Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Magnetic Exchange Coupling
Here we discuss the magnetic exchange coupling as an example of possible applications of the package.
Constucts the basis the elements of which are grouped by conserved quantities | |
Constructs the irreducible basis of the total angular momentum. | |
Constructs the irreducible basis of the total directional angular momentum. | |
Returns an operator corresponding to the total number of particles. | |
Returns an operator describing hoppings. |
Some useful functions used in the example.
Load the package.
In[74]:=
Needs["QuantumMob`Q3`"]
In[75]:=
Let[Fermion,c,d,Spin1/2]Let[Real,t,ϵ,U]
Single quantum dot
In[77]:=
H0=ϵ*FockNumber[d[All]]Hu=U[0]*FockNumber[d[Up]]**FockNumber[d[Down]]HH=H0+Hu
Out[77]=
ϵ+
†
d
↓
d
↓
†
d
↑
d
↑
Out[78]=
†
d
↓
†
d
↑
d
↑
d
↓
U
0
Out[79]=
ϵ++
†
d
↓
d
↓
†
d
↑
d
↑
†
d
↓
†
d
↑
d
↑
d
↓
U
0
In[80]:=
bsNS=FermionBasis[d,"Representation""Generators"];bsNS//PrintFermionBasis
Out[81]=
{0,0} | {|≍〉} |
1, 1 2 | † d ↑ |
{2,0} | - † d ↓ † d ↑ |
Define a utility function to display the matrix representation.
In[82]:=
showMatrix[mat_Association]:=With[{frm=KeyGroupBy[mat,First,Values]},MatrixForm[Map[MatrixForm,Values@frm,{2}],TableHeadings{Keys@frm,Keys@frm}]]
In[83]:=
mNS=MatrixIn[HH,FockKet@bsNS,FockKet@bsNS];mNS//showMatrix
Out[84]//MatrixForm=
{0,0} | 1, 1 2 | {2,0} | ||||
{0,0} | (
| (
| (
| |||
1, 1 2 | (
| (
| (
| |||
{2,0} | (
| (
| (
|
In[85]:=
mNS=MatrixIn[HH,FockKet@bsNS];MatrixForm/@mNS
Out[86]=
{0,0}(
),1,(
),{2,0}(
)
0 |
1
2
ϵ |
2ϵ+ U 0 |
In[89]:=
bsNS=FermionBasis[{c,d},"Representation""Generators"];bsNS//PrintFermionBasis
Out[90]=
{0,0} | {|≍〉} |
1, 1 2 | † d ↑ † c ↑ |
{2,0} | - † c ↓ † c ↑ † c ↓ † d ↑ 2 † c ↑ † d ↓ 2 † d ↓ † d ↑ |
{2,1} | † c ↑ † d ↑ |
3, 1 2 | - † c ↓ † c ↑ † d ↑ † c ↑ † d ↓ † d ↑ |
{4,0} | † c ↓ † c ↑ † d ↓ † d ↑ |
In[91]:=
mNS=MatrixIn[HH,FockKet@bsNS,FockKet@bsNS];mNS//showMatrix
Out[92]//MatrixForm=
{0,0} | 1, 1 2 | {2,0} | {2,1} | 3, 1 2 | {4,0} | |||||||||||||||||||||||||||||||
{0,0} | (
| (
| (
| (
| (
| (
| ||||||||||||||||||||||||||||||
1, 1 2 |
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{2,0} |
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{2,1} | (
| (
| (
| (
| (
| (
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3, 1 2 |
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{4,0} | (
| (
| (
| (
| (
| (
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Construct the Hamiltonian for double quantum dots.
In[93]:=
H0=ϵ*FockNumber[d[1,All],d[2,All]]Ht=-t*PlusDagger@FockHopping[d[1,All],d[2,All]]Hu=U[0]*Sum[FockNumber[d[j,Up]]**FockNumber[d[j,Down]],{j,1,2}]+U[1,2]/2*FockNumber[d[1,All]]**FockNumber[d[2,All]]
Out[93]=
ϵ+++
†
d
1,↓
d
1,↓
†
d
1,↑
d
1,↑
†
d
2,↓
d
2,↓
†
d
2,↑
d
2,↑
Out[94]=
Out[95]=
+++++
†
d
1,↓
†
d
1,↑
d
1,↑
d
1,↓
†
d
2,↓
†
d
2,↑
d
2,↑
d
2,↓
U
0
1
2
†
d
1,↓
†
d
2,↓
d
2,↓
d
1,↓
†
d
1,↓
†
d
2,↑
d
2,↑
d
1,↓
†
d
1,↑
†
d
2,↓
d
2,↓
d
1,↑
†
d
1,↑
†
d
2,↑
d
2,↑
d
1,↑
U
1,2
In[96]:=
HH=H0+Ht+Hu;
In[97]:=
bsNS=FermionBasis[d@{1,2},"Representation""Generators","Conserved"{"Number","Spin"}];bsNS//PrintFermionBasis
Out[98]=
{0,0} | {|≍〉} |
1, 1 2 | † d 2,↑ † d 1,↑ |
{2,0} | - † d 1,↓ † d 1,↑ † d 1,↓ † d 2,↑ 2 † d 1,↑ † d 2,↓ 2 † d 2,↓ † d 2,↑ |
{2,1} | † d 1,↑ † d 2,↑ |
3, 1 2 | - † d 1,↓ † d 1,↑ † d 2,↑ † d 1,↑ † d 2,↓ † d 2,↑ |
{4,0} | † d 1,↓ † d 1,↑ † d 2,↓ † d 2,↑ |
In[99]:=
bsNone=FermionBasis[d@{1,2},"Representation""Generators","Conserved""None"];bsNone//PrintFermionBasis
Out[100]=
{} | ≍, † d 2,↓ † d 2,↑ † d 2,↓ † d 2,↑ † d 1,↓ † d 1,↓ † d 2,↓ † d 1,↓ † d 2,↑ † d 1,↓ † d 2,↓ † d 2,↑ † d 1,↑ † d 1,↑ † d 2,↓ † d 1,↑ † d 2,↑ † d 1,↑ † d 2,↓ † d 2,↑ † d 1,↓ † d 1,↑ † d 1,↓ † d 1,↑ † d 2,↓ † d 1,↓ † d 1,↑ † d 2,↑ † d 1,↓ † d 1,↑ † d 2,↓ † d 2,↑ |
In[101]:=
mNS=MatrixIn[HH,FockKet@bsNS,FockKet@bsNS];mNS//showMatrix
Out[102]//MatrixForm=
{0,0} | 1, 1 2 | {2,0} | {2,1} | 3, 1 2 | {4,0} | |||||||||||||||||||||||||||||||
{0,0} | (
| (
| (
| (
| (
| (
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1, 1 2 |
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{2,0} |
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{2,1} | (
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3, 1 2 |
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{4,0} | (
| (
| (
| (
| (
| (
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In[103]:=
mNone=MatrixIn[HH,FockKet@bsNone,FockKet@bsNone];MatrixForm/@mNone