One Parameter Scaling Theory for Stationary One Parameter Scaling Theory for Stationary States of Disordered Nonlinear Systems States of Disordered Nonlinear Systems One Parameter Scaling Theory for Stationary One Parameter Scaling Theory for Stationary States of Disordered Nonlinear Systems States of Disordered Nonlinear Systems Joshua D. Bodyfelt in collaboration with Tsampikos Kottos & Boris Shapiro Max-Planck-Institute für Physik komplexer Systeme Condensed Matter Division in cooperation with Wesleyan University, CQDMP Group Technion – Israel Institute of Technology supported by U.S. - Israel Binational Science Foundation (BSF) DFG FOR760 - “Scattering Systems with Complex Dynamics” New Perspectives in Quantum Statistics and Correlations – Heidelberg – Mar. 4 th 2010
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One Parameter Scaling Theory for Stationary One Parameter Scaling Theory for Stationary States of Disordered Nonlinear SystemsStates of Disordered Nonlinear Systems
One Parameter Scaling Theory for Stationary One Parameter Scaling Theory for Stationary States of Disordered Nonlinear SystemsStates of Disordered Nonlinear Systems
Joshua D. Bodyfeltin collaboration with
Tsampikos Kottos & Boris Shapiro
MaxPlanckInstitute für Physik komplexer SystemeCondensed Matter Division
in cooperation with
Wesleyan University, CQDMP GroupTechnion – Israel Institute of Technology
supported by
U.S. Israel Binational Science Foundation (BSF)DFG FOR760 “Scattering Systems with Complex Dynamics”
New Perspectives in Quantum Statistics and Correlations – Heidelberg – Mar. 4th 2010
A Brief SynopsisA Brief Synopsis A Brief SynopsisA Brief Synopsis
A. Brief Review of Disorder
1. Motivations from the Laboratory
2. A Characteristic Length for Anderson Localization
3. Modeling Disorder in Quasi1D Systems
3. One Parameter Scaling Theory (OPST)
B. Including Nonlinearity
1. A Brief Mention of Nonlinear Numerical Methods
2. Nonlinear Parametric Evolution
3. Failure of Linear OPST
4. Setting Nonlinear References
5. The Nonlinear OPST
C. Conclusions
Motivations from the LaboratoryMotivations from the Laboratory Motivations from the LaboratoryMotivations from the Laboratory
From Acoustics...
Hu, et al. NaturePhys. 4, 945 (2008)
...to Optics...
...to Atomics...
Lahini, et al. Phys. Rev. Lett. 100, 013906 (2008)
A. Aspect, et al., Nature 453, 891 (2008)
“Does Localization Survive the Nonlinearity?”
A Characteristic Length for Anderson LocalizationA Characteristic Length for Anderson Localization A Characteristic Length for Anderson LocalizationA Characteristic Length for Anderson Localization
N~NExtended
Nref
“Ergodic”
N≃N
Localized
∞≃∞
“Thermodynamic”
N
N=⟨1 /∑∣n∣4⟩
n=1
N
N=g ∞ ,Nref
BRM Modeling Disorder in Quasi1D SystemsBRM Modeling Disorder in Quasi1D Systems BRM Modeling Disorder in Quasi1D SystemsBRM Modeling Disorder in Quasi1D Systems
∞≃b2≫N ∞≃b
2≪N
⟨H nm⟩=0, ⟨H nm2 ⟩=
N11nm
b 2N−b1 ∈[−2,2]
= ∞ / Nref
Casati, Molinari, & Izrailev, Phys. Rev. Lett. 64, 1851 (1990)
One Parameter Scaling Theory (OPST) BasicsOne Parameter Scaling Theory (OPST) Basics One Parameter Scaling Theory (OPST) BasicsOne Parameter Scaling Theory (OPST) Basics
Kawabata, Prog. Theo. Phys. Sup. 84, 16 (1985)
g =d ln g d ln N
N~g ∞ , N ● Question
● Renormalization Group Equation
N NFor , N ∞ ,N g g ∞ , N , does ?
v x= dxdt
form similar to dynamical flow:v
x
OPST – BRM Linear CaseOPST – BRM Linear Case OPST – BRM Linear CaseOPST – BRM Linear Case
ln
ln g
g ≃c∞ /N
ref
1c∞ /Nref
=c1c
Kawabata, Prog. Theo. Phys. Sup. 84, 16 (1985)
Localized
Extended
Ergodic: ,Nref
∞Thermodynamic: g≃N
Nref
Izrailev, Phys. Rep. 196, 299 (1990)
A Brief Mention of Nonlinear Numerical MethodsA Brief Mention of Nonlinear Numerical Methods A Brief Mention of Nonlinear Numerical MethodsA Brief Mention of Nonlinear Numerical Methods
“Continuation” Method
F n=∑ H nmm∣n∣2n−n ; n=1,2,. . , N
m
F N1=∑∣m∣2−1
m
➢ Take the linear eigensolutions
➢ DNLSlike Equation, add small nonlinearity
➢ Solutions found from minimizing the function
using the linear eigensolutions as an initial guess.
~10−4
∑ H nmm=n ; n=1,2,. . , Nm
∑ H nmm∣n∣2n=n ; n=1,2,. . , N
m
➢ Repeat for next step in nonlinearity, using new solution as initial guess.
Lahini, et al. Phys. Rev. Lett. 100, 013906 (2008) Bodyfelt, Kottos, & Shapiro, Phys. Rev. Lett., submitted (2010)
The Failure of Linear OneParameter Scaling TheoryThe Failure of Linear OneParameter Scaling Theory The Failure of Linear OneParameter Scaling TheoryThe Failure of Linear OneParameter Scaling Theory