non-hermitian anderson model (1996) chebyshev polynomial expansion (2015) 1

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)

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3

Anderson Localization

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Anderson Localization

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In Three Dimensions

energy

density of states

localized extended

mobility edgeFermi energyFermi energy

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In One Dimension

Destructive interference

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In One DimensionAlmost all states are

localized.

κ : inverse localization length

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Inverse Localization Length

lower energy→ short localization length → large κ

higher energy→ long localization length → small κ

κ : inverse localization length

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1d tight-binding model

random potentia

l

hopping

0 1 2 3−3 −2 −1

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1d tight-binding model

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Transfer-matrix method

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1d tight-binding model

Non-Hermitian Anderson model (1996)

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

0 1 2 3−3 −2 −1

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

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Non-Hermitian Anderson model

1000 sites, periodic boundary condition

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

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Imaginary Vector Potential

vector potential

imaginaryvector

potential

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

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Gauge Transformation

Gauge Transformation

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

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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

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1d tight-binding model

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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)

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1000 sites1 sample

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Random-hopping model

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

(97) 8651

25periodic boundary condition

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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)

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1000 sites1 sample

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Chebyshev Polynomial Expansion

of the density of statesN×N Hermitian matrix: H

: Chebyshev polynomial

R.N. Silver and H. Röder (1994)

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Chebyshev Polynomial Expansion

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

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Chebyshev Polynomial Expansion

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

Recursive Relation

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Chebyshev Polynomial Expansion

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

(i)

(ii)

(iii)

cutoff

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Chebyshev Polynomial Expansion

of the density of states

1000 sites1 sample

up to 1000th order

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

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Chebyshev Polynomial Expansion

of the inverse localization length

N. Hatano (2015)

(n ≥ 1)

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Chebyshev Polynomial Expansion

of the inverse localization length(i)

(ii)

(iii)

cutoff

N. Hatano (2015)

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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)

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Random Sign Model

0 1 2 3−3 −2 −1

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

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Random Sign Model

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

10000 sites1 sample

E

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

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Random Sign Model

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

0 1 2 3−3 −2 −1

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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