Download Verre de Bragg et diagramme de phase des

Chiral (exciton condensation) transition
in graphene.
Baruch Rosenstein
Nat. Chiao Tung University, Hsinchu
Collaborators:
Hsien-Chong Kao (Nat. Taiwan Normal University, Taipei)
Meir Lewkowicz (Ariel University, Israel),
Tsofar Maniv (Technion, Israel)
KITPC, Beijing , August 10, 2012
Outline
1. Noninteracting electrons in graphene : relativistic massless
fermions in 2+1.
2. Finite conductivity without either carriers or impurities.
Screening.
3. Effect of large electric coupling: how can graphene become an
insulator?
4. Dynamical chiral symmetry breaking.
5. Chiral universality classes.
6. Understanding a conductor with long range Coulomb
interactions.
1. MASSLESS RELATIVISTIC FERMIONS IN GRAPHENE
Carbon atoms in graphene are arranged in honeycomb 2D
crystal by the covalent bonds between nearest neighbours.
E. Andrei et al, Nature
Nano 3, 491(08)
Single graphene sheet on STM
It can be viewed as a prototype of various 0D-3D
structures based on the similar covalent C bonds.
Novoselov et al, Proc. Nat.
Acad. Sci. USA 102 10451 (05)
Geim et al, Nature
Mat. 6, 183 (07)
Graphene shows metallic behaviour even at neutrality point
where resistivity is maximal. Quantum Hall effect at room
temperature clearly shows high mobility. Inhomogeneities or
interaction, definitely present are somehow almost invisible in
transport.
Honeycomb has two hexagonal sublattices.
Two representations of
unit cell:
Two atoms
1/3 each of 6 atoms
= 2 atoms
Tight binding model
Hˆ   


As  Bs
ˆ
a
 rn aˆrn 
sites 1,2,3 s ,
 c.c.
Spectrum consists of two bands
with Fermi surface pinched right
between them: not an obvious
band insulator or a metal.
Fermi surface is located exactly at two non - equivalent points
of the Brillouin zone with sufficient little group to support a
two dimensional representation
K
K


KK, K
K
Around K  the Hamiltonian is that of the left and right two
component Weyl spinors:  


H  vg σ  p
σ    x ,  y 

Wallace, PR71 622 (1947)
3 a
c
vg 

2
300
The Dirac point in band structure was pointed out very early
by Wallace: electrons in graphene should behave like a 2D
analogue of relativistic massless particles (“charged neutrinos”).
The L,R Weyl spinors can
be combined into two 4Dirac spinor described by
L    0 t   1 x   2 y  s
s
that possesses the Z 2 chiral
(sublattice) symmetry
 s   5 s ; 5  i 0 1 2 3
2. FINITE CONDUCTIVITY WITHOUT EITHER CHARGES
OR INPURITIES. SCREENING.
Nonzero pseudo Ohmic conductivity
There is resistivity at Dirac point,
but resistivity is finite
Geim et al, Nature Mat.
6, 183 (07)
SiO2
Yacoby et al, Nature (08)
Si
graphene
Samples on substrate generally correspond to either p or n
doped graphene.
Recently developed suspended samples are clearly undoped and
exhibit the minimal resistivity at zero temperature.

 e2
2 h
E. Andrei et al, Nature
Nano 3, 491(08)
Optical frequencies
conductivity was
measured to accuracy of
1%. And agrees with the
theoretical value
Geim, Novoselov et al, Science 102 10451 (08)
Explanation: Schwinger’s electron – hole pair creation
by electric field
The basic picture of the pseudo –
diffusive resistivity in pure
graphene is the creation of the
electron – hole pairs by electric
field. The pairs carry current that
can be further increased by
reorientation of moving particles

E
   
 e2
2 h

Linear response
Dynamical approach to linear response
Let’s try to understand qualitatively how massless fermions
react on electric field by just switching a homogeneous electric
field and observing the creation of electron - hole pairs and
the induced charge motion.
Lewkowicz, B.R., PRL102, 106802 (09)
hk   ei k

e
 
2
2
2
 h h ' h ' hk 
 2 k
sin




2
2


BZ 
k

*
k k
*
k
 e2

t   2 
2 h

t  / 
One indeed observes that the current stabilizes at finite value. In
DC this requires the use of tight binding model rather than its
Dirac continuum theory.
In addition to the finite term, there is also a linear acceleration
term that vanishes due to cancellation of part near the Dirac
points and far away . It can be presented as a full derivative
2
Et 
J     k  /  k  4k 

4 k 
Et
d
  
Sk  0
4 k dk y
*

y
div
k  Im  hk hk  /  k ;
k  Im  hk*hk ''  /  k
Kao, Lewkowicz, B.R.,
PRB81, 041416 (2010)
Therefore there is no perfect scale
separation when massless fermions
are involved. This is also the
source of parity anomaly .
K
K
The 2D vacuum polarization
   0,q   q
0
Is too weak to exponentially screen the 3D
Coulomb with 2D Fourier transform
0

q
4
2
V q 
q
2
2


The linearity of the current
3  vF eEt 
   2 1  
 ...

dependence on electric field is
 64  t 



quite accidental. It is not an Ohmic
dissipative systems. For example
B.R., H. C. Kao, Lewkowicz,
the third order “correction” is
Kornijenko, PRB81, R041416 (2010)
This means that linear response
breaks down at time scale
tnl 
/ eEv g
Dynamical approach to nonlinear transport
B.R, Lewkowicz, PRL 77, (09) ; B.R., Kao, Lewkowicz, PRB 81, R041416 (10)
2.5
J E
2.0
E0
E
2 6
E0
2 7
E0   / ea
pair creation rate
conductivity
2
E
E
2 8
E0
1.5
1.0
0.5
 1010V / m
0.0
0
20
40
60
80
100
120
time in units of t
Stays there till a crossover time tnl 
/ eEvg
to a linearly increasing Schwinger’s regime J  t  
2

2
1/ 2
g
ev
 eE 
 
 
3/ 2
t
At each k x one solves, using the WKB approximation, a
tunneling problem similar to that in the Landau – Zener
transition.
Gavrilov, Gitman, PRD53, 7162 (1986)
The interband transition
probability at large times is
N t  

ky
k y  eEt
2 kx
k y
k y  eEt
4
eEt /
 2  k 0
2
1 eEt
2
y
  vF k x2 
kx exp   eE 
eE
1
1/ 2
3/ 2
 2  vF   eE  t
vF

Schwinger, PR82, 664 (1951)
Electric field is screened
very effectively beyond
linear response. Therefore
let us assume that the
Schwinger, PR (1962)interaction is effectively
screened and becomes local
When density of created pairs rises, the recombination process
is amplified due to Coulomb forces. The system crosses over to
a true dissipative regime due to radiation friction.
The neutral electron – hole plasma
state is achieved when density
approaches
12
2
N pl
10 cm
Vandecasteele et al, PRB 82,
045416 (08)
3. IS GRAPHENE MORE METALLIC OR INSULATING?
Electron – electron interactions are strong
e2
1
H int     r 
  r '
2 r ,r '
r  r'
  r      r  rn  cˆnAs † cˆnAs    r  rn   3  cˆnBs † cˆnBs 
n
   r   0  r    s †  r  s  r 
They break the (pseudo)
relativistic invariance and are
still 3D, namely less long range
in 2D than that of the 2+1
dimensional massless QED.
Coulomb interactions naively are strong due to coupling
constant of order 1
2
e
c
300
g 
 QED 
vF vF
137
Just rescale the field in
2
e
H  vF   †   
r
2
1
†



r

 0  r ' 


r,r ' 0
r  r'
†
If unscreened, in graphene this would lead to a strongly coupled
electronic system. A point - like charge in graphene would even
supercritical phenomena (fall to the center).
Z
1

 137
Z
supercritical regime
Z
Z
1

 137
Positron emission
Although supercriticality phenomena were never observed
(perhaps due to nonlinear screening discussed above), the
interactions should be strong enough to break chiral
symmetry.
4. CHIRAL SYMMETRY BREAKING in GRAPHENE
Exciton condensation
If experience with relativistic interacting fermions is of any
value, the attractive force that strong should be enough to
create the electron – hole pairs condensate that would also
break the chiral symmetry spontaneously, while the spin
(flavour) rotation symmetry would remain unbroken.
 s s  0
This would make quasiparticle
massive and the phase insulating.
Nothing of that was observed in
graphene as yet, but there might be
experimental reasons (excuses).
1. In graphene on substrate there are metallic
puddles that screen reducing the coupling and in
addition dielectric constant.
2. In suspended samples electrostatics is nontrivial
and one worries about capacitances near
Yaish (12)
suspended contacts.
Moreover in another two phenomena where
interactions are crucial
1. FQHE although observed for   1/ 3 is weaker
than expected
2. DC, AC conductivity nearly coincides with the
noninteracting one.
Simulations
Lattice simulation using staggered fermions indeed
demonstrated formation of the condensate
  

S   d 2 rdt  s vF       t  iA0   0  s

2
1
2
 2  d rdzdt A0  r, z, t 
2e
Simulation predicted
that the critical
coupling is below
 g  
 g  1/ 4 g  
Drut, Lande, PRB 79, 79167425 (09)
Critical exponent however is turned out to be different from
the Ising’s and points towards relativistic chiral universality
class
   c  ; m ;

     c  ; m   1  

m
N f  2 :  1,  2.26  0.06
N f  4 :  1,  2.62  0.11
Drut, Lahde, PRB 23, 345601 (09)
Critical exponent points to the chiral universality class: a
relativistic model with local (screened) effective interaction
Critical exponent however is turned out to be different from
the Ising’s and points towards relativistic chiral universality
class
   c  ; m ;

     c  ; m   1  

m
N f  2 :  1,  2.26  0.06
N f  4 :  1,  2.62  0.11
Drut, Lahde, PRB 23, 345601 (09)
Critical exponent points to the chiral universality class: a
relativistic model with local (screened) effective interaction

F   d x v      g1  
3
s

s
s

s 2
 g 2   
s

s 2

 ...
RG flow calculated by various means, Dyson – Schwinger
equations truncation
Khveshchenko, Veal , PRL 87, 246801 (01); Khveshchenko,
Veal , Nucl. Phys. B687, 323 (04) …
Perturbation theory in couplings, 1/N, various  expansions
Drut, Son, PRB 77,
075115 (08),….
The situation is still not entirely settled, but apparently the
Coulomb interaction is “irrelevant” and at criticality the
interaction becomes local and relativistically invariant, hence
belongs to a chiral universality class.
5. UNIVERSALITY REVISITED
Universality of transitions
A second order phase transition is generally well described
phenomenologically if one identifies:
a. The order parameter field
i ( x )
b. Symmetry group G and its spontaneous breaking pattern.
L.D. Landau ( 1937 )
Rest of degrees of freedom are “irrelevant” sufficiently close to
the critical temperature Tc. Later, the “relevant” part, namely
the symmetry breaking pattern and dimensionality was termed
“the universality class”. Its critical properties (“exponents”)
are described by a renormalizable FT (CFT via its
“deformations”).
Wilson, Fisher, …1960s
a
g
1
2
2
F   d x  (i )  ii  (ii ) 
2
4
2

3
The CFT appears at strong coupling on the critical line
a
g
   a  ac  ;  0.33

UV
Various aspects of physics of transitions of interacting 2D
relativistic fermions were discussed in eighties is connection with
nodal quasi-particles and a weak one to graphite
Chiral universality classes
It turns out that chiral phase transitions involving Dirac point
are qualitatively different from the scalar ones due to presence of
massless excitations in the ordered phase.
 s s

g
s s 2
F   d x   
  

2N f


3
gc
g
Unbroken chiral
symmetry,
Massless fermions
Broken chiral
symmetry,
Massive fermions
How the massless fermions create a strongly coupled
fixed point
It is convenient to consider the 1/N expansions (despite the fact
that other “nonperturbative” methods achieve the same).
g
N
g
N
N
g
N
g r  g  g 2   g 3  2  ...
gc
g

g
1  g
g r1  g 1  
g c1  
From weakly coupled FP to chiral FP
The four Fermi theory is similar to fixed length scalar
models (sigma models).
gc
UV
Gat, Kovner, B.R. Nucl. Phys. B385,
76 ( 1992 )
g
This way one can classify
chiral phase transitions
according to both the
symmetry breaking pattern
and the number of massless
fermions.
Can this be called
“topological” nowdays?
Some critical exponents were calculated analytically and
numerically (A. Li talk on Wednesday)
The flow to the
chiral CFT can be
simulated on lattice
using the Yukawa
model.
6. UNDERSTANDING A CONDUCTOR WITH LONG
RANGE COULOMB INTERACTIONS
Early conflicting results
A first step in estimating the corrections to conductivity would
be a perturbative calculation of corrections, especially when
experimental value almost coincides with the free electron
result.
First attempt in which sharp momentum cutoff was used gave
a disappointingly large correction indicating that interactions
are not weak

 e2
1  C g  ...
2 h
C
1
25 

  0.51
12 2
Herbut, Juricic, Vafek, PRL100 , 046403 (08)
However it was shown that Ward identities were not satisfied.
When the interaction and current were modified one obtains a
very small correction consistent with experiment
C
2
19 
   0.012
12 2
Mischenko, EurPL100 , 046403 (08); Sheehy, Schmalian, PRB80, 193411 (09)
Despite this certain identities were not obeyed and a variant of
dimensional regularization was applied with yet different
answer
C
 3
22 

  0.25
12 2
Juricic,Vafek, Herbut,
PRB82 , 235402 (10)
New calculations appear, most consistent with the prefered
small value
MacDonald…Vignale…, PRB84, 045429 (11)
As was noticed in calculation with noninteracting fermions,
there is no complete scale separation in the model due to
anomaly related zero modes. It is interesting to note that the
results always differ by
1
C 
4
And one might suspect that “incorrect UV regularization
brings in “topological” term. Only calculation with more
physical regularization will clarify this.
Calculation on the lattice
To resolve the ambiguity we have performed the calculation
within the tight binding model and obtained the third result.
Hˆ  
s
s
ˆ
ˆ
a
U
t
a


 rn rn , rn   c.c.
n, , s
1
   cˆns † cˆnsV  rn  rm  cˆms † cˆms  ...
2 n ,m
 e
U n ,  t   exp i
 c

A
r

s

,
t
ds


0 n 

1
The current density operator is defined in such a way
1 
J  r, t   
H
c  A  r, t 
That the Ward identities are satisfied for the lattice model and
one does not have to guess corrections for the continuum
operators.
Results
B.R., Maniv, Lewkowicz, arxiv cond mat (12)
1. For a local, quasilocal and even long range potential
1
V  r    ;  1
r
The correction to DC conductivity is zero
MacDonald…Vignale…, PRB84, 045429 (11)
2. For 1    0 there is an IR divergence and the
perturbation theory is invalid.
3. For the Coulomb interaction one obtains the third value.
25 
 3
C 
  0.25
2 2
Interactions are not accidentally exceptionally small.
One still have to look for explanation of the small interaction
correction in experiment elswhere.
Conclusions
1. Graphene undergoes the chiral symmetry breaking transition,
however several theoretical questions remain:
a. is the critical value of coupling larger than that of
“ideal”=suspended graphene?
b. Relativistic Z 2 ,N f  2 chiral universality class is the
strongest candidate, but more simulations or analytic
calculations are needed to assertain this.
2. The chirally unbroken quasi – Ohmic phase is by now better
understood, still has several puzzles like why the conductivity
is so close to the noninteracting one.
3. The insulating chiral symmetry broken phase still awaits
discovery. It might be a laboratory of massless strongly coupled
QED.