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M. E. Raikh. Localization Properties of 2D Random-Mass Dirac Fermions . Department of Physics. University of Utah. V. V. Mkhitaryan . In collaboration with . Phys. Rev. Lett . 106 , 256803 (2011). Supported by: BSF Grant No. 2006201 . - PowerPoint PPT Presentation
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Localization Properties of 2D Random-Mass Dirac Fermions
Supported by: BSF Grant No. 2006201
V. V. Mkhitaryan
Department of PhysicsUniversity of Utah
In collaboration with
M. E. Raikh
Phys. Rev. Lett. 106, 256803 (2011).
clean Dirac fermions of agiven type are chiral
he
xy 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 ),(
11
)(exp0
)]([0
yxdxVEi
ydyfe
x
11
)(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
xkxk
)sin()cos(
0
0
xkxk
Example: azimuthal symmety: )(),( rMyxM
0)(,0)(,0)( ararar
rMrMrM
2/
2/
)(exp),(
i
ir
a iee
Mdr
r
picks up a phase along a contour arround the origin
Closed contour 0),( yxM
Dirac Hamiltonian in polar coordinates
MrierieM
Hr
ir
i
)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 links2D 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
trrt
one small contour:
trrt
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
trrt
trrtp
I
From the point of view of level statistics
RMT density of states in a sample with size , L
212t
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
coshsinhsinhcosh
T
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
sccs
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 IV0
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,
wtw-t
ti
2yprobabilitwith,121yprobabilitwith,1
2
2
wtw-tri
t
r
RG transformation for bond percolation on the square lattice
322345 121815 pppppppp RG equation bonds
superbond
p p
superbond connectsa bond connects
one bond is removed three bonds are removed
probability that probability that a fixed point
21
pp
212
21)( ppplocalization
radius428.1
)/ln(2ln
21
pdppd
scaling factor
the limit of strong inhomogeneity:
10
01S with probability ;
0110
S 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
trrt
S
tr
rtS
r̂ t̂
r̂
S- matrix of the red supernode
tr
rtS
ˆˆˆˆˆ reproduces the structure of S for
the red node
S- matrix of supernode consisting of four green and one red nodes reproduces the structure
trrt
S
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(ˆ
543213423513
524135314243251
ttttttrrrrrrtttttrrrttrrrttt
))(())(()()1()1(ˆ
543213423513
524133215454321
ttttttrrrrrrrrrrrtttrrtttrrr
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(ˆ
543213423513
524135314243251
ttttttrrrrrrtttttrrrttrrrttt
))(())(()()1()1(ˆ
543213423513
524133215454321
ttttttrrrrrrrrrrrtttrrtttrrr
21 ii rt 21ˆˆ rt fixed point
)( 20 tp
)( 21 tp
2t
distribution remains symmetricand narrows
expanding
21
21ˆ
5
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 exponent212
0 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
212
02 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(ˆ
543213423513
524135314243251
ttttttrrrrrrtttttrrrttrrrttt
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,
wtw-t
ti
2yprobabilitwith,121yprobabilitwith,1
2
2
wtw-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 to 2.0]sin[cos118.0)(
Q
Q
is almost homogeneously distributed over unit circle
u
t1
r
0
4
1
no disorder: initial distribution with flows to insulator 1,0u40
t1
r
1with disorder
t1
r
1
t1
r
1
cww resonances at intermediate sizes
2yprobabilitwith,21yprobabilitwith,
wtw-t
ti
2yprobabilitwith,121yprobabilitwith,1
2
2
wtw-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(ˆ
543213423513
524135314243251
ttttttrrrrrrtttttrrrttrrrttt
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
11 zG
e
r
1t
t
Quantum RG transformation super-saddle point
0( )Q z z0( )Q z z
z0z
insulator
0 1nz
2n
ln 2 2.39 0.01ln
Determination of the critical exponent