Localization Properties of 2D Random-Mass Dirac Fermions

Supported by: BSF Grant No. 2006201

V. V. Mkhitaryan

Department of Physics

University of Utah

In collaboration with

M. E. Raikh

Phys. Rev. Lett. 106, 256803 (2011).

clean Dirac fermions of agiven type are chiral

h

exy 2

2

for energies inside the gap

0xy

for zero energy,

time reversal symmetry is sustained due to two species

of Dirac fermions

from Kubo formula:

they exhibit quantum Hall transition upon EE

Contact of two Dirac systems with opposite signs of mass

in-gap (zero energy) chiral edge states

states with the same chirality bound to y=0, y=-L

pseudospin structure:pseudospin is directedalong x-axis

D-class: no phase accumulated in course of propagation

along the edge

0)(xVE

line f=0 supports an edge state

sign of defines the direction of propagation (chirality)VE

zyyxx yxMppH ),(

1

1)(exp

0

)]([0

yxdxVEi

ydyfe

x

1

1)(exp0

)]([ y

L

xdxVEi

ydyfe

x

)(),( yfyxM

Hamiltonian contains both Dirac species

“vacuum” A A and B correspond to different pseudospin directions

in-gap state

left: Bloch functions

right: Bloch functions

)cos(

)sin(

0

0

xk

xk

)sin(

)cos(

0

0

xk

xk

Example: azimuthal symmety: )(),( rMyxM

0)(,0)(,0)( ararar

rMrMrM

2/

2/

)(exp),(

i

ir

a ie

eMd

rr

picks up a phase along a contour arround the origin

Closed contour 0),( yxM

Dirac Hamiltonian in polar coordinates

Mrie

rieMH

ri

ri

)1(

)1(

zero-mass contour with radius a

zero-energy solution

pseudospin

a

divergence at is multiplied by a small factor 0r

a

Md0

)(exp

0M

0M

Chiral states of a Dirac fermion on the contours M(x,y)=0 constitute chiral network

fluxes through the contours account for the vector structure of Dirac-fermion wave functions

0),( yxM

0xy

no edge state

1xyedge state

scalar amplitudes on the links

2D electron in a strong magnetic field: chiral drift trajectories along equipotential V(x,y)=0 also constitute a chiral network Chalker-Coddington network model J. Phys. C. 21, 2665 (1988)

in CC model delocalization occurs at a single point where 0),( yxV

the same as classical percolationin random potential ),( yxV

can Dirac fermions delocalize at ? 0),( yxM

K. Ziegler, Phys. Rev. Lett. 102, 126802 (2009);Phys. Rev. B. 79, 195424 (2009).

J. H. Bardarson, M. V. Medvedyeva, J. Tworzydlo,A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B. 81, 121414(R) (2010).

The answer: It depends ...on details of coupling between two contours

0),( yxM

tr

rtS

general form of the scattering matrix:

unlike the CC model which has randomphases on the links, sign randomnessin and is crucial for D-classt r

0M

0M

small contour does not support an edge state

Ma /1

0M

0M

the effect of small contours: change of singns of and t rwithout significantly affecting their absolute values

flux through small contour is zero

tr

rtone small contour:

tr

rt

N. Read and D. Green, Phys. Rev. B. 61, 10267 (2000).

N. Read and A. W. W. Ludwig, Phys. Rev. B. 63, 024404 (2000).

M. Bocquet, D. Serban, and M. R. Zirnbauer, Nucl. Phys. B. 578, 628 (2000).

t t

tt

22 )( tete ii

in the language of scattering matrix:

results in overall phase factor 12 ie

elimination of two fluxes

change of sign is equivalent to elimination of fluxes through contacting loops

(2001)

tS rC

arrangement insures a -flux through each

plaquette

- percentage of “reversed” scattering matrices

tr

rt

tr

rtp

I

From the point of view of level statistics

RMT density of states in a sample with size , L

2

12t

Historically

ijjiijjiij cccctH H.c.

tricritical point

bare Hamiltonian with SO pairing

From quasi-1D perspective

new attractive fixed point

Transfer matrix of a slice of width, M, up to M=256

2.04.1

1.12.16.17.1

128,64,32,16Mfrom

128,64Mfrom T

M

coshsinh

sinhcoshT

Lately

zyyxx rMvppvH )(2Dirac

is randomly distributed in the interval )(rM MMMM ,

weak antilocalization

random sign of mass: transition at

MM

Principal question:

12 t2t

how is it possible that delocalization takes place when coupling between neighboring contours is weak?

has a classical analog

classically must be localized

microscopic mechanism of delocalization due to the disorder in signs of transmission coefficient?

Nodes in the D-class network

sc

cs

signs of the S- matrix elements ensure fluxes

through plaquetts

S

S

IV

I II

III

1. change of sign of transforms -fluxes in plaquetts and into -fluxes

c

II IV

0

2. change of sign of transforms -fluxes in plaquetts and into -fluxes

s

I III0

Cho-Fisher disorder in the signs of masses

A. Mildenberger, F. Evers, A. D. Mirlin,

and J. T. Chalker,Phys. Rev. B 75, 245321 (2007).

reflection

transmission

O(1) disorder: sign factor -1 on each link with probability w

2yprobabilitwith,

21yprobabilitwith,

wt

w-tti

2yprobabilitwith,1

21yprobabilitwith,12

2

wt

w-tri

t

r

RG transformation for bond percolation on the square lattice

322345 121815 pppppppp RG equation

bondssuperbond

p p

superbond connectsa bond connects

one bond is removed three bonds are removed

probability that probability that a fixed point

2

1pp

2

12

2

1)( ppp

localizationradius

428.1)/ln(

2ln

2

1

pdppd

scaling factor

the limit of strong inhomogeneity:

10

01S with probability ;

01

10S with probability (1-P)

bond between and connectsII IV bond between and is removedII IV

I

IIIIV

II

2tP

Quantum generalization

second RG step

Quantum generalization

truncation

supernode for the red sublattice

green sublattice red sublattice

t̂

tr

rtS

tr

rtS

r̂ t̂

r̂

S- matrix of the red supernode

tr

rtS

ˆˆ

ˆˆˆ reproduces the structure of S for

the red node

S- matrix of supernode consisting of four green and one red nodes reproduces the structure

tr

rtS

of the green node

-1 emerges in course of truncationand accounts for the missing green node

from five pairs of linear equations we find the RG transformations for the amplitudes

))(())((

)()1()1(ˆ543213423513

524135314243251

ttttttrrrrrr

tttttrrrttrrrttt

))(())((

)()1()1(ˆ

543213423513

524133215454321

ttttttrrrrrr

rrrrrtttrrtttrrr

Evolution with sample size, L

},...,{ˆ)()( 51

5

11 uuuuuPuduP

jjnjn

Zero disorder

five pairs generate a pair ii rt , rt ˆ,ˆ

introducing a vector of a unit length iii rtu ,

with “projections” ii rt ,

RG transformation

nL 2

))(())((

)()1()1(ˆ543213423513

524135314243251

ttttttrrrrrr

tttttrrrttrrrttt

))(())((

)()1()1(ˆ

543213423513

524133215454321

ttttttrrrrrr

rrrrrtttrrtttrrr

21 ii rt 21ˆˆ rt fixed point

)( 20 tp

)( 21 tp

2t

distribution remains symmetricand narrows

expanding

2

1

2

1ˆ5

1i

ii tct

125421 cccc 2233 c

fixed-point distribution :

)21()( 22 ttp

If is centered around , )( 20 tp

the rate of narrowing:

5

1

2222 7.0ˆi

iii ttct no mesoscopic fluctuations at L

Critical exponent

2120 t

)( 2tpn

2120 tn

5

1

122i

ic

critical exponent: 15.1ln

2ln

the center of moves to the left as

exceeds exact by 15 percent1

xMx exp)( 2021 tM

no sign disorder & nonzero average mass insulator

)( 21 tp

)( 20 tp

)( 23 tp

2t45.02

0 t

where

from 1212

0 tn 2

1202 tn

&

Finite sign disorder

choosing and , tti we get

1ˆ t

24321 1 trrrr

identically

25 1 tr

resonant tunneling!

1)())((

2)1()1(ˆ2222

33232

ttrrrr

trtrtt

))(())((

)()1()1(ˆ543213423513

524135314243251

ttttttrrrrrr

tttttrrrttrrrttt

if all are small and , we expect it 1ir2ˆitt

special realization of sign disorder:

Disorder is quantified as

2t

2t 2t

1. resonances survive a spreadin the initial distributon of 2

it

portion of resonances is 24%portion of resonances is 26%

portion of resonances is 27%

2.0w

2.0w 2.0w

origin of delocalization: disorder prevents the flow towards insulator

025.0,2.0 220 tt

15.0,35.0 220 tt

05.0,1.0 220 tt

2. portion of resonances weakly depends on the initial distribution

2yprobabilitwith,

21yprobabilitwith,

wt

w-tti

2yprobabilitwith,1

21yprobabilitwith,12

2

wt

w-tri

Evolution with the sample size

resonances are suppressed, system flows to insulator

resonances drive the systemto metallic phase

universal distributionof conductance,

15.0,2.02 wt 2.0,2.02 wt

difference between twodistributions is small

more resonances for stronger disorder

2t 6.0)]1([

237.0)(

GGGP

2tG

distribution of reflection amplitudes

025.02 twith removed

no difference after the first step

Delocalization in terms of unit vector sin,cos, rtu

metallic phase corresponds to2.0]sin[cos

118.0)(

Q

Q

is almost homogeneously distributed over unit circle

u

t1

r

0

4

1

no disorder: initial distribution with flows to insulator 1,0u

40

t1

r

1with disorder

t1

r

1

t1

r

1

cww resonances at intermediate sizes

2yprobabilitwith,

21yprobabilitwith,

wt

w-tti

2yprobabilitwith,1

21yprobabilitwith,12

2

wt

w-tri

cww

spread homogeneouslyover the circle

upon increasing L

Delocalization as a sign percolation

02.0002.0 2 t

102.0 2 t102.0 2 t

02.0002.0 2 t

2.0w15.0w

)( 2tp)( 2tp

2t2t

32L

ratio of peaks is 8.1

rr

)(rp

)(rp

11 r11 r

01 r01 r

at small L, the difference between and is minimal for ,but is significant in distribution of amplitudes

2.0w 15.0w )( 2tp)(rp

21.0)0( cw

Phase diagram

0xy 1xy

evolution of the portion, , of negative values of reflection coefficient with the sample size

06.0tr w

Critical exponent of I-M transition

as signs are “erased” with L, we have

1)(2 Lr

715.0 2)15.0( cwA

2.1ln2ln 33.5

fully localize after steps

7

fully localize after steps

6 6127.0 2)127.0( cwA

18.0,2.02 cwt

11.0ln

2

Ld

rd: not a critical region

Tricritical point

“analytical” derivation of

))(())((

)()1()1(ˆ543213423513

524135314243251

ttttttrrrrrr

tttttrrrttrrrttt

42133 ]21[1)1(4)1(4 wwwwww

all are small, and are close to 1 it ir

resonance: only one of these brackets is small

probability that only one of the above brackets is small:

2.0cw

w

w

2t

)( 2tp

2t

)( 2tp05.0w 07.0w

06.0tr w

numericsRG

delocalization occurs by proliferation of resonances to larger scales

Conclusions

6.0)]1([

237.0)(

GGGP

metallic phase emerges even for vanishing transmission of the nodes due to resonances

0M

0M

in-gap state

J. Li, I. Martin, M. Buttiker, A. F. Morpurgo, Nature Physics 7, 38 (2011)

Topological origin of subgap conductance in insulating bilayer graphene

Bilayer graphene:

chiral propagation within a given valley

symmetry betweenthe valleys is lifted

at the edge

gap of varying sign is generated spontaneously

1

1 zGe

r

1t

t

Quantum RG transformationsuper-saddle point

0( )Q z z0( )Q z z

z0z

insulator

0 1nz

2n

ln 22.39 0.01

ln

Determination of the critical exponent