Putting Floquet theory to work:Random hopping systemsRandom hopping systems

and topological phases with time dependent Hamiltonians p g p p

Yue (Rick) Zou (Goldman Sachs, Hong Kong)Netanel Lindner (Caltech, IQI)Ryan Barnett (UMD)Ryan Barnett (UMD)Victor Galitski (UMD)Gil Refael (Caltech)

Funding:gPackard Foundation,

Sloan Foundation,NSF

Exotic wave functions seeking parent Hamiltonian

• Motivation: realizing interesting Hamiltonians using temporal modulation.

Inspiring example: Fractional quantum Hall in optical lattices,Sorensen, Demler, Lukin (2004)

• Personal obsession: Dyson singularity in random hopping systems.

• Current fascination: 2d Topological insulators.d opo og c su o s.

Outline

Part 1 – Random hopping

• Dyson singularity in random hopping models.

Part 1 Random hopping

• Dynamic localization.

• Modulated Anderson disorder - properties.p p

• Experimental realization

P t 2 T l i l hPart 2 – Topological phases

• 2d Topological phases crash course

• How can we topologize the trivial?

• Experimental realization (?)

R d h i d lRandom hopping models

Particle-hole symmetric localizationR d h i i 1d• Random hopping in 1d:

sjsiij cctH ,,ij

t

2 3 4 5 6 7

Particle-hole symmetric localizationR d h i i 1d• Random hopping in 1d:

sjsiij cctH ,,ij

• Chiral (sublattice) symmetry – delocalized state at band center in 1d and 2d.Gade, (1993), Motrunich, Damle, Huse, (2001).

• Random hopping localization vs. interactions in 2d?

See M. Foster, A. Ludwig. Etc.

• Mott glass phase for random bosons (incompressible and gapless).

Altman, Kafri, Polkovnikov, GR (2006), Orignac, Le Doussal, Giamarchi (1998).

Properties of a random hopping model

D i f )(E

Dyson (1953)Thouless (1972)Fisher (1994)

• Denstity of states:

kuE cos2

Pure chain:

)(E

kuEk cos2

22421)(

EuE

Random hopping chain:1)(

E

EEE 3ln

1~)(

• End-to-end wave function decay:End to end wave function decay:

/1 ~ L

L e

||ln~ EE

Experimental realization? • Disorder typically produced by a speckle potential:

From Fallani , Fort, Inguscio, 08

See also exp’s by:Aspect,

D MDeMarco, etc.

)(xV• Effective potential:

)(xV

sjsiij bbuH ,, iii bbvx

random hoppingProblem:Anderson!

ij

sjsiij ,, i

random potential

Anderson insulator vs. random hopping• Denstity of states:

)(E )(E

E E• End-to-end average correlations:

||ln~ EE constE ~

• Dyson singularity Anderson insulator

I t ti bProblem:Anderson!

• Interacting bosons: Mott glass Bose glass

(incompressible, gapless) (compressible, gapless)

)(xV

Dynamic localization

• Periodically Tilted chain:

Holthaus, Arimondo, and others

)(xV• Periodically Tilted chain:

i

iiii bbbbuH 11i

iii

i bbtFx )cos(

x

ftfVV

fAA

iii chbbtFiuH ..sinexp 1

1

0sin )sin()()(

mm

tix tmxJxJe • Use:fAA

tFxtxf sin),(

iii chbbFuJHH ..)( 10

)(xV

Dynamic localization• Periodically Tilted chain: )(

i

iiii bbbbuH 11

iii

i bbtFx )cos(

x

iii chbbFuJHH ..)( 10

W

F

Fully localized flat-band

u2u2

F

band widthW

u2u2

Eckart, Holthaus,…, Arimondo (2008)

)(xV

Modulated on-site disorder

• Modulated disorder: )(xV• Modulated disorder:

i

iiii bbbbuH 11i

iii

i bbv )cos( t

x

iii

ii chbbtvviuH ..sin)(exp 11

1

0sin )sin()()(

mm

tix tmxJxJe • Use:

i

iiii chbbvvuJHH ..)( 110

More honest analysis: Floquet Hamiltonians

• Periodically varying Hamiltonian:

)(tVHH )(0 tVHH

)()( TtVtV with .0)(0

T

tVand:

• Hamiltonian

0

T

Floquet operator:

tHdtiTU

0

)(exp)(

• Energy eigenstates Quasi-energy states:

TiTU )( 0 2n

Tin

neTU )( 0,0 n T 2

Floquet vs. Random hopping: DOS• Compare exact Floquet quasi energy states with averaged random hopping H:

i

iiii chbbvvuJH ..)( 110

• Compare exact Floquet quasi-energy states with averaged random hopping H:

vs. )cos(ˆ)( 0 tnvHtHi

ii

Density of states:

H i Di d idth

- Floquet

- Time averaged HHopping:1u

Disorder width:10|| iv

- Time averaged H

Floquet vs. Random hopping: DOS• Compare exact Floquet quasi energy states with averaged random hopping H:

i

iiii chbbvvuJH ..)( 110

• Compare exact Floquet quasi-energy states with averaged random hopping H:

vs. )cos(ˆ0 tnvHHi

ii

Localization Length

Effectivedisorder

FloquetHopping:

1uDisorder width:

10|| iv

- Floquet

- Time averaged H

Low frequency de-localization?• Surprising trend appears in the low frequency regime:• Surprising trend appears in the low frequency regime:

FFrequency

Hopping:1u

Disorder width:10|| iv

Possible experimental realization

• Speckle potential interaction:

)(3)(2

rIcrV

)(2

)( 30

rIrV

0

• To produce modulation: f(t)

To produce modulation:

)(~ 0 tf t

f(t)

• Use Bragg spectroscopy to obtain DOS.

Recent fascinations:

Topological phases

First observed topological insulator: CdTe, HgTe heterostructures

HgTe

CdTe

nmdd c

6X •

CdTe

Bernevig, Hughes, Zhang (2006)Koenig,.., Molenkamp.., Zhang (2007)

First observed topological insulator: HgTe• HgTe dispersion (per spin): Dirac cone with small band gap g p (p p ) g p

ˆ0 dH

zppbmypxpd ˆ)(ˆˆ 22 zppbmypxpd yxyx )(

0,0 bm (Trivial) 0,0 bm (Topological)E E

xp

yp xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

)(3.0,2.0 topononbm

Ep

Wrapping the unit sphere?

spheren ˆ z

E

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topononbm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topononbm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

EWrapping the unit sphere?

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topononbm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topononbm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

)(3.0,2.0 topobm

E

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?

Ep

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topobm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topobm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topobm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topobm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topobm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

What makes a phase topological?

dH

0 ddn

ˆ )( 22 ppbmppd

• TKNN invariant:

dH 0 ddn )(,, yxyx ppbmppd (map BZ to unit sphere)

E

)(3.0,2.0 topobm

yxBZp

yxxy pn

pnndpdp

he ˆˆˆ

4

2

Wrapping the unit sphere?E

spheren ˆ z

p

spheren z

xx

xpyp

Can we induce a topological phase with radiation effects?S i h l i l H T ll• Start with non-topological HgTe wells:

n

ztA ˆcos

ddn

ˆ

ˆ0 dH 0 dH

zppbmypxpd yxyx ˆ)(ˆˆ 22

0, bm

Irradiation effectsS i h l i l H T ll• Start with non-topological HgTe wells:

n

ztA ˆcos

ddn

ˆ

ˆ0 dH

ˆ)(ˆ

dT

0 dH

zppbmypxpd yxyx ˆ)(ˆˆ 22

0, bm)(ˆexp 21

0

pnssHdtiS S

Irradiation effectsS i h l i l H T ll• Start with non-topological HgTe wells:

ztA ˆcos

)(ˆexp

pnssHdtiST

)(exp 210

pnssHdtiS S

Floquet topological phase

ˆ dH 0 dH

zppbmypxpd yxyx ˆ)(ˆˆ 22

ztA ˆcos

E

xpyp

Equilibration in a Floquet topological phase? (Work in progress)

• Phonons can provide bath for low-energy, low momentum relaxation.p gy,

Experimental realization In (Hg,Cd)Te wells?

HgTe

CdTe(Initially Non- ztB ˆcos HgTe

CdTeTopological)

tB cos

(Tera-Hertz frequencies)

• Using Zeeman coupling:mTB 10~

K1.0~

• Electric field: • Electric field: Linear Stark effect enhancement

mVE 1~ mVE 1~

K10~ kiEErrV

)(

Summary and conclusions

• Periodically modulated systems could be used to stabilize unique

quantum statesquantum states.

R i f d h i M d l d kl l• Recipe for random hopping: Modulated speckle potential.

• Topological insulator on demand: modulate the subband gap.

• Open questions:

Relaxation into a steady state – will the unique properties shine out?y q p p