Chebyshev polynomial expansion (2015)

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Presentation transcript:

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

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

Anderson Localization

Anderson Localization

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

In One Dimension Destructive interference

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

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

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

1d tight-binding model

Transfer-matrix method

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

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

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

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

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

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

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

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

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

1d tight-binding model

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

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

Random-hopping model

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

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

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

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

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

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

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

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

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

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

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

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)

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

Random Sign Model E 10000 sites 1 sample MOTHRA: https://en.wikipedia.org/wiki/Mothra Random Sign Model J. Feinberg and A. Zee, PRE 59 (1999) 6433 E 10000 sites 1 sample

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

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