Two methods of numerically

computing the inverse

localization length in one dimension

Naomichi HatanoUniversity of Tokyo

Collaborators: David R. Nelson, Ariel Amir

Non-Hermitian Anderson model (1996)

Chebyshev polynomial expansion (2015)

2

3

Anderson Localization

4

Anderson Localization

5

In Three Dimensions

energy

density of states

localized extended

mobility edgeFermi energyFermi energy

6

In One Dimension

Destructive interference

7

In One DimensionAlmost all states are

localized.

κ : inverse localization length

8

Inverse Localization Length

lower energy→ short localization length → large κ

higher energy→ long localization length → small κ

κ : inverse localization length

9

1d tight-binding model

random potentia

l

hopping

0 1 2 3−3 −2 −1

10

1d tight-binding model

11

Transfer-matrix method

12

1d tight-binding model

Non-Hermitian Anderson model (1996)

13

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

0 1 2 3−3 −2 −1

14

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

15

Non-Hermitian Anderson model

1000 sites, periodic boundary condition

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

16

Imaginary Vector Potential

vector potential

imaginaryvector

potential

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

17

Gauge Transformation

Gauge Transformation

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

18

Imaginary Gauge Transformation

Imaginary Gauge Transformation

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

19

Non-Hermitian Anderson model

1000 sites, periodic boundary condition

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

Imaginary Gauge TransformationN. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56

(97) 8651

20

21

1d tight-binding model

22

Non-Hermitian Anderson model

1000 sites, periodic boundary condition

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

Non-Hermitian Anderson model (1996)

23

1000 sites1 sample

24

Random-hopping model

Imaginary Gauge TransformationN. Hatano and D.R. Nelson, PRL 77 (96) 570; PRB 56

(97) 8651

25periodic boundary condition

26

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

Non-Hermitian Anderson model (1996)

Chebyshev polynomial expansion (2015)

27

1000 sites1 sample

28

Chebyshev Polynomial Expansion

of the density of statesN×N Hermitian matrix: H

: Chebyshev polynomial

R.N. Silver and H. Röder (1994)

29

Chebyshev Polynomial Expansion

of the density of statesR.N. Silver and H. Röder (1994)

30

Chebyshev Polynomial Expansion

of the density of statesR.N. Silver and H. Röder (1994)

Recursive Relation

31

Chebyshev Polynomial Expansion

of the density of statesR.N. Silver and H. Röder (1994)

(i)

(ii)

(iii)

cutoff

32

Chebyshev Polynomial Expansion

of the density of states

1000 sites1 sample

up to 1000th order

33

Thouless FormulaD.J. Thouless, J. Phys. C 5 (1972) 77

34

Chebyshev Polynomial Expansion

of the inverse localization length

N. Hatano (2015)

(n ≥ 1)

35

Chebyshev Polynomial Expansion

of the inverse localization length(i)

(ii)

(iii)

cutoff

N. Hatano (2015)

36

Chebyshev Polynomial Expansion

of the inverse localization length

N. Hatano (2015)

1000 sites1 sample

up to 1000th orderNon-Hermitian Anderson model (1996)

Chebyshev polynomial expansion (2015)

37

Random Sign Model

0 1 2 3−3 −2 −1

J. Feinberg and A. Zee, PRE 59 (1999) 6433

38

Random Sign Model

J. Feinberg and A. Zee, PRE 59 (1999) 6433

10000 sites1 sample

E

MOTHRA: https://en.wikipedia.org/wiki/Mothra

39

Random Sign Model

A. Amir, N. Hatano and D.R. Nelson, work in progress

0 1 2 3−3 −2 −1

40

Random Sign Model

A. Amir, N. Hatano and D.R. Nelson, work in progress

g=0.010000 sites1 sample

g=0.110000 sites1 sample

κ = 0.1

E