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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

rrg ergodic extended localized ergodic rrg localized extended lnk states transition localization disorder ergodicity cuevas order eigenstates altshuler typical

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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

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