i.l. aleiner ( columbia u, nyc, usa ) b.l. altshuler ( columbia u, nyc, usa ) k.b. efetov (...
TRANSCRIPT
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