Single Parameter Scaling Theories for Stationary Solutions in Nonlinear Disordered Media Single Parameter Scaling Theories for Stationary Solutions in Nonlinear Disordered Media 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” Albert-Ludwigs-Universität Freiburg Quantum Optics and Statistics Physikalisches Institut November 18 th, 2010

Motivations from the Laboratory From Organic Chemistry......to Atomics. Lahini, et al. Phys. Rev. Lett. 100, (2008) A. Aspect, et al., Nature 453, 891 (2008)...to Optics... Eilbeck, et al. Phys. Rev. B 30, 4703 (1984) Feddersen, Phys. Lett. A 154, 391 (1991)

A Brief Synopsis A Brief Synopsis A. Linear Disordered Systems 1. Characterizing Anderson Localization 2. Single Parameter Scaling Theory (SPST) 3. A Quasi-1D Model of Disorder 4. Universality of SPST B. Nonlinear Disordered Systems 1. Nonlinear Parametric Evolution 2. Linear SPST in Nonlinear Systems – Failure 3. Correcting the Nonlinear Scalings 4. Nonlinear SPST Theory – Success C. Conclusions and Outlook

Characterizing Anderson Localization – The Problem Characterizing Anderson Localization – The Problem ● IDEALLY build a Green's function: ● Given: ● with solutions: Correlations of solutions depend on parameter V/W: and apply Kubo-Greenwood-Peierls formula V/W < 1 (0.3) – 'Metallic Phase': V/W > 1 (3.0) – 'Insulator Phase':

Characterizing Anderson Localization – A Solution Characterizing Anderson Localization – A Solution Thouless Conductance:.

Single Parameter Scaling Theory (SPST) Single Parameter Scaling Theory (SPST) The “Gang of Four” Abrahams, Anderson, Licciardello, & Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979) For block of length Energy Levels: Spectral Width: In the continuous limit γ 1 : g IS the single parameter! N

A Quasi-1D Model of Disorder – BRM Casati, Molinari, & Izrailev, Phys. Rev. Lett. 64, 1851 (1990) In Thouless-Anderson: What about β (g) ? What about β (g) ?

A Quasi-1D Model of Disorder – BRM A Quasi-1D Model of Disorder – BRM “Ergodic” Two Limiting Cases “Thermodynamic” Inverse Participation

Universality of SPST Universality of SPST Include: 1) BRM Thouless Scaling 2) Anderson IPR Scaling 3) Delay Time Scalings ALL LINEAR.

PST – BRM Linear Case SPST – BRM Linear Case Localized Extended Ergodic:, Thermodynamic: Izrailev, Phys. Rep. 196, 299 (1990)

A Brief Mention of Nonlinear Numerical Methods “Continuation” Method ➢ Take the linear eigensolutions ➢ DNLS-like Equation, add small nonlinearity ➢ Solutions found from minimizing the function using the linear eigensolutions as an initial guess. ➢ Repeat for next step in nonlinearity, using new solution as initial guess.

Parametric Nonlinear Spectra

Parametric Wavefunction Evolution Lahini, et al. Phys. Rev. Lett. 100, (2008) Bodyfelt, Kottos, & Shapiro, Phys. Rev. Lett., submitted (2010)

The Failure of Linear SPST in Nonlinear Systems Bodyfelt, Kottos, & Shapiro, Phys. Rev. Lett., submitted (2010)

Setting NonlinearReferences - Localization Length Setting Nonlinear References - Localization Length Needs Bodyfelt, Kottos, & Shapiro, Phys. Rev. Lett., submitted (2010) b=4

Setting Nonlinear References - Localization Length ● Small Nonlinear Limit ● Large Nonlinear Limit

SettingNonlinear References - Localization Length Setting Nonlinear References - Localization Length Bodyfelt, Kottos, & Shapiro, Phys. Rev. Lett., submitted (2010)

Setting Nonlinear References – Full Ergodic ● Linear Case Maximize entropy : Under the constraint : Gaussian Distribution ● Nonlinear Case Minimize free energy :

Constructing the Nonlinear OPST Linear Nonlinear

One Parameter Scaling Theory: Nonlinear Case Bodyfelt, Kottos, & Shapiro, Phys. Rev. Lett., submitted (2010)

Conclusion Conclusion ➢ Nonlinear Scaling Theory Established for the Single Parameter ➢ Future Possibilities ● Nonlinear Scaling applied to Thouless Conductance ● Nonlinearity Effect on 3D Metal-Insulator Transition ● Field Theories to Incorporate Nonlinearity?

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