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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)
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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
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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)
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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
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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)
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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
/
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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
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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)
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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)
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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
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Single dipole spectrum
+ -
+
-
+-+
-+ ++
+ -
+
-
+-+
-- -+
+ -+--
+
- +
--
Degeneracy protected by the lattice symmetry
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Single dipole spectrum
Degeneracy protected by the lattice symmetry
Alone does nothing
Qualitatively change E-branch
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Single dipole long-range hops
+ -
+
-
Second order coupling:
Fourier transform:
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Single dipole spectrum
Degeneracy protected by the lattice symmetry lifted by long-range hops
Similar to the transverse-longitudinalsplitting in exciton or phonon polaritons
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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
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Dipole two band model and disorder
disorder
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… and disorder and magnetic field
disorder
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Approach from metallic side
Only important new parameter:
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Scaling results
1
ansatz
No magnetic field (GOE)
Used to be for A=0
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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)
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Scaling results
In magnetic field (GUE)
Used to be for A=0
1
ansatz
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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:
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Qualitative consideration
1) Long hops (Levy flights) Consider two wave-packets
(1)(2)
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Qualitative consideration
1) Long hops (Levy flights) Consider two wave-packets
(1)(2)
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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
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2) Weak localization (first loop) due to the short-range hops[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
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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)
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3) Weak localization (second loop) short hops;
In magneticfield; Wegner (1979)
0 no magnetic field (GOE)
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4) New interference term: Second loop: short hops and Levy flight interference:
No magnetic field (GOE)
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Scaling results
1
ansatz
No magnetic field (GOE)
A>0
is not renormalized
Stable critical fixed point
Accessible by perturbative RGfor
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Scaling results
1ansatz
In magnetic field (GUE)
A>0
is not renormalized
Unstable critical fixed point
Accessible by perturbative RGfor
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Standard non-linear s-model for localization
See textbook by K.B. Efetov, Supersymmetry in disorder and chaos, 1997
- supersymmetry
Any correlation function
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Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Standard non-linear s-model for localization
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Beyond standard non-linear s-model for localization (long range hops)
- supersymmetryAny correlation
function
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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