## Abstract

Graphene is a two-dimensional crystal of carbon atoms with fascinating electronic and morphological properties. The low-energy excitations of the neutral, clean system are described by a massless Dirac Hamiltonian in (2+1) dimensions, which also captures the main electronic and transport properties. A renormalization group analysis sheds light on the success of the free model: owing to the special form of the Fermi surface that reduces to two single points in momentum space, short-range interactions are irrelevant and only gauge interactions such as long-range Coulomb or effective disorder can play a role in the low-energy physics. We review these features and briefly discuss other aspects related to disorder and to the bilayer material along the same lines.

## 1. Introduction

For the last few decades, the Landau Fermi liquid (LFL) has been the standard model in condensed matter to describe most metals [1,2]. A renormalization group (RG) analysis of the effective continuum model of non-relativistic electrons allows us to understand this universal behaviour as a consequence of the marginal character of electron–electron interactions in the presence of a Fermi surface [3]. Strongly correlated materials, such as heavy fermions or high-temperature superconductors, have given rise to very important new ideas in the last 20 years because of their ‘non-Fermi liquid’ behaviour that arise from their strongly interacting nature [4]. The appearance of graphene in 2004 [5] has originated a ‘second revolution’ in the field because of its very many exotic electronic and morphological properties. Unlike the cuprates where there is not currently an agreement on the model to describe the basic material, graphene resembles the situation in quantum field theory (QFT) where the starting point is clear: the low-energy excitations can be modelled by the massless Dirac Hamiltonian in two spatial dimensions. From the RG point of view, graphene has defied the LFL paradigm by confronting us with a material described by a universal non-interacting Hamiltonian that shows agreements and contradictions with the usual LFL behaviour. From many other points of view, it is a fascinating system that lives in between different branches of physics as condensed matter, QFT, statistical physics or the theory of elasticity.

Graphene is a two-dimensional crystal of carbon atoms arranged in a honeycomb lattice: a single layer of graphite. Its synthesis [6,7], amazing properties (it is flexible like plastic but stronger than diamond, and it conducts electricity like a metal but is transparent like glass) [8] and potential applications [9,10] have granted the 2010 Nobel prize in Physics to A. Geim and K. Novoselov.

One of the most interesting aspects of graphene physics from a theoretical point of view is the deep and fruitful relation that it has with quantum electrodynamics (QED) and other QFT ideas [11–14]. The connection arises from the abovementioned fact that the low-energy excitations can be modelled by the massless Dirac equation in two spatial dimensions—a fact that makes most of the phenomenological expressions for transport properties in Fermi liquids useless. From the morphological point of view, the mere existence of two-dimensional crystals has been argued not to be possible [9]. Moreover, despite being one of the most rigid materials with a Young’s modulus of the order of terapascals, the samples show ripples of various sizes whose origin is unknown. The intrinsic relation between the morphology (honeycomb lattice) and the electronic properties constitutes a great playground for physical ideas as well as one of the sources of potential applications as a single molecule detector.

From the QFT point of view, graphene has given rise to very interesting developments: the so-called axial anomaly [11,12] has acquired special relevance in relation to the recently discussed topological insulators [15,16] that provide a condensed matter realization of the axion electrodynamics [17,18]. Charge fractionalization has also been explored in the honeycomb lattice with special defects [19,20] and QFT in curved space [21,22], and cosmological models [23,24] have been used to explore the electronic properties of the curved material. A very interesting development has been associated with the generation of various types of vector fields from the elastic properties or from disorders that couple with the electrons in the form of gauge fields [25].

Among the unexpected properties related to interactions on the condensed matter point of view are the anomalous behaviour of the quasi-particles decaying linearly with frequency [26], and the renormalization of the Fermi velocity at low energies [27]. These properties were predicted theoretically in early work [13,28] from RG analysis, which will be described in this work. We will review the RG aspects of graphene in the presence of Coulomb interactions with special emphasis on the similarities and discrepancies with three-dimensional QED (QED(3))Q1. We will show that the system has a non-trivial infrared fixed point, where Lorentz covariance is restored, appearing as an emergent property [29]. The RG analysis allows us to classify graphene as a new type of electron liquid described by a Lorentz covariant infrared fixed point whose effective coupling constant is the fine structure constant. We will focus on the general, conceptual features, referring to the specific bibliography for the technical details. The differences and similarities with the usual QFT models will also be emphasized.

## 2. Continuum model for the low-energy excitations of graphene

The graphene structure is made of carbon atoms located in a honeycomb lattice. Three of the four available electrons of the carbon atom: 2*s*, 2*p*_{x} and 2*p*_{y} orbitals hybridize in a so-called *sp*^{2}, a strong covalent bond that ‘draws’ the honeycomb lattice and has typical energies around 3 eV. The fourth 2*p*_{z} orbital perpendicular to the plane formed by the sigma bonds remains delocalized and is responsible for the metallic properties of the material. The *σ* bonds give rigidity to the structure, while the *π* bonds give rise to the valence and conduction bands [30]. The electronic properties around the Fermi energy of a graphene sheet can be described by a tight binding model with only one orbital per atom, the so-called *π*-electron approximation and were obtained in earlier work [31,32]. The nearest-neighbour tight binding approach reduces the problem to the diagonalization of the one-electron Hamiltonian,
2.1
where the sum is over pairs of nearest-neighbour atoms *i*,*j* on the lattice and *a*_{i}, *a*^{+}_{j} are the usual creation and annihilation operators. What is special about the honeycomb lattice shown in figure 1 is the fact that, even though all carbon atoms are equal, their positions in the lattice make them topologically non-equivalent. It is easy to see that to generate all the points in the lattice, we must take a pair of points. The lattice can be seen as two interpenetrating triangular sublattices of these points or, more technically, it is a lattice with a basis. This special topology of the honeycomb lattice is at the heart of all the electronic peculiarities of graphene. The Bloch trial wave function has to be built as a superposition of the atomic orbitals from the two atoms forming the primitive cells *A* and *B*,
2.2

The eigenfunctions and eigenvalues of the Hamiltonian are obtained from the equation
2.3
where **u**_{j} is a triad of vectors connecting an *A* atom with its *B* nearest neighbours and **v**_{j} the triad of their respective opposites, *a* is the distance between carbon atoms and *ϵ* is the 2*p*_{z} energy level taken as the origin of the energy. The tight binding parameter *t*, estimated to be of the order of 3 eV in graphene, sets the bandwidth (6 eV) and is a measure of the kinetic energy of the electrons. The eigenfunctions are determined by the the coefficients *C*_{A} and *C*_{B} solutions of equation (2.3). The eigenvalues of the equation give the energy levels whose dispersion relation is
2.4
in which the two signs define two energy bands: the lower half called the bonding *π* band and the upper half called the antibonding *π** bands, which are degenerate at the **K** points of the Brillouin zone. The dispersion relation is shown in figure 2. Within the *π* electron approximation, each site of the graphite honeycomb lattice yields one electron to the Fermi sea and the band is at half-filling. Since each level of the band may accommodate two states owing to the spin degeneracy, and the Fermi level turns out to be at the midpoint of the band, instead of a whole Fermi line, the two-dimensional honeycomb lattice has six isolated Fermi points, which are the six vertices of the hexagonal Brillouin zone. Only two of them are inequivalent and can be chosen as and **K**_{2}=−**K**_{1} [33].

A continuum model can be defined for the low-energy excitations around any of the Fermi points, say *K*_{1}, by expanding the dispersion relation around it: **k**=**K**_{1}+*δ***k** gives the Hamiltonian from equation (2.3),
2.5
The limit defines the continuum Hamiltonian
2.6
where ** σ** are the Pauli matrices and the parameter

*v*

_{F}is the Fermi velocity of the electrons estimated to be

*v*

_{F}∼3

*ta*∼

*c*/300. Hence, the low-energy excitations of the system are massless, charged spinors in two spatial dimensions moving at a speed

*v*

_{F}. We must notice that the physical spin of the electrons has been neglected in the analysis, the spinorial nature of the wave function has its origin in the sublattice degrees of freedom and is called pseudospin in the graphene literature. The stability of Fermi points in graphene has been studied in [34]. There, it was shown that the Fermi points are topologically stable and cannot be lifted by smooth continuum perturbations of the Hamiltonian if there are no short-range interactions or disorder effects that connect the two points. The same expansion around the other Fermi point gives rise to a time-reversed Hamiltonian: . The degeneracy associated with the Fermi points (valleys in semiconductor language) is taken as a flavour. Together with the real spin, the total degeneracy of the system is 4.

The calculation of the density of states (DOS) for the linear dispersion relation *E*(** k**)=

*v*

_{F}|

**| gives 2.7 where**

*k**g*

_{S}=4 is the degeneracy. Equation (2.7) shows one of the peculiarities of graphene as opposed to a usual two-dimensional electron gas (2DEG): the DOS grows linearly with the energy and vanishes at the Fermi energy (in the standard 2DEG, the DOS is a constant in two dimensions). This fact has very important phenomenological consequences that have been tested experimentally. For our purposes, it implies that the Coulomb interactions between the electrons are not screened in graphene.

We finish this section by mentioning that the massless spinorial nature of the graphene quasi-particles, the linearity of the DOS, the degeneracy of 4 and the estimated value of the Fermi velocity have been experimentally tested [9], which makes the present model universally accepted in the community. We also mention that many electronic and transport properties can be described by the non-interacting model and that disorder, which usually leads to localization in two dimensions, does not seem to play an important role in the graphene system.

## 3. Renormalization group analysis: tree level

As discussed by Shankar [3], the main issue of the RG ideas as applied to condensed matter systems is that for special systems (critical, renormalizable), the low-energy physics is governed by an effective Hamiltonian made of a few marginal interactions that can be obtained from the microscopic high-energy Hamiltonian in a well-prescribed manner [34,35]. Following the nomenclature of critical systems, interactions are classified as relevant, irrelevant or marginal according to their scale dimensions. These dimensions determine whether they grow, decrease or acquire at most logarithmic corrections and at low energies. The effective coupling constants of a model at intermediate energies can be obtained by ‘integrating out’ high-energy modes, even if there is no stable fixed point at the end of the RG flow. Luttinger and Fermi liquids are identified as infrared fixed points of the RG applied to an interacting metallic system in one or more dimensions, respectively.

In this context, the universal LFL behaviour of most electronic systems can be established in the form of almost a theorem: a system of electrons will behave as a Fermi liquid if: (i) the spatial dimension is *D*≥2, (ii) the system has a smooth extended Fermi surface, and (iii) the interactions are repulsive in the Cooper channel, non-singular and short-ranged.

The key ingredient in the result is the existence of a finite Fermi surface that

— ensures a finite DOS at the Fermi energy and hence the screening of the electron–electron interaction making it effectively short-ranged,

— makes the tree-level scaling analysis effectively one dimensional as only the momentum component perpendicular to the Fermi surface scales like the energy, and

— provides the kinematics necessary for the Landau channels to work.

Most of the non-Fermi liquid behaviours searched for in two-dimensional systems in the last few years have been associated with either specific shapes of the Fermi surface as in the so-called Van Hove scenario [36] or with singular interactions [37]. The case of graphene is especially interesting for the reduction of the Fermi line to two single points. This fact prompted the early RG analysis [13], which at the time aimed to find non-Fermi liquid behaviour from the fact that Fermi points were characteristics of one-dimensional systems with different universal behaviour (Luttinger liquid).

Next, we will classify the local interactions in the graphene system at tree level following [34,35]. From the Hamiltonian (2.6), we can construct the action
3.1
which allows the scale dimension of the fields to be fixed to [*Ψ*]=−2 in units of energy. Notice that for usual non-relativistic fermions, the dispersion relation is also linearized around the Fermi surface. The crucial difference between the two cases is that, because the Fermi surface is a point, there is an isotropic scaling of the momentum so that under a rescaling of the energy *ω*∼*sω*, **d**^{2}**k**∼*s*^{2}**k**. In the presence of an extended Fermi line, we would have d^{2}**k**=d**k**_{∥} d**k**_{⊥}∼d**k**_{∥}*s*d**k**_{⊥} and the dimension of the fields would be [*Ψ*]=−3/2 in any number of dimensions.

Having established the different scale dimension of the fields, it is obvious that four (or more) Fermi interactions—i.e. all local interactions in the condensed matter approach—are irrelevant. (Notice that they were also already generically irrelevant in the non-relativistic case and only the Landau channels corresponding to special kinematics around the Fermi line were made marginal.) As we cannot find any marginal local interactions in the graphene case, this finishes the analysis. No magnetic or superconducting instabilities driven from local interactions are to be expected in the clean graphene system assuming that the couplings are small and perturbation theory applies, a fact that also seems to be confirmed by most experiments.

Although the only possible local interactions in non-relativistic electrons are made of polynomials in the fields, we know that relativistic fermions interact through gauge fields and these interactions will be marginal. We will describe how they arise in the graphene system in the next section.

## 4. Coulomb interaction: graphene versus quantum electrodynamics

In QFT, the Coulomb interaction among the fermions is realized by coupling the fermionic current to the (photon) gauge field *A*_{μ} through the minimal coupling
4.1
The gauge field has a free Lagrangian, *L*_{0}=*F*_{μν}*F*^{μν}, where *F*_{μν} is the electromagnetic tensor, *F*_{μν}=∂_{μ}*A*_{ν}−∂_{ν}*A*_{μ}. Hence, the **k**-dependence of the propagator of the gauge field is 1/**k**^{2} in any number of dimensions. In the case of graphene owing to the presence of the Fermi velocity, *v*_{F}≠*c*, the electronic current has the form
4.2
Notice that although the electrons are confined to live in the two-dimensional surface, the Coulomb interaction among them lives in the three-dimensional space. We then face the problem of coupling the two-dimensional current with a three-dimensional gauge potential. In González *et al.* [13], this difficulty was solved by integrating the usual photon propagator over the *z*-component, which produces an effective two-dimensional propagator for the gauge field with a 1/|*k*|-dependence. Since, by construction, the vector potential scales like the derivative, the interaction (4.1) is marginal. The full action
4.3
is scale invariant. *γ*^{μ} are a set of Dirac matrices constructed with the Pauli matrices.

The unscreened Coulomb interaction is usually written as
4.4
This static, non-local interaction can be obtained as a one-loop effective action from equation (4.3) by integrating out the vector field with the 1/**k** propagator and taking the limit *v*/*c*→0.

Action (4.3) looks like that of (non-relativistic) QED in two spatial dimensions, but there are important differences that affect the discussion of the one-loop renormalization of the coupling. It is the anomalous photon propagator that makes the interacting graphene system different from QED(2+1) and the reason why the results found there cannot be translated directly to the physics of graphene. The coupling constant of QED(2+1) has a dimension of mass () and the theory is called ‘superrenormalizable’. It means that it has less divergences than its four-dimensional counterpart. In fact, there are no ultravioletQ4 infinities in QED(2+1). By construction, the model given by equation (4.3) is fully scale invariant, the coupling constant is dimensionless and, in this sense, the system resembles more what happens in QED(3+1), which is strictly renormalizable—or at a critical point. This is also what induces a renormalization of the electron self-energy in the graphene system not present in planar QED.

## 5. Renormalization group at the one-loop level

Once we have identified the marginal character of the Coulomb—and in general any generalized gauge-type interaction of the electronic current (4.2) with a vector field of weight 1—at tree level, we have to establish the behaviour of the renormalized coupling constant.

From the structure of the perturbative series, we see that the effective—dimensionless—coupling constant is
where *e* is the electron charge and *v*_{F} is the Fermi velocity. This is the equivalent of the graphene fine structure constant, where *v*_{F} replaces the speed of light *c*. We will proceed as in the Fermi liquid analysis and perform a perturbative renormalization at the one-loop level. This amounts to assuming, for the time being, that the effective coupling constant is small, something which is not guaranteed in the suspended samples. We will comment on that later. (The bare value of this constant at energies around 1 eV is of the order of 2–3, corresponding to 300*α*_{QED}.) The RG analysis of the model (4.3) has been performed in full detail in earlier studies [13,28,38,39]. Here, we will summarize the results obtained there.

The very special structure of the model discussed in the previous section makes the discussion of the coupling constant renormalization at the one-loop level more subtle than the case of the non-relativistic condensed matter fermions or QED. The building blocks of the Feynman graphs are the electron and photon propagators
5.1
and
5.2
and the tree-level vertex *Γ*^{μ}=*ie*(*γ*^{0},*v*** γ**).

Owing to the non-covariant form of the electron propagator (5.1), the system has four free parameters, the electric charge *e*, the Fermi velocity *v*_{F} and the electron and photon wave function renormalization. A Ward identity associated with charge conservation relates the electron propagator and the vertex function so that the renormalization of the model can be done as in QED by renormalizing the electron and photon propagators only.

The one-loop diagrams are shown in figure 3. It can be shown, to all orders, in perturbation theory that the photon propagator is finite and hence the electric charge is not renormalized. All the renormalization of the system comes from the electron propagator. The renormalization of the Fermi velocity and the electron wave function is obtained from the electron self-energy *Σ* computed from figure 3*a* and from the relations
5.3
The wave function renormalization
5.4
defines the anomalous dimension of the field
5.5
(*l* is the RG parameter that for a hard cutoff *Λ* is ), and hence the asymptotic behaviour of the fermion propagator,
5.6
The function *Z*_{v} defines the running of the coupling *v* whose beta function is defined as
5.7
The fixed points of the system are determined by the zeros of this beta function, and the running of the velocity is given by the local behaviour of the beta function around the fixed point [40].

The one-loop computation of the electron propagator (figure 3*a*) gives the result [13]
5.8
The equation *β*_{v}=0 has the solution *v*=1, which in our units means that we have a fixed point of RG where the electronic velocity equals the speed of light.

Since the electric charge *e* is not renormalized, the value of the coupling constant at the fixed point *g** is the fine structure constant of QED
5.9

The first derivative of the velocity beta function at the fixed point is negative, hence the velocity decreases as the energy is increased from the infrared fixed point. The growth of the velocity as the energy decreases can be observed in the experiments and there have already been some reports in this respect [27]. The initial value of the velocity can be taken as the Fermi velocity measured in the experiments [41]
5.10
The analysis of the running of the velocity at the energies accessible to the experiments can be simplified by taking this limit *v*/*c*<1, where we get the simple expressions
5.11
from where we can relate the Fermi velocity at two different energies,
5.12
Since the parameter *e* remains arbitrary (it is not renormalized), the value of *g* has to be fixed by a different experiment [42]. The anomalous dimension is found to be
5.13
The anomalous dimension at the fixed point implies the departure of the system from the Fermi liquid behaviour.

Summarizing, the results are the following:

— The Fermi velocity grows as the energy is decreased. This result implies a breakdown of the relation between the energy and momentum scaling, a signature of a quantum critical point.

— The electron–photon vertex and the photon propagator are not renormalized at the one-loop level. This means that the electric charge is not renormalized, a result that could be predicted by gauge invariance, and it also implies that the effective coupling constant,

*g*=*e*^{2}/4*πv*_{F}decreases at low energies defining an infrared free fixed point of the RG.— The beta function of the Fermi velocity has a non-trivial zero at the value

*v*=*c*. Hence, the infrared fixed point defines a weak coupling model ruled by the fine structure constant of QED, which is Lorentz covariant.— There is a critical exponent that determines the universality class of the model. The anomalous exponent means that the fixed point is not a Fermi liquid and resembles more a Luttinger liquid.

## 6. Marginal Fermi liquid?

Because of the smallness of the ratio, *v*_{F}/*c* and for simplicity, the Coulomb interactions in graphene are often modelled with a static scalar field that couples only the the charge density. This model was analysed by González *et al.* [38]. The original motivation of this work was to explore if the gapless free infrared fixed point encountered in the study of González *et al.* [13] within the perturbative RG scheme would survive in the simplest non-perturbative resummation consisting of computing the one-loop electron self-energy with an effective interaction given by the bubble sum in the photon propagator (1/*N* approximation). The logarithmic renormalization of the electron wave function found in this case has led to some confusion on the marginal Fermi liquid (MFL) nature of the system that it is worth clarifying.

The so-called MFL behaviour [43] was set as a phenomenological model to explain some anomalous behaviour of the cuprate high-*T*_{c} superconductors. It has two prominent features: a linear behaviour of the scattering rate with the energy near the Fermi surface
6.1
where , and more important, a vanishing quasi-particle residue at the Fermi energy
6.2
and
6.3
The two are related by causality as they are extracted from the real and imaginary part of the electron Green function. The quasi-particle weight *Z*_{k} vanishes logarithmically at the Fermi surface *E*_{k}=0 and the Green function is entirely incoherent. As pointed out in the original MFL reference, such a behaviour is possible only if perturbation theory, starting from a non-interacting Fermi gas, breaks down at some energy. We must also notice that the postulated MFL behaviour in the high-*T*_{c} superconductors is expected to arise from the strongly correlated nature of the electron–electron interaction in these compounds. In the case of graphene, the linear scattering rate comes from the singular nature of the unscreened Coulomb interaction (*N*(0)=0 in equation (6.1)).

The (instantaneous) Coulomb interaction in graphene is described by the time component of the gauge field taken as a scalar field *ϕ*,
6.4
where the field *ϕ* has the propagator
6.5
Inserting the random phase approximation sum in the electron propagator, we get the divergent contributions to the electron self-energy from which we can extract the following RG equations:^{1}
6.6
and
6.7
where *Λ* is the ultraviolet cut-off and *g* is proportional to the effective coupling constant, *g*=*e*^{2}/16*v*_{F}. The simultaneous solution of these equations shows that the quasi-particle residue goes asymptotically to a non-zero constant value [38].

The infrared behaviour of the system is that of a strange Fermi liquid: although the scattering rate grows linearly with the energy, the wave function renormalization *Z* runs to a finite value at the Fermi point. This paradoxical behaviour can be seen as another manifestation of the ‘incompleteness’ of the static model, which cannot be used to establish the infrared nature of the system.

## 7. Other renormalization group aspects

### (a) Disorder and interactions

In graphene, many classes of lattice defects can be described by gauge fields coupled to the two-dimensional Dirac equation [25]. The standard techniques of disordered electrons [44] can be applied to rippled graphene by averaging over the random effective gauge fields induced by curvature or elastic deformations. A random distribution of defects leads to a random gauge field, with variance related to the type of defect and its concentration. There is extensive literature on the problem, as the model is also relevant to fractional quantum Hall states [45] and to disorder in *d*-wave superconductors. A random field, when treated perturbatively, leads to a renormalization of the Fermi velocity, which decreases it at lower energies opposing the upward renormalization induced by the long-range Coulomb interaction. The simultaneous presence of interaction and disorder gives rise to new interesting fixed points. The issue was analysed by Stauber *et al*. [46]. The most interesting case arises when considering a random gauge potential that models elastic distortions and some topological defects. There is a line of fixed points with Luttinger-like behaviour for each disorder correlation strength *Δ* given by . An extensive analysis of the issue in the large-*N* limit was carried out by Foster & Aleiner [47].

### (b) Short-range interactions

As we have seen, short-range interactions, such as the onsite Hubbard term *U* are irrelevant, a fact due to the vanishing DOS at the Fermi level. As mentioned above, the DOS at low energies is increased by the presence of disorder. This, in turn, enhances the effect of short-range interactions that were discussed in the earlier work of González *et al.* [39] and have attracted some attention recently. These interactions can be relevant in the strong coupling regime of the hexagonal electronic and optical lattices [48,49].

A summary of the situation following the study of González *et al.* [39] is as follows: in the absence of disorder, an onsite Hubbard term favours antiferromagnetism. An antiferromagnetic phase, however, is likely to be suppressed by disorder, especially by the presence of odd-numbered rings in the lattice. Then, the next leading instability that such an interaction can induce is towards a ferromagnetic phase.

If a magnetic phase does not appear, electron–electron interactions, even when they are repulsive, will lead to an anisotropic ground state. The existence of two inequivalent Fermi points in the Brillouin zone suggests that the superconducting order parameter induced by a repulsive interaction will have opposite signs at each point. The corresponding symmetry is *p*-wave. Disorder, in addition to the enhancement of the DOS mentioned already, will lead to pair-breaking effects in an anisotropic superconducting phase.

### (c) Bilayer graphene

The synthesis characterization and analysis of the bilayer material (bilayer graphene) occurred simultaneously with its monolayer counterpart [50]. The band structure was also theoretically described in the early publication [32]. The most common structure is made of two graphene layers arranged in the so-called Bernal stacking, as shown in figure 4. It was soon realized that the bilayer graphene was even more promising than the monolayer because of the possibility of opening and controlling a gap in the system [51,52]. From a QFT point of view, it constitutes an interesting example of a chiral system with quadratic dispersion relation.

A tight binding approach carried out to the minimal Bernal stacking bilayer shows that the system has four bands: two low-energy bands touching at a Fermi point and two at higher energy. An effective low-energy continuum model [53] keeping only the lower bands can be described by the Hamiltonian
7.1
where **k**=(*k*_{x}+*ik*_{y}). The dispersion relation is quadratic, but still has a non-trivial chirality. Unlike the free monolayer system whose effective description corresponds to regular massless fermions, there is no QFT with quadratic propagator and Dirac structure. In this sense, although being chiral, the Fermi points of the bilayer cannot be termed as Dirac points. The interacting model has been subjected to great interest recently owing to the enhanced quality of the samples [54,55].

The role of interactions and possible instabilities in the bilayer system was first analysed by Castro *et al*. [56], and has received renewed attention more recently [57–60]. RG studies have been performed earlier [61–63]. The interest of this case relies on the fact that while the quadratic dispersion makes short-range interactions marginal, the existence of Fermi points (although with finite DOS) can give rise to departures from the Fermi liquid behaviour. MFL behaviour similar to the one discussed for the monolayer in §6 has been advocated in the study of Barlas & Yang [61], and a fairly complete list of magnetic and pairing instabilities has been described in earlier studies by Nandkishore & Levitov [63] and Vafek [64].

## 8. An attempt of a summary

The infrared properties of graphene described in this work explain the success of the non-interacting model to describe most of the low-energy electronic and transport properties of the system. We have seen that the only interactions that can play a role in the system are of the form of gauge couplings. Of these, the most important is the unscreened Coulomb interaction that runs in the infrared to a very weakly interacting fixed point. In the non-relativistic model, the interaction is marginally irrelevant.

A very interesting theoretical issue that makes the model differ from both its QFT and its condensed matter counterparts is the fact that the coupling constant renormalization at the one-loop level is owing to the upward renormalization to the Fermi velocity to the infrared. This very fact sets a lower bound on the validity of the widely used instantaneous model for the Coulomb interactions, which ceases to be valid when the Fermi velocity approaches the speed of light. Although this bound is so small that it can be ignored for most practical matters [42], it is conceptually important that the infrared nature of the system has to be decided with the full retarded model.

Another interesting issue is that the running of the Fermi velocity stops at a non-trivial fixed point where its value equals the speed of light so that the Lorentz covariance appears as an emergent property [29]. Unlike most infrared stable QFT, where the fixed point is trivial (*g*=0), the graphene system has a non-trivial fixed point characterized by the fine structure constant.

Within the static approximation, a two-loop calculation of the electron inverse lifetime shows that it is linear in energy, a behaviour that persists in the 1/*N* approximation and that is anomalous for a metallic system.

Experimental indications of both the behaviour of the inverse lifetime of the electron linear in the energy [26,65] and of the Fermi velocity renormalization [27] have been recently reported. The irrelevance of the short-range interactions excludes *a priori* any intrinsic superconductivity or magnetism in the samples. The RG results also exclude the possibility of opening a gap in the weak coupling regime, a fatal situation for the electronic applications. The possibility of opening a gap in the spectrum with or without the help of a magnetic field is crucial for the possible electronic applications and was the main concern in the strong coupling analysis. A recent update of the issue and the main references can be found in the work of Khveshchenko [66]. A non-perturbative RG analysis of the system in the lattice has been carried out by Giuliani *et al.* [67].

## Acknowledgements

I thank A. Cortijo, A. G. Grushin, F. de Juan and B. Valenzuela for very useful conversations and critical reading of the manuscript. Support by MEC (Spain) through grant FIS2008-00124 is acknowledged.

## Footnotes

↵1 Notice that there is an erratum in eqn (10) of González

*et al.*[38], where the derivative of appears instead of*Z*.One contribution of 11 to a Theme Issue ‘New applications of the renormalization group in nuclear, particle and condensed matter physics’.

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