Download Levitation of delocalized states in weak magnetic fields

Localization Properties of 2D
Random-Mass Dirac Fermions
M. E. Raikh
Department of Physics
University of Utah
In collaboration with V. V. Mkhitaryan
Phys. Rev. Lett. 106, 256803 (2011).
Supported by: BSF Grant No. 2006201
time reversal symmetry is
sustained due to two species
of Dirac fermions
clean Dirac fermions of a
given type are chiral
for energies inside the gap
they exhibit quantum
Hall transition upon E   E
 xy  0
 xy
from Kubo formula:
for zero energy,
e2

2h
Contact of two Dirac systems with opposite signs of mass
in-gap (zero energy)
chiral edge states
M ( x, y )  f ( y )
x
e

i [ E V ( x )] dx
0
H   x p x   y p y  M ( x, y ) z
x
line f=0 supports an edge state
e
 y
1


exp   f ( y )dy  
 0
1
pseudospin structure:
pseudospin is directed
along x-axis
sign of E  V defines the direction of propagation (chirality)

i [ E V ( x )] dx
0
y
 1 
exp   f ( y)dy 
 L
  1
states with the same chirality
bound to y=0, y=-L
D-class: E  V (x)  0  no phase
accumulated in course of propagation
along the edge
Hamiltonian contains both Dirac species
“vacuum” A
A and B correspond to different
pseudospin directions
in-gap state
 sin( k0 x) 
left: Bloch functions  cos(k x) 
0


right: Bloch functions
 cos(k0 x) 


sin(
k
x
)
0


Closed contour M ( x, y )  0
Example: azimuthal symmety: M ( x, y )  M (r )
Dirac Hamiltonian in polar coordinates

M
e i (i r  1 r  ) 

H   i

M
 e (i r  1 r  )

zero-mass contour with radius a
M 0
M 0
a
M (r ) r a  0, M (r ) r a  0, M (r ) r a  0
zero-energy solution
r
  e  i / 2 

 (r ,  ) 
exp   d M (  ) i / 2 
r

a
 ie
picks up a  phase along a
contour arround the origin
 a

divergence at r  0 is multiplied by a small factor exp   d M (  )
 0

pseudospin
Chiral states of a Dirac fermion on the contours M(x,y)=0 constitute chiral network
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)
M ( x, y )  0
no edge state

 xy  0
fluxes through the contours
account for the vector structure
of Dirac-fermion wave functions
can Dirac fermions delocalize at
M ( x, y)  0 ?
J. H. Bardarson, M. V. Medvedyeva, J. Tworzydlo,
A. R. Akhmerov, and C. W. J. Beenakker,
Phys. Rev. B. 81, 121414(R) (2010).
edge state
 xy  1
K. Ziegler, Phys. Rev. Lett. 102, 126802 (2009);
Phys. Rev. B. 79, 195424 (2009).
in CC model delocalization occurs
at a single point where V ( x, y)  0
the same as classical percolation
in random potential V ( x, y )
The answer: It depends ...
on details of coupling between two M ( x, y )  0 contours
M 0
t
general form of the scattering matrix: S  
r
M 0
small contour a  1/ M does
not support an edge state


t
M 0
results in overall phase factor
e i 2  1
elimination of two
t t
M 0
change of sign is equivalent
to elimination of  fluxes
through contacting loops
in the language of scattering matrix:
t
one small contour: 
r

flux through small
contour is zero
the effect of small contours: change of singns of t and r
without significantly affecting their absolute values
ei t 2ei (t ) 2
t
r 

t
r 

t
 t

 r
 fluxes
unlike the CC model which has random
phases on the links, sign randomness
in t and r is crucial for D-class
N. Read and D. Green, Phys. Rev. B. 61, 10267 (2000).
r

t 
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).
(2001)
S t
Cr
arrangement insures a
 -flux through each
plaquette
“reversed”  t

scattering matrices
r
p - percentage of
I
r 

t
 t

 r
r

t 
From the point of view of level statistics
RMT density of states in a
sample with size , L
Historically
1

  t2  
2

pairing
bare Hamiltonian with SO
H   tij ci c j  ij ci c j  H.c.
ij
tricritical point
From quasi-1D perspective
Lately
  1.4  0.2
T
from M  64,128
  1.7 1.6 1.2 1.1
from M  16, 32, 64,128
M
 cosh 
T  
 sinh 
sinh  

cosh  
Transfer matrix of a slice of
width, M, up to M=256
new attractive fixed point

H Dirac  v x px   y p y   v 2 M (r ) z

M (r ) is randomly distributed in the interval M   M , M   M 
weak antilocalization
random sign of mass:
transition at
 M  M
Principal question:
how is it possible that delocalization takes place when
coupling between neighboring contours is weak?
classically must be localized
t 2  1
 t2
has a classical analog
microscopic mechanism of delocalization due to the disorder in signs of
transmission coefficient?
Nodes in the D-class network
transmission
S
I
II
reflection
t
 s c

 
 c s
r
S
IV
III
1. change of sign of c
transforms  -fluxes in
plaquetts II and IV
into 0 -fluxes
2. change of sign of s
transforms  -fluxes in
plaquetts I and III
into 0 -fluxes
signs of the S- matrix
elements ensure
fluxes
through plaquetts

Cho-Fisher disorder in the signs of masses
t , with probabilit y 1- w 2
ti  
  t , with probabilit y w 2
A. Mildenberger, F. Evers, A. D.
Mirlin,
and J. T. Chalker,
Phys. Rev. B 75, 245321 (2007).

 1  t 2 , with probabilit y 1- w 2
ri  
2


1

t
, with probabilit y w 2

O(1) disorder: sign factor -1
on each link with probability w
the limit of strong S   1
0
inhomogeneity:

0
0 1
 with probability P  t 2 ; S  
 with probability (1-P)
 1
1 0
bond between II and IV connects
bond between II and IV is removed
RG transformation for bond percolation on the square lattice
I
II
IV
III
bonds
RG equation
superbond
p  p 5  5 p 4 1  p   8 p 3 1  p   2 p 2 1  p 
2
p
probability that
a bond connects
localization
radius
p
one bond is removed
probability that a
superbond connects
1

 ( p)   p  
2


scaling factor
1

 2 p  
2

three bonds are removed
fixed point p  p 

3
 
1
2
ln 2
ln( dp / dp)
 1.428
1
p
2
Quantum generalization
Quantum generalization
supernode for the
red sublattice
truncation
green sublattice
t r

S   
r t 
second RG step
tˆ
r̂
r̂
 tˆ
red sublattice
t r 

S  
r t
 tˆ rˆ  reproduces the structure of S for
ˆ
S- matrix of the red supernode S  

the red node
 rˆ  tˆ 
-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
S- matrix of supernode consisting of four green
and one red nodes reproduces the structure
t r

S   
r t 
of the green node
t t (r r r  1)  t 2t 4 (r1r3r5  1)  t3 (t1t 4  t 2t5 )
tˆ  1 5 2 3 4
(r3  r1r5 )( r3  r2 r4 )  (t3  t1t 2 )(t3  t 4t5 )
rˆ 
r1r2 (t3t 4t5  1)  r4 r5 (t1t 2t3  1)  r3 (r1r4  r2 r5 )
(r3  r1r5 )( r3  r2 r4 )  (t3  t1t 2 )(t3  t 4t5 )
Evolution with sample size, L
five pairs
ti , ri 
generate a pair
introducing a vector of a unit length
L2
n
tˆ, rˆ

ui  ti , ri 
t t (r r r  1)  t 2t 4 (r1r3r5  1)  t3 (t1t 4  t 2t5 )
tˆ  1 5 2 3 4
(r3  r1r5 )( r3  r2 r4 )  (t3  t1t 2 )(t3  t 4t5 )
rˆ 
r1r2 (t3t 4t5  1)  r4 r5 (t1t 2t3  1)  r3 (r1r4  r2 r5 )
(r3  r1r5 )( r3  r2 r4 )  (t3  t1t 2 )(t3  t 4t5 )
ti , ri
with “projections”


 5 

    

Pn 1 (u )     du j Pn (u j )   u  uˆ{u1 ,..., u5 }
 j 1

RG transformation
ti  ri  1
Zero disorder
tˆ  rˆ  1
2
2
fixed point
distribution remains symmetric
and narrows
2
p1 (t )
p0 (t 2 )
expanding
t2
fixed-point distribution :
p (t 2 )   (t 2  1 2)
5
1
1 

ˆt 
  ci  ti 

2 i 1 
2
c1  c2  c4  c5  2 1
c3  3  2 2
5
the rate of narrowing:
 tˆ   ci2  ti 2  0.7  ti 2
2
no mesoscopic fluctuations at
i 1
L 
Critical exponent
 t  1 2
If p0 (t 2 ) is centered around t02  1 2 ,
n
2
the center of pn (t ) moves to the left as
2
0
from
p3 (t 2 )
p1 (t 2 )
where  
 t 
n
2
0
1
2
1
critical exponent: 
t02  0.45
t2
no sign disorder & nonzero average mass
c
i 1
&

p0 (t 2 )
exceeds exact
5
 1
 2 2 1
  2  t 
n
2
0
ln 2
 1.15
ln 
by 15 percent
( x)  exp M x
insulator
i

M  1 2  t02

1
2


Finite sign disorder
t t (r r r  1)  t 2t 4 (r1r3r5  1)  t3 (t1t 4  t 2t5 )
tˆ  1 5 2 3 4
(r3  r1r5 )( r3  r2 r4 )  (t3  t1t 2 )(t3  t 4t5 )
if all
ti are small and ri  1, we expect
tˆ  ti2
special realization of sign disorder:
choosing ti  t and r1  r2  r3  r4  1  t 2 , r5   1  t 2
2
3
2
3
3
t
(
1

r
)

t
(
1

r
)

2
t
tˆ 
1
2
2
2 2
(r  r )( r  r )  (t  t )
tˆ  1 identically
resonant tunneling!
we get
t , with probabilit y 1- w 2
ti  
  t , with probabilit y w 2
Disorder is quantified as
w  0.2

 1  t 2 , with probabilit y 1- w 2
ri  
2

  1  t , with probabilit y w 2
portion of resonances is 27%
t 02  0.1, t 2  0.05
2. portion of resonances weakly depends
t2
1. resonances survive a spread
in the initial distributon of t i2
on the initial distribution
w  0.2
w  0.2
portion of resonances is 26%
portion of resonances is 24%
t02  0.2, t 2  0.025
t02  0.35, t 2  0.15
t2
origin of delocalization: disorder prevents the flow towards insulator
t2
t 2  0.2, w  0.15
Evolution with the sample size
t 2  0.2, w  0.2
no difference after
the first step
distribution of
reflection amplitudes
difference between two
distributions is small
with t 2  0.025 removed
more resonances for
stronger disorder
universal distribution
of conductance, G  t 2
t2
resonances are suppressed,
system flows to insulator
P(G ) 
0.237
[G (1  G )]0.6
resonances drive the system
to metallic phase

Delocalization in terms of unit vector u  t, r   cos , sin  
0.118
metallic phase corresponds to Q( ) 
[cos  sin  ]0.2
no disorder: initial distribution

with 0  4 flows to insulator u  0,1
r
Q
r

1

u
4
1
1
0
1
t , with probabilit y 1- w 2
ti  
  t , with probabilit y w 2
 1  t 2 , with probabilit y 1- w 2
ri  
  1  t 2 , with probabilit y w 2
is almost homogeneously
distributed over unit circle
t
t
w  wc
r
with disorder

r
resonances at intermediate sizes
spread homogeneously
over the circle
upon increasing L
1
1
1
t
w  wc
1
t
Delocalization as a sign percolation
w  0.15
p (t 2 )
w  0.2
L  23
0.002  t 2  0.02
p (t 2 )
0.002  t 2  0.02
0.02  t 2  1
0.02  t 2  1
t2
t2
at small L, the difference between w  0.2 and w  0.15 is minimal for
p (r )
but is significant in distribution of amplitudes
p (t 2 ),
ratio of peaks is 1.8
p(r )
1  r  1
p(r )
1  r  1
1  r  0
1  r  0
r
r
evolution of the portion,

, of negative values of reflection coefficient with the sample size
Phase diagram
wc (0)  0.21
 xy  0
 xy  1
wtr  0.06
Critical exponent of I-M transition
as signs are “erased” with L, we have
r 2 ( L)  1
t 2  0.2, wc  0.18
fully localize after
fully localize after
7
6
steps
steps
 0.15  A( wc  0.15)   27
 0.127  A( wc  0.127)   26
  ln 2 ln 5.33   1.2
d r2
d ln L
 0.11 : not a critical region
p (t 2 )
w  0.05
Tricritical point
p (t 2 )
w  0.07
wtr  0.06
t2
t2
all
ti
are small, and are
ri
“analytical” derivation of wc  0.2
close to 1
w
t t (r r r  1)  t 2t 4 (r1r3r5  1)  t3 (t1t 4  t 2t5 )
tˆ  1 5 2 3 4
(r3  r1r5 )( r3  r2 r4 )  (t3  t1t 2 )(t3  t 4t5 )
resonance: only one of these brackets is small
probability that only one of the above brackets is small:

w  4w(1  w)3  4w3 (1  w)  12 1  [1  2w]4

w
Conclusions
P(G ) 
0.237
[G (1  G )]0.6
RG
numerics
metallic phase emerges even for vanishing
transmission of the nodes due to resonances
delocalization occurs by proliferation of
resonances to larger scales
Bilayer graphene:
gap of varying sign is generated
spontaneously
in-gap state
M 0
chiral propagation within a
given valley
M 0
symmetry between
the valleys is lifted
at the
edge
Topological origin of subgap conductance in
insulating bilayer graphene
J. Li, I. Martin, M. Buttiker, A. F. Morpurgo,
Nature Physics 7, 38 (2011)
Quantum RG transformation
super-saddle point
t
1
r
G
1
1  ez
t
Determination of the critical exponent
 n z0  1
Q ( z  z0 ) Q ( z   z )
0
  2n
insulator
z0
z

ln 2
 2.39  0.01
ln