I.L. Aleiner (Columbia U, NYC, USA)B.L. Altshuler (Columbia U, NYC, USA)K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)

Localization and Critical Diffusion of Quantum Dipoles in Two Dimensions

Windsor Summer School August 25, 2012

Phys. Rev. Lett. 107, 076401 (2011)

2

Outline:1) Introduction: a) “dirty” – Localization in two dimensions

b) “clean” – Dipole excitations in clean system

2) Qualitative discussion and results for localization of dipoles:Fixed points accessible by perturbative renormalization group.

3) Modified non-linear -s model for localization

4) Conclusions

1. Localization of single-electron wave-functions:

extended

localized

d=1; All states are localized

M.E. Gertsenshtein, V.B. Vasil’ev, (1959)

Exact solution for one channel:

D.J. Thouless, (1977)

Exact solutions for multi-channel:

Scaling argument for multi-channel :

K.B.Efetov, A.I. Larkin (1983)O.N. Dorokhov (1983)

“Conjecture” for one channel:Sir N.F. Mott and W.D. Twose (1961)

Exact solution for ( )s w for one channel:V.L. Berezinskii, (1973)

1. Localization of single-electron wave-functions:

extended

localized

d=1; All states are localized

d=3; Anderson transitionAnderson (1958); Proof of the stability of the insulator

1. Localization of single-electron wave-functions:

extended

localized

d=1; All states are localized

d=3; Anderson transition

d=2; All states are localized

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979)

Thouless scaling + ansatz:

If no spin-orbit interaction

Instability of metal with respect to quantum(weak localization) corrections:L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)

First numerical evidence:A Maccinnon, B. Kramer, (1981)

d=2; All states are localized

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979)

Thouless scaling + ansatz:

If no spin-orbit interaction

Conductivity

Density of state per unit

area

Diffusion coefficient

Dimensionless conductance

Thouless energy

Level spacing

/

d=2; All states are localized E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979)

Thouless scaling + ansatz:

If no spin-orbit interaction

First numerical evidence:A Maccinnon, B. Kramer, (1981)

1

ansatz

Locator expansion

d=2; All states are localized E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979)

Thouless scaling + ansatz:

If no spin-orbit interaction

Instability of metal with respect to quantum(weak localization) corrections:L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)

1

ansatz

No magnetic field (GOE)

d=2; All states are localized E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979)

Thouless scaling + ansatz:

If no spin-orbit interaction

Instability of metal with respect to quantum(weak localization) corrections: Wegner (1979)

1

ansatz

In magnetic field (GUE)

2. Quantum dipoles in clean 2-dimensional systems

Simplest example:Each site can be in four excited states, a

+ -

+

-

+-+

-

Short-range part # of dipoles is not conserved

Square lattice:z

x

Single dipole spectrum

+ -

+

-

+-+

-+ ++

+ -

+

-

+-+

-- -+

+ -+--

+

- +

--

Degeneracy protected by the lattice symmetry

Single dipole spectrum

Degeneracy protected by the lattice symmetry

Alone does nothing

Qualitatively change E-branch

Single dipole long-range hops

+ -

+

-

Second order coupling:

Fourier transform:

Single dipole spectrum

Degeneracy protected by the lattice symmetry lifted by long-range hops

Similar to the transverse-longitudinalsplitting in exciton or phonon polaritons

Single dipole spectrum

Goal: To build the scaling theory of localization including long-range hops

Similar to the transverse-longitudinalsplitting in exciton or phonon polaritons

Dipole two band model and disorder

disorder

… and disorder and magnetic field

disorder

Approach from metallic side

Only important new parameter:

Scaling results

1

ansatz

No magnetic field (GOE)

Used to be for A=0

Scaling results

1

ansatz

No magnetic field (GOE)

A>0

is not renormalized

Instability of insulator,L.S.Levitov, PRL, 64, 547 (1990)

Stable critical fixed point

Accessible by perturbative RGfor

Critical diffusion (scale invariant)

Scaling results

In magnetic field (GUE)

Used to be for A=0

1

ansatz

Scaling results

1ansatz

In magnetic field (GUE)

A>0

is not renormalized

Unstable critical fixed point

Accessible by perturbative RGfor

“Metal-Instulator” transition (scale

invariant)

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Orthogonal ensemble: universal conductance (independent of disorder)

Unitary ensemble: metal-insulator transition

Summary of RG flow:

Qualitative consideration

1) Long hops (Levy flights) Consider two wave-packets

(1)(2)

Qualitative consideration

1) Long hops (Levy flights) Consider two wave-packets

(1)(2)

Qualitative consideration

1) Long hops (Levy flights) Consider two wave-packets

(1)(2)Rate: R

Does not depend on the shape of the wave-function

Levy flights

2) Weak localization (first loop) due to the short-range hops[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]

Constructive interference

Destructive interference

2) Weak localization (first loop) due to the short-range hops[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]

Constructive interference

No magnetic field (GOE)

0 in magnetic field (GUE)

3) Weak localization (second loop) short hops;

In magneticfield; Wegner (1979)

0 no magnetic field (GOE)

4) New interference term: Second loop: short hops and Levy flight interference:

No magnetic field (GOE)

Scaling results

1

ansatz

No magnetic field (GOE)

A>0

is not renormalized

Stable critical fixed point

Accessible by perturbative RGfor

Scaling results

1ansatz

In magnetic field (GUE)

A>0

is not renormalized

Unstable critical fixed point

Accessible by perturbative RGfor

Standard non-linear s-model for localization

See textbook by K.B. Efetov, Supersymmetry in disorder and chaos, 1997

- supersymmetry

Any correlation function

Free energy functional (form fixed by symmetries) (GOE):

Only running constant (one parameter scaling)

Standard non-linear s-model for localization

Beyond standard non-linear s-model for localization (long range hops)

- supersymmetryAny correlation

function

Beyond standard non-linear s-model for localization (long range hops)

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Orthogonal ensemble: universal conductance (independent of disorder)

Unitary ensemble: metal-insulator transition

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Conclusions.

1. Dipoles move easier than particles due to long-range hops.

2. Non-linear sigma-model acquires a new term contributing to RG.

3. RG analysis demonstrates criticality for any disorder for the orthogonal ensemble and existence of a metal-insulator transition for the unitary one.

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Renormalization group in two dimensions.

Integration over fast modes:~

0

~

VQVQ

0Q~

Vfast, slow

Expansion in and integration over V 0Q

New non-linear -model with renormalized and ~

D~

Gell-Mann-Low equations:

~

A consequence of the supersymmetry

Physical meaning: the density of states is constant.

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1Dt 232 12

1

ttd

dtt

1,0,1 For the orthogonal, unitary and symplectic ensembles

Orthogonal: localization Unitary: localization but with a much larger localization length Symplectic: “antilocalization”

Unfortunately, no exact solution for 2D has been obtained.

)/ln(1 00

0

t

tt

Reason: non-compactness of the symmetry group of Q.

Renormalization group (RG) equations.

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The explicit structure of Q

UUQQ 0

v

uU

0

0 u,v contain all Grassmann variables

All essential structure is in 0Q

^^

^^

0

cossin

sincos^

i

i

ie

ieQ

i0

0^

(unitary ensemble)

Mixture of both compact and non-compact symmetries rotations: rotations on a sphere and hyperboloid glued by the anticommuting variables.

0

0