圆环系综(COE、CUE、...)
圆环系综是在不同酉变换下分布不变的酉矩阵家族. 典型应用于统计力学、数论、组合数学和核物理学中.
由圆环实系综(CRE)得到的矩阵是正交矩阵. »
In[1]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_1.png)
cre = RandomVariate[CircularRealMatrixDistribution[5]];
In[2]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_2.png)
OrthogonalMatrixQ[cre]
Out[2]=
![](assets.zh/circular-ensembles-coe-cue-/O_1.png)
由圆环酉系综(CUE)得到的矩阵是酉矩阵. »
In[3]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_3.png)
cue = RandomVariate[CircularUnitaryMatrixDistribution[5]];
In[4]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_4.png)
UnitaryMatrixQ[cue]
Out[4]=
![](assets.zh/circular-ensembles-coe-cue-/O_2.png)
由圆环正交系综(COE)得到的矩阵是对称的酉矩阵. »
In[5]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_5.png)
coe = RandomVariate[CircularOrthogonalMatrixDistribution[5]];
In[6]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_6.png)
SymmetricMatrixQ[coe] && UnitaryMatrixQ[coe]
Out[6]=
![](assets.zh/circular-ensembles-coe-cue-/O_3.png)
由圆环辛系综(CSE)得到的矩阵是自对偶四元数酉矩阵. »
显示完整的 Wolfram 语言输入
In[8]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_8.png)
cse = RandomVariate[CircularSymplecticMatrixDistribution[5]];
In[9]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_9.png)
UnitaryMatrixQ[cse] && selfdualQuaternionicQ[cse]
Out[9]=
![](assets.zh/circular-ensembles-coe-cue-/O_4.png)
由圆环四元数系综 (CQE) 得到的矩阵是辛酉矩阵. »
显示完整的 Wolfram 语言输入
In[11]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_11.png)
cqe = RandomVariate[CircularQuaternionMatrixDistribution[5]];
In[12]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_12.png)
UnitaryMatrixQ[cqe] && symplecticMatrixQ[cqe]
Out[12]=
![](assets.zh/circular-ensembles-coe-cue-/O_5.png)
由 CUE、COE 和 CSE 得到的矩阵特征值具有单位长度并且在相位上均匀分布.
显示完整的 Wolfram 语言输入
Out[13]=
![](assets.zh/circular-ensembles-coe-cue-/O_6.png)
对来自二维 CUE 的特征值相位联合分布进行可视化,并将其与实际密度比较.
In[14]:=
![Click for copyable input](assets.zh/circular-ensembles-coe-cue-/In_14.png)
evs\[ScriptCapitalD] =
MatrixPropertyDistribution[Arg[Eigenvalues[x]],
x \[Distributed] CircularUnitaryMatrixDistribution[2]];
\[CurlyPhi]s =
RandomSample /@ RandomVariate[evs\[ScriptCapitalD], 10^5];
显示完整的 Wolfram 语言输入
Out[15]=
![](assets.zh/circular-ensembles-coe-cue-/O_7.png)