Assa Auerbach, Ilya Finkler, Bert Halperin and Amir Yacoby Self Organization in Non Equilibrium: The Quantum Hall Zero Conductance State Assa Auerbach, Ilya Finkler, Bert Halperin and Amir Yacoby Phys. Rev. Lett. 94, 196801 (2005) Organizing Principle: Lyapunov Functional Zero Conductance Domains Phase. Introducing Long Range Disorder . Experimental effects of disorder potentials.

ZRS Dark magneto resistance: Shubnikov-de-Haas oscillations at large B Hall resistance (classical) Dark magneto resistance: Shubnikov-de-Haas oscillations at large B Microwave photo resistance. oscillations with period: ZRS

Microscopic Theory: negative conductivity Ando+Uemura. No radiation, weak magnetic field: (SCBA) Durst et. al. (Yale). Displacement Photoconductivity (DP) ---> negative conductivity Dimitriev, Mirlin, Polyakov, Aleiner, Vavilov Distribution Function (DF) photoconductivity.

Magneto Transport theory j(E) is determined by microscopics at lengthscales < lc If: Hall current drops out! Since there exists a scalar functional

The Lyapunov functional If exists, then 1. Extrema of G are Kirchoff Steady States. 2. G is a spontaneously decreasing function of time! (like a ‘Free Energy’). Electrostatics:

The Global Lyapunov Minimum: (Ground Steady-State)

The Zero Conductance state (no long range disorder) Negative conductivity is unstable toward creation of finite field domains corbino Free motion of domain wall under external bias: Zero Conductance

Domains in two dimensions: Clean system unstable regime Lyapunov function domain walls bisect field directions photoinduced electric potential What are the effects of a long range potential disorder?

Coupling a slowly varying internal field electrochemical field photocurrent Microscopics: the different role of electric and electrochemical fields in the current equation: More Generally: a weak internal field modifies g: The induced field would align with the disorder, but it is frustrated by the stability constraint

White noise disorder (Imry-Ma argument) In D=2 white noise disorder is irrelevant to the Zero Conductance State. Mesoscopic current fluctuations

One dimensional disorder Resistance is finite: 2. Disorder induces a photovoltaic effect:

stability constraints on domain walls with disorder 1. The field magnitude at domain walls is marginally stable . 2. E is a gradient: domain walls locally constrained by domain wall 3. Currents at domain walls are 4. Domains enclose zero ``2D charge’’ of the disorder potential

The separable `Egg Carton’ potential Charge density

correlated disorder f(x)+g(y)

2D disorder (non separable) radiation Field E cannot fully align with disorder field Ed: Frustration

Charge and dissipative currents 1. Domain walls are pinned. 2. Frustration drives circulating dissipative currents. 3. I(V) is discontinuous, walls jump between steady states. ==> glassiness.

conclusions For constant Hall conductivity, there is a Lyapunov functional which organizes the stability of steady states The Zero Conductance State survives weak 2D white noise disorder. The disorder fields produces a photo voltage and circulating dissipative currents. 1D and correlated or strong 2D disorder pins domain walls and produce finite resistivity. 5. Soon: A fuller microscopic theory for j(E): the magnitude of induced fields, detailed phase diagram and frequency dependent magnetoresistance.