0. Collaborators: David R. Nelson, Ariel Amir
Two methods of numerically computing the inverse localization length in one dimension
Naomichi Hatano
University of Tokyo
Collaborators: David R. Nelson, Ariel Amir

1. Chebyshev polynomial expansion (2015)
Non-Hermitian Anderson model (1996)

2. Anderson Localization

3. Anderson Localization

4. In Three Dimensions density of states localized extended energy
Fermi energy
Fermi energy
mobility edge

5. In One Dimension
Destructive interference

6. In One Dimension κ : inverse localization length
Almost all states are localized.
κ : inverse localization length

7. Inverse Localization Length
higher energy
→ long localization
length → small κ
lower energy
→ short localization
length → large κ
κ : inverse localization length

8. 1d tight-binding model −3 −2 −1 1 2 3
hopping
random potential

9. 1d tight-binding model

10. Transfer-matrix method

11. Non-Hermitian Anderson model (1996)
1d tight-binding model
Non-Hermitian Anderson model (1996)

12. Non-Hermitian Anderson model
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651 −3 −2 −1 1 2 3

13. Non-Hermitian Anderson model
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651

14. Non-Hermitian Anderson model
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651
1000 sites, periodic boundary condition

15. Imaginary Vector Potential
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651
imaginary
vector potential
vector potential

16. N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651
Gauge Transformation
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651
Gauge Transformation

17. Imaginary Gauge Transformation
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651
Imaginary Gauge Transformation

18. Non-Hermitian Anderson model
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651
1000 sites, periodic boundary condition

19. Imaginary Gauge Transformation
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651

20. 1d tight-binding model

21. Non-Hermitian Anderson model
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651
1000 sites, periodic boundary condition

22. Non-Hermitian Anderson model (1996)
1000 sites
1 sample

23. Random-hopping model

24. Imaginary Gauge Transformation
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651
periodic boundary condition

25. Non-Hermitian Anderson model
N. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56 (97) 8651

26. 1000 sites 1 sample Chebyshev polynomial expansion (2015)
Non-Hermitian Anderson model (1996)
1000 sites
1 sample

27. Chebyshev Polynomial Expansion of the density of states
R.N. Silver and H. Röder (1994)
N×N Hermitian matrix: H
: Chebyshev polynomial

28. Chebyshev Polynomial Expansion of the density of states
R.N. Silver and H. Röder (1994)

29. Chebyshev Polynomial Expansion of the density of states
R.N. Silver and H. Röder (1994)
Recursive Relation

30. Chebyshev Polynomial Expansion of the density of states
R.N. Silver and H. Röder (1994)
(i)
(ii)
cutoff
(iii)

31. Chebyshev Polynomial Expansion of the density of states
1000 sites
1 sample
up to 1000th order

32. Thouless Formula
D.J. Thouless, J. Phys. C 5 (1972) 77

33. Chebyshev Polynomial Expansion of the inverse localization length
N. Hatano (2015)
(n ≥ 1)

34. Chebyshev Polynomial Expansion of the inverse localization length
N. Hatano (2015)
(i)
(ii)
cutoff
(iii)

35. Chebyshev Polynomial Expansion of the inverse localization length
N. Hatano (2015)
Chebyshev polynomial expansion (2015)
1000 sites
1 sample
up to 1000th order
Non-Hermitian Anderson model (1996)

36. J. Feinberg and A. Zee, PRE 59 (1999) 6433
Random Sign Model
J. Feinberg and A. Zee, PRE 59 (1999) 6433 −3 −2 −1 1 2 3

37. Random Sign Model E 10000 sites 1 sample
MOTHRA:
Random Sign Model
J. Feinberg and A. Zee, PRE 59 (1999) 6433 E
10000 sites
1 sample

38. A. Amir, N. Hatano and D.R. Nelson, work in progress
Random Sign Model
A. Amir, N. Hatano and D.R. Nelson, work in progress −3 −2 −1 1 2 3

39. A. Amir, N. Hatano and D.R. Nelson, work in progress
Random Sign Model
A. Amir, N. Hatano and D.R. Nelson, work in progress E
κ = 0.1
g=0.0
10000 sites
1 sample
g=0.1
10000 sites
1 sample