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QuantumMob`Q3mini`
WickEntropy  |  |
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Examples
(2)
Basic Examples
(2)
In[1]:=
$n=6;
Generate a statistical ensemble of Wick states. Make sure to normalize each state.
In[2]:=
SeedRandom[362];
In[3]:=
ens=Table[RandomWickState[$n],4]
Out[3]=
WickState
,WickState
,WickState
,WickState
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In[4]:=
Norm/@ens
Out[4]=
{1,1,1,1}
Calculate the average of the Green's functions.
In[5]:=
grn=Mean[WickGreenFunction/@ens]grn//ArrayShort
Out[5]=
NambuGreen
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Out[5]=
,
0.478141 | -0.0384557+0.0282099 | -0.0522505+0.0221816 | -0.115205+0.0625819 | … |
-0.0384557-0.0282099 | 0.626679 | 0.0732485+0.0189654 | -0.0038312+0.0300497 | … |
-0.0522505-0.0221816 | 0.0732485-0.0189654 | 0.567864 | -0.0894801+0.0800685 | … |
-0.115205-0.0625819 | -0.0038312-0.0300497 | -0.0894801-0.0800685 | 0.470713 | … |
… | … | … | … | … |
0 | 0.021475-0.00611407 | -0.0514683-0.0120041 | 0.0054456-0.00921281 | … |
-0.021475+0.00611407 | 0 | 0.0233473-0.067923 | -0.00310981-0.00153415 | … |
0.0514683+0.0120041 | -0.0233473+0.067923 | 0 | 0.040589+0.0635759 | … |
-0.0054456+0.00921281 | 0.00310981+0.00153415 | -0.040589-0.0635759 | 0 | … |
… | … | … | … | … |
In[6]:=
WickEntropy[grn]
Out[6]=
4.94228
In[1]:=
$n=6;
Consider a pure Wick state.
In[2]:=
SeedRandom[362];
In[3]:=
ws=RandomWickState[$n]
Out[3]=
WickState
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Calculate the Green's functions.
In[4]:=
full=WickGreenFunction[ws];full//ArrayShort
Out[4]=
,
0.301525 | -0.0397476-0.0381598 | -0.0510844+0.0846824 | -0.221221-0.0323891 | … |
-0.0397476+0.0381598 | 0.676175 | 0.0219312+0.00543513 | -0.172542-0.040477 | … |
-0.0510844-0.0846824 | 0.0219312-0.00543513 | 0.579745 | -0.0738395-0.0616823 | … |
-0.221221+0.0323891 | -0.172542+0.040477 | -0.0738395+0.0616823 | 0.51448 | … |
… | … | … | … | … |
0 | -0.0136839+0.0823599 | -0.045609-0.027744 | 0.0274149-0.11059 | … |
0.0136839-0.0823599 | 0 | 0.14+0.00194123 | -0.143893+0.10174 | … |
0.045609+0.027744 | -0.14-0.00194123 | 0 | 0.0405121+0.237186 | … |
-0.0274149+0.11059 | 0.143893-0.10174 | -0.0405121-0.237186 | 0 | … |
… | … | … | … | … |
Consider a mixed state of subsystem consisting of the first half of the fermion modes that results from tracing out the rest modes. The mixed state is characterized by the sub-block of the full matrix of Green's functions.
In[5]:=
kk=Range[$n/2]
Out[5]=
{1,2,3}
In[6]:=
{gg,ff}=First[full];grn=NambuGreen[{gg〚kk,kk〛,ff〚kk,kk〛}];grn//ArrayShort
Out[6]=
,
0.301525 | -0.0397476-0.0381598 | -0.0510844+0.0846824 |
-0.0397476+0.0381598 | 0.676175 | 0.0219312+0.00543513 |
-0.0510844-0.0846824 | 0.0219312-0.00543513 | 0.579745 |
0 | -0.0136839+0.0823599 | -0.045609-0.027744 |
0.0136839-0.0823599 | 0 | 0.14+0.00194123 |
0.045609+0.027744 | -0.14-0.00194123 | 0 |
In[7]:=
WickEntropy[grn]
Out[7]=
2.49201
Note that the above quantity is nothing but the entanglement entropy between the subsystem and the rest.
In[8]:=
WickEntanglementEntropy[full,kk]
Out[8]=
2.49201
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