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