Introduction
The model
Conductivity
Main results
Sketch of the proof
Universal conductivity in graphene with
short-range interactions
Alessandro Giuliani - Università di Roma Tre
joint work with V. Mastropietro and M. Porta
Bressanone, MMQM, February 15, 2011
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Introduction
The model
Conductivity
Outline
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Introduction
2
The model
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Conductivity
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Main results
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Sketch of the proof
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Conclusions
Main results
Sketch of the proof
Conclusions
Introduction
The model
Conductivity
Outline
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Introduction
2
The model
3
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Sketch of the proof
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Conclusions
Main results
Sketch of the proof
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Introduction
The model
Conductivity
Main results
Sketch of the proof
Graphene
Graphene consists of a single layer of graphite.
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Introduction
The model
Conductivity
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Sketch of the proof
Conclusions
Graphene
Graphene has been first experimentally realized in
2004 by Geim and Novoselov (Nobel prize 2010).
It displays several unusual properties:
remarkably stable against thermal fluctuations
very high electron mobility
insensitivity to localization effects
anomalous integer quantum Hall effect
(observable at room temperature)
which make it an exciting and promising material
for future nano-technological applications.
Introduction
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Conductivity
Main results
Sketch of the proof
Conclusions
Degenerate Fermi surface
What distinguishes graphene from most
conventional 2D electron gases is that at half-filling
the density of charge carriers vanishes and the
Fermi surface is highly degenerate.
The low-energy
excitations around the
Fermi points behave like
2D Dirac fermions with
“speed of light” vF ∝ t.
Introduction
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Conductivity
Main results
Sketch of the proof
Conclusions
Minimal conductivity
An important feature is that graphene’s zero-field
conductivity does not disappear in the limit of
vanishing density of charge carriers (half-filling)
but instead exhibits values close to the conductivity
quantum e 2 /h, both in the presence and in the
absence of disorder.
Introduction
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Conductivity
Main results
Sketch of the proof
Conclusions
Minimal conductivity
For clean samples, the minimal conductivity appears
2
to be σ0 = π2 eh , an apparently universal value,
independent of the material parameters (e.g., of the
hopping strength), very easy to measure: it can be
obtained from the visible transparency of graphene,
t(ω) =
1
.
(1 + 2πσ(ω)/c)2
in the frequency range kB T ω vF .
Introduction
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Conductivity
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Sketch of the proof
Conclusions
Minimal conductivity
The observed value of σ0 agrees with the theoretical
predictions based on the description in terms of
massless non-interacting Dirac particles.
A very natural question:
is the value of the minimal conductivity affected by
electron-electron interactions?
This question is entirely analogous to the one
concerning universality in the QHE and it is of
enormous importance for metrology.
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Minimal conductivity
The observed value of σ0 agrees with the theoretical
predictions based on the description in terms of
massless non-interacting Dirac particles.
A very natural question:
is the value of the minimal conductivity affected by
electron-electron interactions?
This question is entirely analogous to the one
concerning universality in the QHE and it is of
enormous importance for metrology.
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Interaction effects
The effects of electron-electron interactions on the
conductivity have been investigated in the Dirac
approximation by perturbation theory both in the
presence of long- and of short-ranged interactions
(Gusynin, Herbut, Mischchenko, Sharapov, ...)
Explicit lowest order computations produce different
results, depending on the regularization scheme
(momentum cut-off or dimensional regularization)
chosen to cure the spurious UV divergences
introduced by the Dirac approximations.
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Interaction effects
The effects of electron-electron interactions on the
conductivity have been investigated in the Dirac
approximation by perturbation theory both in the
presence of long- and of short-ranged interactions
(Gusynin, Herbut, Mischchenko, Sharapov, ...)
Explicit lowest order computations produce different
results, depending on the regularization scheme
(momentum cut-off or dimensional regularization)
chosen to cure the spurious UV divergences
introduced by the Dirac approximations.
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Interaction effects
In this talk I report a Theorem on the non-existence
of interaction corrections to the graphene minimal
conductivity in the presence of weak short range
interactions. The proof is based on a combination
of constructive RG and lattice WIs.
The result is a non-perturbative analogue of the
Adler-Bardeen theorem in QED and represents the
first two-dimensional example of a universality
phenomenon in condensed matter that can be
established on firm mathematical grounds.
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Interaction effects
In this talk I report a Theorem on the non-existence
of interaction corrections to the graphene minimal
conductivity in the presence of weak short range
interactions. The proof is based on a combination
of constructive RG and lattice WIs.
The result is a non-perturbative analogue of the
Adler-Bardeen theorem in QED and represents the
first two-dimensional example of a universality
phenomenon in condensed matter that can be
established on firm mathematical grounds.
Introduction
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Conductivity
Outline
1
Introduction
2
The model
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Sketch of the proof
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Conclusions
Main results
Sketch of the proof
Conclusions
Introduction
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Conductivity
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Sketch of the proof
The model
A basic model for half-filled single-layer graphene
with screened Coulomb interactions is the
2D Hubbard model on the honeycomb lattice
X X
+ −
H = −
t a~x ,σ b~x +~δ ,σ + c.c. +
i
~x ∈ΛA σ=↑↓
i=1,2,3
+U
X
a~x+,↑ a~x−,↑ −
1
1 + −
a~x ,↓ a~x ,↓ −
+
2
2
b~x+,↑ b~x−,↑ −
1 + −
1
b~x ,↓ b~x ,↓ −
2
2
~x ∈ΛA
+U
X
~x ∈ΛB
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Introduction
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Conclusions
The model
Here a± and b ± are creation/annihilation operators
associated to the two periodic triangular sublattices
ΛA = Λ and ΛB = Λ + ~δi .
Introduction
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Conclusions
The non-interacting gas
H
U=0
= H0 = −
X X
t
a~x+,σ b~x−+~δ ,σ
i
+ c.c.
~x ∈ΛA σ=↑↓
i=1,2,3
If U = 0, the energy bands are
±v0 |Ω(~k)|, with v0 = 32 t and
√
2
3
Ω(~k) = (1+2e −i3k1 /2 cos
k2 )
3
2
Around the Fermi points ~pF± ,
Ω(~k 0 + ~pF± ) ' ik10 ± k20
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The interacting gas
At half-filling and U small enough, the interacting
ground state is analytically close to the free one
(Giuliani-Mastropietro 2009). E.g., as |~x − ~y | → ∞,
Z ~
Tre −βH a~x−,σ a~y+,σ
1
d k −i ~k(~x −~y )
ik0
∼
e
,
−βH
Tre
Z B |B|
k02 + vF2 |Ω(~k)|
where vF = vF (U) and Z = Z (U) are analytic
functions of U, with the meaning of interacting
Fermi velocity and inverse quasi-particle weight.
Introduction
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Sketch of the proof
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Main results
Sketch of the proof
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Introduction
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Conductivity
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Lattice currents
If we modify the hopping parameter along the bond
(~x , ~x + ~δj ) as
~ = t e ie
t → t~x ,j (A)
R1
0
~ x +s ~δj )·~δj ds
A(~
and denote by H(A) the modified Hamiltonian, the
lattice current is defined as
Z
∂H(A)
(A)
~J
b ~p,~q A
~ ~q + O(A2 ) ,
=−
= ~J~p + d~q ∆
~p
~
∂ A~p
where ~J~p is the paramagnetic current and
b ~p,~q the diamagnetic tensor.
∆
Introduction
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Conductivity
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Sketch of the proof
Conductivity
The average current in the Gibbs state at inverse
temperature β is
(A)
(A)
hJ~p,i i
=
Tre −βH(A) J~p,i
Tre −βH(A)
so that the variation of the current to an external
vector potential in the linear response regime is
(A)
∂hJ~p,i i b ~p,−~p ]ij i .
= βhJ~p,i ; J−~p,j i + h[∆
∂A−~p,j
A=0
Conclusions
Introduction
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Conductivity
Main results
Sketch of the proof
Conclusions
Conductivity
Let us now make the vector potential a dynamical
~ 0 , ~x ), with an associated electric field:
field A(x
~ (x0 , ~x ) = ∂x A(x
~ 0 , ~x ) ,
E
0
~ (ω,~p) = −iω A
~ (ω,~p) .
E
Kubo formula for the electric conductivity is:
(A)
∂hJ
i
(ω,~p ),i β,Λ
Σij (ω, ~p ) =
=
∂E(−ω,−~p),j A=0
Z
i
1h ∞
−iωx0
b
=
dx0 e
hJ(x0 ,~p),i ; J(0,−~p),j i + h[∆~p,−~p ]ij i
iω −∞
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Conductivity
It is customary to investigate the thermodynamic
and response functions in the imaginary time
formalism, in which case, after the Wick rotation
x0 → ix0 and ω → −iω,
Σβ,Λ
p ) → σijβ,Λ (ω, ~p ) =
ij (ω, ~
Z
i
1 h β/2
−iωx0
b
=−
dx0 e
hJ(x0 ,~p),i ; J(0,−~p),j i + h[∆~p,−~p ]ij i
ω −β/2
It is expected that σ := limβ,|Λ|→∞ σ β,Λ can be
obtained from Σ by analytic continuation.
Introduction
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Conductivity
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Introduction
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Sketch of the proof
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Main results
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Introduction
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Conclusions
Main results
Theorem. There exists a constant U0 > 0 such
that, for |U| ≤ U0 and any fixed ω, the ground state
conductivity in the thermodynamic limit σ(ω, ~p ) is
analytic in U. Moreover,
e2 π
~
lim σij (ω, 0) =
δij .
ω→0+
h 2
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Remarks
The limit we computed corresponds to the
universal optical conductivity of graphene, which
in fact matches with the experimental measure
in the optical range β −1 ω t.
This is the first rigorous universality result in a
system of 2D interacting electrons.
Unfortunately, we cannot prove that, as
expected, limω→0+ σij (ω, ~0) = limω→0+ Σij (ω, ~0).
Introduction
The model
Conductivity
Main results
Sketch of the proof
Remarks
The proof is based on two main ingredients:
(1) exact lattice Ward Identities;
(2) the fact that the interaction-dependent
corrections to the Fourier transform of the
current-current correlations are differentiable
with continuous derivative, uniformly in β, Λ.
Conclusions
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Remarks
The proof of the uniform analyticity of σ β,Λ (ω)
and of its regularity properties in ω follows from
the non-perturbative estimates found by
Giuliani-Mastropietro (2009) on the basis of
constructive fermionic RG methods due to
Gawedski-Kupiainen
Battle-Brydges-Federbush
Lesniewski
Benfatto-Gallavotti
Feldman-Magnen-Rivasseau-Trubowitz
Key fact: the short range interaction is
irrelevant.
Introduction
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Conductivity
Outline
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Introduction
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Sketch of the proof
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Conclusions
Main results
Sketch of the proof
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Introduction
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Conclusions
The fermionic field and the density operator
Let p = (ω, ~p ) and k = (ω 0 , ~k). We let
− a
+
+
+
−
ψk,σ
= (ak,σ
, bk,σ
),
ψk,σ
= k,σ
−
bk,σ
be the two-component fermionic field operator and
X Z dk
+
−
ρp =
ψk+p,σ
ψk,σ
B |B|
σ=↑↓
the density operator.
Introduction
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Conductivity
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Conclusions
Green, vertex and response functions
We let Jp,µ , µ ∈ {0, 1, 2}, be the space-time vector
with components (eρp , Jp,1 , Jp2 ) and define:
Green function :
−
+
S(k) = hψk,σ
ψk,σ
i
Vertex function :
−
+
G2,1;µ (k, p) = hJp,µ ; ψk+p,σ
ψk,σ
i
Response function : Kµν (p) = hJp,µ ; J−p,ν i
Introduction
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Conclusions
Exact lattice Ward identities
A key remark is that the continuity equation
−ie∂x0 ρ(x0 ,~p) + ~p · ~J(x0 ,~p) = 0
implies the validity of exact lattice Ward Identities
p µ G2,1;µ (k, p) = −e S(k + p) − S(k)
ˆ ~p,−~p i
p µ Kµj (p) = − ~p · h∆
j
where p 0 = −iω, p 1 = p1 , p 2 = p2 and summation
over repeated indices is understood.
Introduction
The model
Conductivity
Main results
Sketch of the proof
Conclusions
Exact lattice Ward identities
A key remark is that the continuity equation
−ie∂x0 ρ(x0 ,~p) + ~p · ~J(x0 ,~p) = 0
implies the validity of exact lattice Ward Identities
p µ G2,1;µ (k, p) = −e S(k + p) − S(k)
ˆ ~p,−~p i
p µ Kµj (p) = − ~p · h∆
j
where p 0 = −iω, p 1 = p1 , p 2 = p2 and summation
over repeated indices is understood.
Introduction
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Conductivity
Main results
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Conclusions
Exact lattice Ward identities
We use that
|hJx,µ ; Jy,ν i| ≤
C
1 + |x − y|4
so that Kµν (p) is continuous at p = 0.
Using the second WI and picking, e.g., p2 = 0,
ω
ˆ (p ,0),(−p ,0) ]1j i
i K̂0j (ω, p1 , 0) = K̂1j (ω, p1 , 0) + h[∆
1
1
p1
we find that, taking first ω → 0 and then p1 → 0
we get (by the continuity of K̂µν (p)),
ˆ ~ ~ ]1j i
K1j (0) = lim lim K1j (ω, p1 , 0) = −h[∆
p1 →0 ω→0
0,0
Introduction
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Conductivity
Main results
Sketch of the proof
Conclusions
Exact lattice Ward identities
We use that
|hJx,µ ; Jy,ν i| ≤
C
1 + |x − y|4
so that Kµν (p) is continuous at p = 0.
Using the second WI and picking, e.g., p2 = 0,
ω
ˆ (p ,0),(−p ,0) ]1j i
i K̂0j (ω, p1 , 0) = K̂1j (ω, p1 , 0) + h[∆
1
1
p1
we find that, taking first ω → 0 and then p1 → 0
we get (by the continuity of K̂µν (p)),
ˆ ~ ~ ]1j i
K1j (0) = lim lim K1j (ω, p1 , 0) = −h[∆
p1 →0 ω→0
0,0
Introduction
The model
Conductivity
Main results
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Exact lattice Ward identities
Similarly,
ˆ ~ ~ ]ij i
Kij (0) = −h[∆
0,0
for all i, j ∈ {1, 2} and, therefore,
Kij (ω, ~0) − Kij (0)
σij (ω, ~0) = − lim+
,
ω→0
ω
where, by symmetry, Kij (ω, ~0) is even in ω.
Therefore, if Kij (p) had continuous derivative at
p = 0, then limω→0+ σij (ω, ~0) = 0.
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Conclusions
Absolutely convergent contributions
But: Kij (p) is non-continuously differentiable.
Among the contributions to Kij (p), we distinguish:
1
the 1st order contribution in renormalized
perturbation theory, which is a non-absolutely
convergent (but explicit!) integral;
2
the higher order terms, which are absolutely
convergent and continuously differentiable, due
to the irrelevance of the short range interaction.
Introduction
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A convenient rewriting
Correspondingly, we decompose:
Kij (p) =
Zi Zj
hJp,i ; J−p,j i0,vF + Rµν (p) ,
v02 Z 2
where:
1
Zi is the vertex function (amputeated of the
external propagators) at the Fermi points;
2
h·i0,vF is the average associated to a non
interacting system with Fermi velocity vF (U);
3
Rµν (p) is the sum of the higher order terms and
is continuously differentiable in p.
Introduction
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Another exact cancellation
Since R(ω, ~0) is even and differentiable,
lim σ(ω, ~0) =
ω→0+
hJ(ω,~0),i ; J(−ω,~0),j i
− hJ0,i ; J0,j i0,vF
Zi Zj
0,vF
= − 2 2 lim+
ω
v0 Z ω→0
Finally, using the first WI:
p µ G2,1;µ (k, p) = −e S(k+p)−S(k)
⇒
which implies
lim+ σ(ω, ~0) = lim+ σ0,vF (ω, ~0)
ω→0
ω→0
with σ0,vF the free conductivity at v = vF .
Zi = vF Z
Introduction
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Conductivity
The only contribution to the conductivity we are left
with is the free one, which reads
Z
2 2
4e
v
dk0 d ~k
F
~
lim σij (ω, 0) = √ lim+
·
ω→0+
2π |B|
3 3 ω→0
n S (k) − S (k + (ω, ~0))
o
0,vF
0,vF
·Tr
σj S0,vF (k)σi ,
ω
with σi the Pauli matrices. The integral can be
evaluated explicitly and leads to
e2 π
δij
σij =
h 2
(independent of vF and of the lattice!)
Introduction
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Conductivity
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Introduction
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Conclusions
Main results
Sketch of the proof
Conclusions
Introduction
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Conductivity
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Sketch of the proof
Conclusions
Conclusions
We considered the 2D Hubbard model on the
honeycomb lattice as a model for graphene with
screened Coulomb interactions. At half-filling
and weak coupling, the ground state is
analytically close to the non-interacting one.
The interacting correlations decay with the same
exponents as the free ones, modulo a finite
renormalization of some physical parameters
(quasi-particles weight, Fermi velocity and
vertex functions).
Introduction
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Conclusions
The zero temperature and zero frequency limit
of the conductivity is controlled thanks to lattice
Ward Identities, which imply the exact vanishing
of the corrections due to the local interaction.
A crucial role is played by the regularity
properties of the Fourier transform of the
response functions, which follow from improved
dimensional estimates.
Introduction
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Conductivity
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Conclusions
Conclusions
The proof is based on constructive fermionic
Renormalization Group methods and on exact
lattice Ward Identities. Our results provide the
first rigorous proof of universality, in a realistic
2D interacting electron system.
The correctness of the Wick rotation procedure
remains to be proved. Reconstruction theorems
(a’la Osterwalder-Schrader) may allow one to
establish the analytic extendability of σ(ω, ~0).
Introduction
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Conductivity
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Conclusions
References
A.G., V.Mastropietro: Rigorous construction of
ground state correlations in graphene, PRB 2009.
A.G., V.Mastropietro: The 2D Hubbard model
on the honeycomb lattice, CMP 2010.
A.G., V.Mastropietro, M.Porta: Absence of
interaction corrections in graphene conductivity,
arXiv:1010.4461.
A.G., V.Mastropietro, M.Porta: Universality of
conductivity in interacting graphene,
arXiv:1101.5236.
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Summer school on current topics in Mathematical Physics
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