40行代碼計算驗證安德森局域化的一個小觀點(diǎn)：在一維體系上加一點(diǎn)點(diǎn)無序都將導(dǎo)致擴(kuò)展態(tài)到局域態(tài)

先看閻守勝《固體物理基礎(chǔ)》（第三版）第216頁。

• 當(dāng)隨機(jī)均勻分布的width=0的時候：

• 當(dāng)width=5%時：

• 當(dāng)width=10%時：

Python 源碼

1. import numpy as np

2. import matplotlib.pyplot as plt

3. 
   

4. # Name: 1-D lattice with disorder

5. # Date: 2023/4/11

6. # Author: GHz

7. 
   

8. # Parameters

9. width = 0.3

10. center = 10

11. hopping1 = -1

12. hopping2 = -1

13. particleNum = 1000

14. 
    

15. # Create Hamiltonian

16. disorderDiag = (np.random.uniform(size = particleNum)-0.5) * width + center

17. 
    

18. disorderDiag = np.diag(disorderDiag)

19. oneDirectionHopping = np.diag([hopping1]*(particleNum-1), -1)

20. anotherDirectionHopping = np.diag([hopping1]*(particleNum-1), 1)

21. 
    

22. Hamiltonian = disorderDiag + oneDirectionHopping + anotherDirectionHopping

23. 
    

24. # Diagonalize Hamiltonian

25. eigenvalues, eigenvectors = np.linalg.eig(Hamiltonian)

26. xAxis = np.arange(len(eigenvalues))

27. idx = np.argsort(eigenvalues)#[::-1]

28. eigenValues = eigenvalues[idx]

29. eigenVectors = eigenvectors[:,idx]

30. 
    

31. # Plot Results

32. plt.plot(xAxis, eigenValues)

33. plt.title(label="Energy levels array", fontdict={'family':'Times New Roman', 'size':19})

34. plt.show()

35. '''

36. for i in range(len(eigenValues)):

37. ? ? plt.plot(xAxis, eigenVectors[:, i] * np.conj(eigenVectors[:, i]))

38. ? ? plt.title(label="Energy={}".format(eigenValues[i]), fontdict={'family':'Times New Roman', 'size':19})

39. ? ? plt.show()

40. '''

41. plt.contourf(eigenVectors * np.conj(eigenVectors), cmap='RdBu', levels=100)

42. plt.colorbar()

43. plt.show()