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Two methods of numerically
computing the inverse
localization length in one dimension
Naomichi HatanoUniversity of Tokyo
Collaborators: David R. Nelson, Ariel Amir
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Non-Hermitian Anderson model (1996)
Chebyshev polynomial expansion (2015)
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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
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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 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
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Non-Hermitian Anderson model (1996)
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1000 sites1 sample
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Random-hopping model
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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
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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