7 * OppCA0 ) * Opp3 Opp4.Name = "My fantastic Operator" print(Opp4)

The output is:

- Operators.out
Operator: Creation operator on site 0 QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 1 (largest number of product of lader operators) NFermionic modes = 6 (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 1 QComplex = 0 (Real==0 or Complex==1) N = 1 (number of operators of length 1) C 0 | 1.000000000000000E+00 Operator: Operator QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (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 2 QComplex = 0 (Real==0 or Complex==1) N = 1 (number of operators of length 2) C 0 A 0 | 1.000000000000000E+00 Operator: Operator QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (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) | 1.000000000000000E+00 Operator of Length 2 QComplex = 0 (Real==0 or Complex==1) N = 1 (number of operators of length 2) C 0 A 0 | 0.000000000000000E+00 Operator: Operator QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (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) | 1.000000000000000E+00 Operator: CrAn QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (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 2 QComplex = 0 (Real==0 or Complex==1) N = 1 (number of operators of length 2) C 0 A 0 | 1.000000000000000E+00 Operator: Cr QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 1 (largest number of product of lader operators) NFermionic modes = 6 (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 1 QComplex = 0 (Real==0 or Complex==1) N = 3 (number of operators of length 1) C 0 | 1.000000000000000E-01 C 1 | 2.000000000000000E-01 C 2 | 3.000000000000000E-01 Operator: CrAn QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (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 2 QComplex = 0 (Real==0 or Complex==1) N = 3 (number of operators of length 2) C 0 A 0 | 1.000000000000000E-01 C 1 A 1 | 2.000000000000000E-01 C 2 A 5 | 3.000000000000000E-01 Operator: My fantastic Operator QComplex = 1 (Real==0 or Complex==1 or Mixed==2) MaxLength = 5 (largest number of product of lader operators) NFermionic modes = 6 (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 = 1 (Real==0 or Complex==1) N = 1 (number of operators of length 0) | 1.000000000000000E+00 0.000000000000000E+00 Operator of Length 2 QComplex = 1 (Real==0 or Complex==1) N = 1 (number of operators of length 2) C 0 A 0 | 2.700000000000000E-01 0.000000000000000E+00 Operator of Length 3 QComplex = 1 (Real==0 or Complex==1) N = 4 (number of operators of length 3) C 1 C 0 A 0 | 2.000000000000000E-03 2.000000000000000E-03 C 2 C 0 A 0 | 3.000000000000000E-03 3.000000000000000E-03 C 1 C 0 A 1 | -4.000000000000001E-03 -4.000000000000001E-03 C 2 C 1 A 1 | 1.200000000000000E-02 1.200000000000000E-02 Operator of Length 4 QComplex = 1 (Real==0 or Complex==1) N = 2 (number of operators of length 4) C 1 C 0 A 1 A 0 | -5.400000000000000E-01 -0.000000000000000E+00 C 2 C 0 A 5 A 0 | -8.100000000000001E-01 -0.000000000000000E+00 Operator of Length 5 QComplex = 1 (Real==0 or Complex==1) N = 3 (number of operators of length 5) C 2 C 1 C 0 A 1 A 0 | -1.200000000000000E-02 -1.200000000000000E-02 C 2 C 1 C 0 A 5 A 0 | 1.200000000000000E-02 1.200000000000000E-02 C 2 C 1 C 0 A 5 A 1 | -1.200000000000000E-02 -1.200000000000000E-02