igor aleiner (columbia) · igor aleiner (columbia) collaborators: b.l. altshuler (columbia, nec...
TRANSCRIPT
![Page 1: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/1.jpg)
Igor Aleiner (Columbia)
Collaborators: B.L. Altshuler (Columbia, NEC America)
D.M. Basko (Columbia, Trieste, Grenoble)
G.V. Shlyapnikov
Many body localizationMany body localization
G.V. Shlyapnikov (Orsay)
Detailed paper (fermions): Annals of Physics 321 (2006) 1126-1205
Shorter version: cond-mat/0602510; chapter in “Problems of CMP”
Bosons: NATURE PHYSICS 6 (2010) 900-904
Lewiner Institute of Theoretical Physics, Colloquium, December 23rd, 2010
![Page 2: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/2.jpg)
I
V
Metal
Transport in solids
Superconductor
Conductivity:
Conductance:
Insulator
?????
![Page 3: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/3.jpg)
Outline:Outline:
• Formulation and history of the problem
• Results for fermionic system
• Effective model
• Technique• Technique
• Stability of the metal
• Stability of the many-body insulator
• Metal insulator transition
• Extension for weakly interacting bosons in 1D.
![Page 4: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/4.jpg)
can e-e interaction alonesustain finite conductivity
in a localized system?
1. All one-electron states are localized
2. Electrons interact with each other
Problem:
Given:
2. Electrons interact with each other
3. The system is closed (no phonons)
4. Temperature is low but finite
Find: DC conductivity σ(T,ω=0)
(zero or finite?)
![Page 5: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/5.jpg)
1. Localization of single-electron wave-functions:
extended
Disorder IV
localized
V
Conductance
![Page 6: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/6.jpg)
Most of the knowledge is based on extensions and
Improvements of:
![Page 7: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/7.jpg)
1. Localization of single-electron wave-functions:
extendedd=1; All states are localizedExact solution for one channel:
localized
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
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 σ(ω)σ(ω)σ(ω)σ(ω) for one channel:
V.L. Berezinskii, (1973)
![Page 8: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/8.jpg)
1. Localization of single-electron wave-functions:
extendedd=1; All states are localized
d=2; All states are localized
localized
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)
![Page 9: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/9.jpg)
Instability of 2D metal with respect to quantum(weak localization) corrections:L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
![Page 10: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/10.jpg)
1. Localization of single-electron wave-functions:
extendedd=1; All states are localized
d=2; All states are localized
localized
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)
![Page 11: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/11.jpg)
“All states are localized “
means
Probability to find an extended state:Probability to find an extended state:
System size
![Page 12: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/12.jpg)
1. Localization of single-electron wave-functions:
extendedd=1; All states are localized
d=2; All states are localized
localized
d=2; All states are localized
d>2; Anderson transition
![Page 13: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/13.jpg)
Anderson Model
• Lattice - tight binding model
• Onsite energies εεεεi - random
• Hopping matrix elements Iijj i
Iij
I =
I i and j are nearest neighbors{
-W < εεεεi <Wuniformly distributed
Iij =neighbors
0 otherwise{ Critical hopping:
![Page 14: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/14.jpg)
all states are
localized
I < IcI > Ic
Anderson Transition
Coexistence of the localized and extended states is not possible!!!
DoS DoS
localized
extended
- mobility edges (one particle)
Rules out first order phase transition
![Page 15: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/15.jpg)
Temperature dependence of the Temperature dependence of the
conductivity (I)conductivity (I)
Assume that all the states
DoS DoSDoS
Assume that all the states
are localized
![Page 16: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/16.jpg)
Inelastic processes )transitions between localized states
α
β energy
mismatch
(inelastic lifetime)–1
(any mechanism)
![Page 17: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/17.jpg)
Phonon-induced hopping
energy difference can be matched by a phonon
α
β
Variable Range HoppingSir N.F. Mott (1968)
Any bath with a continuous spectrum of delocalized
excitations down to ω ω ω ω = 0 will give the same exponential
Sir N.F. Mott (1968)
Without Coulomb gapA.L.Efros, B.I.Shklovskii (1975)
Optimized
phase volume
Mechanism-dependent
prefactor
![Page 18: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/18.jpg)
Q: Can we replace phonons with
e-h pairs and obtain phonon-less VRH?
A#1: Sure
1) Recall phonon-less AC conductivity:Sir N.F. Mott (1970)
Easy stepsEasy steps:: Person from the street (2005)
3) Use the electric noise instead of phonons.
2) Calculate the Nyquist noise
(fluctuation dissipation Theorem).
4) Do self-consistency (whatever it means).
![Page 19: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/19.jpg)
Q: Can we replace phonons with
e-h pairs and obtain phonon-less VRH?
A#1: Sure [Person from the street (2005)]
A#2: No way(for Coulomb interaction in 3D – may be)
[L. Fleishman. P.W. Anderson (1980)]
is contributed by rare Thus, the matrix element vanishes !!!is contributed by rare
resonances
δα
βγγγγ
R
Thus, the matrix element vanishes !!!
0 *
![Page 20: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/20.jpg)
Q: Can we replace phonons with
e-h pairs and obtain phonon-less VRH?
A#1: Sure [Person from the street (2005)]
A#2: No way [L. Fleishman. P.W. Anderson (1980)]
A#3: Finite T MetalMetal--Insulator TransitionInsulator Transition
Drude
[Basko, Aleiner, Altshuler (2005)]
insulator
metal
Interaction strength(Perfect Ins)
![Page 21: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/21.jpg)
ManyMany--body mobility thresholdbody mobility threshold
[Basko, Aleiner, Altshuler (2005)]
insulator
metal
Many body DoS
All STATES LOCALIZED
All STATES EXTENDED -many-body
mobility threshold
![Page 22: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/22.jpg)
“All states are localized “
meansProbability to find an extended state:
System volume
![Page 23: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/23.jpg)
Localized oneLocalized one--body wavebody wave--functionfunction
Means, in particular:
localized
extended
We define localized manyWe define localized many--body wavebody wave--function as:function as:We define localized manyWe define localized many--body wavebody wave--function as:function as:
extended
localized
![Page 24: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/24.jpg)
All STATES EXTENDED
States always thermalized!!!
Entropy
Many body DoS
All STATES LOCALIZED
States never thermalized!!!
![Page 25: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/25.jpg)
Is it similar to Anderson transition?
Why no activation?
Many body DoS One-body DoS
![Page 26: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/26.jpg)
0σ ==== Physics: Many-body excitations turn out
to be localized in the Fock space
![Page 27: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/27.jpg)
Anderson Model
• Lattice - tight binding model
• Onsite energies εεεεi - random
• Hopping matrix elements Iijj i
IijCritical hopping:
Interpretation:
W
In fact, i,j can be states in any space (not necessarily
coordinate)
W – maximal energy mismatch;
2d – number of coupled neighbors;
(connectivity)
At I>Ic there will be always level mismatchedfrom given by
and the resonance transport will occur
![Page 28: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/28.jpg)
Fock space localization in quantum dots (AGKL, 1997)
´
No spatial structure ( “0-dimensional” )
- one-particle level spacing;
![Page 29: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/29.jpg)
Fock space localization in quantum dots (AGKL, 1997)
1-particle excitation
3-particle excitation
5-particle excitation
Cayley tree mapping
![Page 30: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/30.jpg)
Fock space localization in quantum dots (AGKL, 1997)
1-particle excitation
3-particle excitation
5-particle excitation
1. Coupling between states:
- one-particle level spacing;
1. Coupling between states:
2. Maximal energy mismatch:
3. Connectivity:
![Page 31: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/31.jpg)
Vs. finite T MetalMetal--Insulator Transition Insulator Transition in the bulk systemsin the bulk systems[Basko, Aleiner, Altshuler (2005)]
MetalMetal--Insulator “Transition” Insulator “Transition” in zero dimensionsin zero dimensions
[Altshuler, Gefen, Kamenev,Levitov (1997)]
In the paper:
insulator
metal
Interaction strength
![Page 32: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/32.jpg)
Vs. finite T MetalMetal--Insulator Transition Insulator Transition in the bulk systemsin the bulk systems[Basko, Aleiner, Altshuler (2005)]
MetalMetal--Insulator “Transition” Insulator “Transition” in zero dimensionsin zero dimensions
[Altshuler, Gefen, Kamenev,Levitov (1997)]
- one-particle level spacing;
1-particle level spacing inlocalization volume;localization volume;
1) Localization in Fock space = Localization in the coordinate space.
2) Interaction is local;
![Page 33: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/33.jpg)
Vs. finite T MetalMetal--Insulator Transition Insulator Transition in the bulk systemsin the bulk systems[Basko, Aleiner, Altshuler (2005)]
MetalMetal--Insulator “Transition” Insulator “Transition” in zero dimensionsin zero dimensions
[Altshuler, Gefen, Kamenev,Levitov (1997)]
- one-particle level spacing;
1-particle level spacing inlocalization volume;localization volume;
1,2) Locality:
3) Interaction matrix elements
strongly depend on the energy transfer, ωωωω:
![Page 34: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/34.jpg)
Effective Hamiltonian for MITMIT.We would like to describe the low-temperatureregime only.
Spatial scales of interest >>
1-particle localization length
Otherwise, conventional perturbation theory fordisordered metals works.
Altshuler, Aronov, Lee (1979); Finkelshtein (1983) – T-dependent SC potential Altshuler, Aronov, Khmelnitskii (1982) – inelastic processes
![Page 35: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/35.jpg)
Details: Seminar #1
December 26
![Page 36: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/36.jpg)
{{{{ }}}} (((( ))))ρ ξ ρr r
ζρr
ξ
dνζδζ
1≡
ar
Reproduces correct behavior of the tails of one particle wavefunctions
{{{{ }}}} (((( )))), jjρ ξ ρr r
O
ζ No spins
![Page 37: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/37.jpg)
ξj1
l1
l2
j2
Interaction only within the same cell;
![Page 38: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/38.jpg)
Statistics of matrix elements?
![Page 39: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/39.jpg)
Parameters:j
lσ randomsigns
![Page 40: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/40.jpg)
Parameters:
j
lσ randomsigns
Ensemble averaging over:
Level repulsion: Only within one cell.
Probability to find n levels in the energy interval of the width E:
/
( , ) x!
e p
nE
e E
En
nFE
nP
ζζ
δ
ζ
δ
δ
−−−− ====
−−−−
(((( ))))
limx
F x
x→∞→∞→∞→∞= ∞= ∞= ∞= ∞
![Page 41: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/41.jpg)
What to calculate?
– random quantity
Idea for one particle localization Anderson, (1958);
MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
No interaction:
Metal InsulatorMetal Insulator
![Page 42: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/42.jpg)
What to calculate?
– random quantity
Idea for one particle localization Anderson, (1958);
MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
No interaction:
Metal InsulatorMetal Insulator
![Page 43: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/43.jpg)
What to calculate?
– random quantity
Idea for one particle localization Anderson, (1958);
MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
No interaction:
Metal InsulatorMetal Insulator
![Page 44: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/44.jpg)
What to calculate?
– random quantity
Idea for one particle localization Anderson, (1958);
MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
No interaction:
Metal InsulatorMetal Insulator
![Page 45: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/45.jpg)
What to calculate?
– random quantity
Idea for one particle localization Anderson, (1958);
MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)
No interaction:
η!0metal
insulator
behavior for a
given realization
metal
insulator
∼ η
probability distribution
for a fixed energy
![Page 46: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/46.jpg)
Probability Distribution
metal
insulator
Note:
Look for:
![Page 47: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/47.jpg)
How to calculate?How to calculate?
Parameters:
non-equilibrium (arbitrary occupations) → Keldysh
allow to select the most
relevant series
SCBA
relevant series
Find the distribution function of each diagram
![Page 48: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/48.jpg)
Iterations:Iterations:
Cayley tree structure
![Page 49: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/49.jpg)
after standard simple tricks:
Nonlinear integral equation with random coefficients
Decay due to tunneling
Decay due to e-h pair creation
+ kinetic equation for occupation function
![Page 50: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/50.jpg)
Stability of metallic phaseStability of metallic phase
Assume is Gaussian:
>>
( )2
![Page 51: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/51.jpg)
Probability Distributions“Non-ergodic” metal [discussed first in AGKL,97]
![Page 52: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/52.jpg)
Probability Distributions
Drude metal
![Page 53: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/53.jpg)
Kinetic Coefficients in Metallic PhaseKinetic Coefficients in Metallic Phase
![Page 54: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/54.jpg)
Kinetic Coefficients in Metallic PhaseKinetic Coefficients in Metallic Phase
WiedemannWiedemann--Frantz law ?Frantz law ?
![Page 55: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/55.jpg)
Non-ergodic+Drude metal
So far, we have learned:
Trouble !!!Trouble !!!
Insulator
???
![Page 56: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/56.jpg)
Nonlinear integral equation with random coefficients
Stability of the insulatorStability of the insulator
Notice: for is a solution
Linearization:
![Page 57: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/57.jpg)
Recall:
# of interactions# of hops in space
metal
insulator
η
probability distribution
for a fixed energyunstable
STABLE
![Page 58: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/58.jpg)
So, we have just learned:
Insulator
Metal
Non-ergodic+Drude metal
![Page 59: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/59.jpg)
Extension to non-degenerate system
I.A. and B.L. Altshuler , unpublished (2008)
For 1D it leads to:
![Page 60: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/60.jpg)
Estimate for the transition temperature for general case
¢ t (Tc) ' U(Tc)N1(Tc)
2) Identify elementary (one particle) excitations and prove that they are localized.
3) Consider a one particle excitation at finite T and the possible paths of its decays:
1) Start with T=0;
¢ t (Tc) ' U(Tc)N1(Tc)
Energy mismatch
Interaction matrix element
# of possible decay processes of an excitationsallowed by interaction Hamiltonian;
![Page 61: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/61.jpg)
Weakly interacting bosons in
one dimension
![Page 62: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/62.jpg)
Details: Seminar #2
December 28
![Page 63: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/63.jpg)
![Page 64: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/64.jpg)
Phase diagram
1Crossover????No finite T phase transition
1
No finite T phase transitionin 1D
See e.g.Altman, Kafri, Polkovnikov, G.Refael, PRL, 100, 170402 (2008); 93,150402 (2004).
G.M. Falco, T. Nattermann, & V.L. Pokrovsky, PRB,80, 104515 (2009) ????.
![Page 65: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/65.jpg)
( )1 32
ctκ γ=
1 3
ctκ =
( )ctκ
E ngκ ∗≡
~1κ
Finite temperature phase transition in 1D
1 γ1 γ1
( )ctκ γ=
t T ng≡
~1c
κ
I.A., Altshuler, ShlyapnikovarXiv:0910.434; Nature Physics (2010)
![Page 66: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/66.jpg)
Disordered interacting bosons in two dimensionsDisordered interacting bosons in two dimensions
![Page 67: Igor Aleiner (Columbia) · Igor Aleiner (Columbia) Collaborators: B.L. Altshuler (Columbia, NEC America)D.M. Basko (Columbia, Trieste, Grenoble)G.V. Shlyapnikov Many body localization](https://reader033.vdocuments.us/reader033/viewer/2022051807/60038bf585c3f10c882adf07/html5/thumbnails/67.jpg)
Conclusions:
• Existence of the many-body mobility
threshold is established.
• The many body metal-insulator transition is • The many body metal-insulator transition is
not a thermodynamic phase transition.
• It is associated with the vanishing of the
Langevine forces rather the divergences in
energy landscape (like in classical glass)
• Only phase transition possible in one
dimension (for local Hamiltonians)