Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
QuantumMob`Q3mini`
WickGreenFunction  |  |
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Details and Options
Examples
(4)
Basic Examples
(2)
Set the number of fermion modes to consider.
In[1]:=
$n=4;
Choose a Wick state.
In[2]:=
SeedRandom[354];
In[3]:=
ws=RandomWickState[$n]
Out[3]=
WickState
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Make sure that the state is normalized.
In[4]:=
Norm[ws]
Out[4]=
1
Calculate the Green's functions associated with respect to the Wick state.
In[5]:=
gg=WickGreenFunction[ws];gg//ArrayShort
Out[5]=
,
0.576289 | -0.103601+0.00354606 | -0.284443+0.0628991 | -0.0243118-0.109142 |
-0.103601-0.00354606 | 0.336377 | -0.0410471-0.0741597 | -0.131615+0.0373883 |
-0.284443-0.0628991 | -0.0410471+0.0741597 | 0.577372 | -0.234171+0.190114 |
-0.0243118+0.109142 | -0.131615-0.0373883 | -0.234171-0.190114 | 0.338214 |
0 | 0.0231364+0.320681 | -0.0407901-0.105615 | -0.0208461-0.139435 |
-0.0231364-0.320681 | 0 | 0.100234-0.121517 | -0.145058+0.193261 |
0.0407901+0.105615 | -0.100234+0.121517 | 0 | 0.0687233-0.136495 |
0.0208461+0.139435 | 0.145058-0.193261 | -0.0687233+0.136495 | 0 |
Now, let us verify the above result by computing the Green's functions in the conventional way.
In[6]:=
Let[Fermion,c]cc=c[Range@$n]cd=Join[cc,Dagger@cc]
Out[6]=
{,,,}
c
1
c
2
c
3
c
4
Out[6]=
,,,,,,,
c
1
c
2
c
3
c
4
†
c
1
†
c
2
†
c
3
†
c
4
In[7]:=
alt=ExpressionFor[Matrix[ws],cc]//Chop
Out[7]=
0.41526+(0.287588-0.177)+(0.296626-0.0161201)-(0.162554-0.4185)+(0.193373-0.0448297)-(0.050484-0.0891589)+(0.00325422-0.526565)-(0.220105+0.219277)
0
c
1
0
c
2
0
c
3
1
c
4
0
c
1
0
c
2
1
c
3
0
c
4
0
c
1
1
c
2
0
c
3
0
c
4
0
c
1
1
c
2
1
c
3
1
c
4
1
c
1
0
c
2
0
c
3
0
c
4
1
c
1
0
c
2
1
c
3
1
c
4
1
c
1
1
c
2
0
c
3
1
c
4
1
c
1
1
c
2
1
c
3
0
c
4
In[8]:=
EchoTiming[new=Dagger[alt]**Outer[Multiply,cd,Dagger@cd]**alt;]new//ArrayShort
⌚
2.52748
Out[8]//MatrixForm=
0.576289 | -0.103601+0.00354606 | -0.284443+0.0628991 | -0.0243118-0.109142 | … |
-0.103601-0.00354606 | 0.336377 | -0.0410471-0.0741597 | -0.131615+0.0373883 | … |
-0.284443-0.0628991 | -0.0410471+0.0741597 | 0.577372 | -0.234171+0.190114 | … |
-0.0243118+0.109142 | -0.131615-0.0373883 | -0.234171-0.190114 | 0.338214 | … |
… | … | … | … | … |
Compare the results.
In[9]:=
Normal[gg]-new//Chop//ArrayShort
Out[9]//MatrixForm=
0 | 0 | 0 | 0 | … |
0 | 0 | 0 | 0 | … |
0 | 0 | 0 | 0 | … |
0 | 0 | 0 | 0 | … |
… | … | … | … | … |
In[10]:=
Normal[gg]-new//ArrayZeroQ
Out[10]=
True
In[1]:=
$n=4;$N=50;
Consider an array of Wick states.
In[2]:=
SeedRandom[362];
In[3]:=
data=Array[RandomWickState[$n]&,{$N,$N}];Dimensions[data]data〚;;2,;;2〛
Out[3]=
{50,50}
Out[3]=
WickState
,WickState
,WickState
,WickState
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Calculate the Green's functions for individual Wick states.
In[4]:=
grn=WickGreenFunction[data];
In[5]:=
Dimensions[grn]
Out[5]=
{50,50}
Generalizations & Extensions
(2)
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""