Ergodic phases in strongly disordered random regular graphs VEKravtsov ICTP Trieste Collaboration Boris Altshuler Columbia U Lev Ioffe Paris and Rutgers Ivan Khaymovich Aalto ID: 538942
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Slide1
Non-Ergodic phases in strongly disordered random regular graphs
V.E.KravtsovICTP, Trieste
Collaboration:
Boris
Altshuler, Columbia U.Lev Ioffe, Paris and RutgersIvan Khaymovich, AaltoEmilio Cuevas, Murcia
International Workshop on Localization, Interaction and Superconductivity
Chernogolovka
June 27, 2016Slide2
Real space analogues of MBL?
Bethe lattice?
No loops
Exponential number of sitesSlide3
Anderson model on random regular graphs (RRG)
site disorder:
e
r
Disorder strength W
Finite fraction of states on the boundary
RRGSlide4
Localization transition in 3D and on RRG
disorder
A
T
E
T
Search for extended multifractal states
i
n a finite range of
paramters
AT
G.
Biroli
,
A. Ribeiro-Teixeira, and M. Tarzia,arXiv:1211.7334.Slide5
Why to bother?
A new state of matter – BAD METAL:* No
Bolzmann’s statistics:* No energy equipartition principle* No “Eigenstates Thermalization Hypothesis (ETH)” No ERGODICITY:
Ergodicity of eigenstates in the Hilbert Space results in the ETH that ensures equivalence of time- and Boltsmann ensemble averagesSlide6
Function f(a)Slide7
Search for ergodic transition on RRG
De Luca,
Altshuler, V.E.K.,
Scardicchio al. PRL v.113, 046806 ( 2014) Extrapolation from N=2000-32000
Localized critical AT
m
ultifractal
k
c
=1/2
localized
k<1/2
}
Gap as signature of extended state
Ergodic stateContinuous deformation from localized to extended states
A wide region of non-ergodic extended statesSlide8
IPR and Shannon entropy of eigenstates
EXACT DIAGONALIZATION,
NO EXTRAPOLATION!Slide9
D2 close to AT
Non-stop towards D
2 =1 ?(Tikhonov, Mirlin,
Skvortsov, 2016)Slide10
Exact diagonalization on large RRG
N=128 000
=Slide11
Singularity close to W=10Slide12
Jump in D2(W)Slide13
Evolution of Jump with N
N=128K
N=36K
N=64KDe Luca et al
Tikhonov et alPresent workSlide14
First order Ergodic Transition on K=2 RRG
AT
ET
Altshuler
, Cuevas,
Ioffe
, V.E.K ArXiv:1605.02295Slide15
g=1
g=2
D
1
Rosenzweig
-Porter
Second
order Ergodic Transition in RP RMT
AT
ET
V.E.K.,
Khaymovich
, Cuevas,
Amini
, New J. Phys., 17, 122002 (2015)
D=2-
gSlide16
Self-consistent theory of localization
Extended phase:
Typical
Localized phase: typical ImG->0 as d->0
i
j
j
’
Exact for:
*
Bethe lattice:
no loops
!*RP random matrix: infinite K
n =
lnN
/
lnK
D=
L/
lnK
RP
BLSlide17
Large K limit: ReS
=0
L
= lnK
D=2ln(Wc/W)/lnK
L
=1/2
lnKSlide18
Typical and average Green function
Ergodicity criterion:
Localization criterion (infinite system):Slide19
M.
Aizenman
& S.
Warzel
W
E
Ergodic extended
localized
localized
localized
localized
AT
ET
?
?
(published as
pp. 239-253 in:
XVIIth
International Congress on Mathematical Physics Ed.: A. Jensen World Scientific 2013Slide20
Conclusion
First order ergodic transition at W=10.0 on K=2 RRG with on-site disorderWide region 10.0 < W< 18.2 of non-ergodic extended statesSimilar features are likely for MBLSlide21
RRG W=9.5Slide22
3D Anderson W=10Slide23
N total sites
M ~ N
D
q
ERGODICITY
in the (Hilbert) space
M “occupied” sites