Graphene and non-Abelian quantization
In this article we employ a simple nonrelativistic model to describe the low energy excitation of graphene. The model is based on a deformation of the Heisenberg algebra which makes the commutator of momenta proportional to the pseudo-spin. We solve the Landau problem for the resulting Hamiltonian which reduces, in the large mass limit while keeping fixed the Fermi velocity, to the usual linear one employed to describe these excitations as massless Dirac fermions. This model, extended to negative mass, allows to reproduce the leading terms in the low energy expansion of the dispersion relation for both nearest and next-to-nearest neighbor interactions. Taking into account the contributions of both Dirac points, the resulting Hall conductivity, evaluated with a -function approach, is consistent with the anomalous integer quantum Hall effect found in graphene. Moreover, when considered in first order perturbation theory, it is shown that the next-to-leading term in the interaction between nearest neighbor produces no modifications in the spectrum of the model while an electric field perpendicular to the magnetic field produces just a rigid shift of this spectrum.
PACS: 03.65.-w, 81.05.ue, 73.43.-f
Key words and phrases:Quantum Mechanics, Graphene, Quantum Hall effect
Several theoretical ideas have produced a fruitful exchange of interpretations and methods between fundamental physics and condensed matter theory, such as spontaneously broken symmetry [1, 2, 3, 4] or renormalization group methods [5, 6, 7, 8, 9].
The recent experimental construction of graphene  opens a new connection between condensed matter and quantum field theory, since its low energy excitations can be represented as massless planar fermions and described by means of a (pseudo) relativistic theory.
Gaphene (See [11, 12] and references therein) is a two-dimensional, one-atom-thick, allotrope of carbon which has attracted great attention in the last years. The carbon atoms are arranged on a honeycomb structure made out of hexagons, a structure with a great versatility. Carbon nanotubes, for example, can be obtained by rolling the graphene plane along a given direction and reconnecting the carbon bonds at the boundaries, giving rise to essentially one-dimensional object. Also, the replacement of an hexagon by a pentagon in this lattice introduces a positive curvature defect; this allows to wrap-up graphene to give fullerenes, molecules where carbon atoms are arranged on spherical structures. Notice that the accumulation of graphene layers, weakly coupled by van der Waals forces, constitute the well-known graphite.
The electronic properties of graphene are the result of the hybridization between one and two orbitals, which leads to a trigonal planar lattice with a bond between carbon atoms separated by Angstrom. This filled band gives the lattice its robustness. The third orbital of the carbon atom, oriented perpendicularly to the plane of the graphene, gives rise to the band through covalent bounds with neighboring atoms. Since this orbital contributes with only one electron, the band is half-filled in neutral graphene.
The low energy excitations of graphene accept a description as states of chiral massless Dirac fermions with a pseudo-relativistic linear dispersion relation, in which the speed of light is replaced by the Fermi velocity, . Then, the Lagrangian describing these low energy states in the presence of an Electromagnetic field is similar to that of QED for massless fermions, shearing therefore some of its peculiarities. In particular, when a magnetic field is applied perpendicularly to the plane of graphene, an anomalous integer quantum Hall effect [15, 16] takes place, which has been experimentally measured [17, 18].
It is not the goal of the present paper to discuss the physics of graphene from first principles. Rather, we will consider a simple effective non-relativistic Hamiltonian, suggested by a particular deformation of the Heisenberg algebra (non-commutativity of momenta, consistent with the introduction of an external constant non-Abelian magnetic field), which could be useful to describe the low energy excitations produced by the dominant nearest and next-to-nearest neighbor interactions in graphene.
The non-commutativity of spacetime is an old idea , the first example of which was probably discussed by Landau in 1930 . It has been revived in recent years within the context of string theory  and since then, non-commutative field theories have attracted much attention in various fields such as Mathematics, Theoretical Physics , [22, 23, 24, 25], and Phenomenology .
The breaking of commutativity of the position operators and the representations of the algebra of the non-commutative space-time coordinates has been studied in . The non-commutativity in the momenta algebra we are interested in can be related to the deformation quantization of Poissonian structures developed in  and considered as a kind of magnetic quantization [29, 30].
All these researches have stimulated the construction of new models in quantum mechanics , which have opened new routes to explore, for example in superconductivity . Also, a massless Dirac-like Hamiltonian in a generalized noncommutative space and its relation with graphene has been considered in .
Recently, some models based on a kind of nonstandard deformation of the Heisenberg algebra, which can be realized by shifting the dynamical variables with the spin, have been studied in [33, 34]. In the following, we will consider a similar deformation, but concentrated in the commutators among momenta, which can be interpreted as the introduction of a constant non-Abelian magnetic field .
In the next Section we present the model and derive the Hamiltonian. In Section 3 we study the free case and in Section 4 we introduce a constant magnetic field and solve the corresponding Landau problem. In Section 5 we apply our results to describe the low energy states associated to the leading nearest and next-to-nearest interactions in graphene.
In Section 6 we evaluate the associated Hall conductivity employing a -function approach, finding that the result is consistent with the anomalous integer quantum Hall effect present in this material. In Section 7 we are also able to show that the next-to-leading (quadratic) term in the nearest neighbor interaction does not change the spectrum at first order in perturbation theory. In Section 8 we comment on the case where crossed electric and magnetic fields are present and, finally, in Section 9 we establish our conclusions.
At the end of the paper, Appendix A is dedicated to the Lagrangian of the model and studies its symmetry, conserved current and the generating functional of Green’s functions. It also comments on the weak field and gradient expansion of the generating functional, its relation with the Hall conductivity and the topological considerations involved.
2. Deformation of the Heisenberg algebra
We consider particles leaving on a plane whose dynamical variables satisfy the deformed Heisenberg algebra given by
where the momenta commutator is proportional to the pseudospin and is a parameter with dimensions of momentum (For convenience, we take and return to full units when necesary).
Notice that these particles are described by wave functions with two components, , and these operators have the structure of matrices on .
The deformed algebra in Eq. (2.1) can be realized by defining
with the usual (commutative) coordinates on the plane and , operators on , and the first two Pauli matrices. We will also write . For notational convenience, from now on we will avoid the explicit indication of the symbol , which can lead to no confusion.
Our aim is to consider the direct generalization of the Hamiltonian of a (nonrelativistic) particle of charge and mass , minimally coupled to an external (2+1-dimensional) electromagnetic field, , which is constructed by means of the replacements , .
For the time being, we make . Then,
where the electromagnetic field is taken in the Coulomb gauge, .
The Hamiltonian can also be written as
where we have defined the Fermi velocity
and subtracted the constant .
In the limit, with fixed , the resulting linear Hamiltonian is appropriate to describe the conducting effective particles in graphene around the Fermi points [13, 11, 12, 15], which justify our proposal. Notice that this limit does not correspond to a small but rather to a large deformation of the commutator .
Notice also that the modification of the Hamiltonian in Eq. (2.4) can also be interpreted as the introduction of an non-Abelian constant and uniform “magnetic” field . Indeed, the transformations relate both components of the wave functions, and the commutator of covariant derivatives gives
In this sense, if we take a constant magnetic field , the system we are considering is a kind of non-Abelian version of the Landau problem .
In Appendix A we describe the Lagrangian of this model, study its symmetry and conserved current and discuss the relation of the asymptotic expansion of its generating functional with the Hall conductivity.
3. The free case
In this Section we consider the free case, with . The Hamiltonian reduces to
We propose solutions of the form
with , which replaced in
Nontrivial solutions require
Then, we get the following dispersion relation (approximately linear for small , see Figure 1)
which, replaced in Eq. (3.4) for , shows that the pseudo-spinor has definite pseudo-helicity,
On the other hand, for the two linearly independent vectors and are just constants solutions with vanishing eigenvalue.
The Hamiltonian in Eq. (3.1) commutes with the effective angular momentum, the generator of a symmetry,
Under a rotation on the plane, the wavefunction in Eq. (3.2) changes into
For we have and . So, we get
Therefore, these particles can be considered as fermions .
4. Constant magnetic field perpendicular to the plane
In this Section we consider the electromagnetic vector potential of a constant magnetic field orthogonal to the plane,
In this case the Hamiltonian can be written as
which clearly commutes with . This allows to look for generalized eigenfunctions of the form
The eigenvalue equation for the Hamiltonian, , reduces to the pair of coupled differential equations
At this point, it is convenient to change the remaining variable in favor of . So, , with , and the system in Eq. (4.4) writes as
For simplicity, in the following we will take . The case can be obtained from the previous one by simply interchanging the components of the solutions, , as can be easily seen from Eq. (4.5). So, we consider
The presence of twice the Hamiltonian of a harmonic oscillator of frequency 1 in the brackets on the left hand side of Eqs. (4.6), operator with eigenfunctions ( the Hermite polynomials ) and eigenvalues , for , suggests that and , for some . Moreover, since the Hermite functions satisfy
one concludes that these solutions are of the form
for , with . Replaced in Eq. (4.5), we get the homogeneous system of algebraic equations
The eigenvalues are determined by the condition
which implies that
Appropriately normalized, the generalized eigenfunctions are written as
Indeed, taking into account the orthogonality relations for the Hermite functions it can be easily verified that
Finally, notice that there is another solution of Eq. (4.6) whose components are given by
Indeed, since , we have
and Eq. (4.5) reduces to
This implies that , which is independent of .
One can also verify that
As mentioned in Section 2, these results can be interpreted as the solution of a non-Abelian version of the Landau problem, in which we have also introduced a constant and uniform non-Abelian magnetic field. Moreover, in the limit, these eigenfunctions are similar to the solutions found  for the Dirac equation in a constant magnetic background.
where are the Hermite functions. Then, the set of generalized eigenvectors of we found is also complete, which ensures that in our analysis we got the whole spectrum of the Hamiltonian.
In the following, we consider the large- (large-) limit we are interested in111For we have equally spaced levels in each branch, with slightly different slopes, (4.24) For the eigenfunctions we get (4.25) and (4.26) The -limit is consistent with the usual Landau levels problem..
For the energies are given by
and the eigenfunctions reduce to
Since we are interested in the description of low energy states and the construction of the solutions of Eq. (4.4) is independent of the sign of , we can explore the behavior of the system for . Indeed, it is safe to change the sign of in the solutions, which corresponds to the replacement , with , in the last line in Eq. (4.12) maintaining . We get (See Figure 2)
5. The relation with graphene
The structure of graphene  can be seen as a triangular lattice with a basis of two atoms per unit cell. The lattice vectors are
where is the lattice constant (distance between nearest neighbor carbon atoms). The vectors characterizing the reciprocal lattice (whose elementary cell is the Brillouin zone) are given by
The superposed triangular lattices form an hexagonal honeycomb array of carbon atoms where the nearest-neighbor vectors are
The tight-binding model for graphene, where it is assumed that electrons can only hop to both nearest () and next-to-nearest () neighbor atoms, is described by the Hamiltonian 
where and are the hoping energies, and are the creation and annihilation operators of electrons in the site with spin belonging to the triangular sublattice , and similarly for the sublattice . By Fourier transforming these operators and diagonalizing the resulting expression, one gets the band structure (dispersion relations) 
The minima of in the Brillouin zone correspond to the Dirac points
where . Then, the bands corresponding to each sign in the dispersion relations, Ec. (5.5), touch each other at the Dirac points. Notice that these two independent Dirac points have been chosen so that they are related by the reflection of about the -axis () .
A series expansion of around leads to
where and .
The expansion around the second Fermi point, , leads to the same expression with reflected about the -axis. This corresponds to the change , which changes the sign of the the second term in the first bracket in the right hand side of Eq. (5.8).
We now turn to the comparison with the model developed in the previous Sections. From Eqs. (5.8) and (3.7), it is seen that we can identify . Moreover, the next-to-nearest neighbor contribution can be represented in our (free) model by the mass term through the identification , up to a rigid displacement of the spectrum in an energy (which can be subtracted to restore the particle-hole symmetry at linear order in ). This means that we must consider a negative mass in our model, 333According to the values of these parameters reported in , we have (5.9) Then, (5.10) or, in natural units (, ), (5.11) This is a rather large value for the (negative) mass of the quasiparticles, since . For example, for an external magnetic field Tesla, we get .
On the other hand, the second (quadratic) term in the nearest neighbor contribution depends on the direction of . We can not reproduce this behavior within the framework of our model since it is rotationally invariant. Rather, we must treat it as a perturbation on the solutions we found for our model. This will be considered later.
Then, we will assume that the effective Hamiltonian for the low energy states of this system around in the presence of an electromagnetic field, , is obtained though minimal coupling and coincides with the one given in Eq. (2.4).
and for hole states
Moreover, there is an additional (hole) state at energy , slightly below the zero energy level (see Eq. (4.20)). Indeed, the difference in energy for the two first hole states is 444For and Tesla, we have in full units (5.14) This leads to and ..
5.1. The second Fermi point contribution
As previously mentioned, the dispersion relations near the Fermi points are related by the reflection of momenta about the -axis. Then, for the linear terms in the effective free Hamiltonian describing the states around the second Fermi point we can write 
Accordingly, we will assume that the effective Hamiltonian describing the quasi-particles near the second Fermi point in our model, , can be obtained from through the transformation
Evidently, the spectrum of is obtained from that of by a reflection about the origin and its (generalized) eigenfunction are just times those of . This interchanges quasi-particles with holes. Indeed,
with an energy . This state corresponds to a quasi-particle of energy slightly above zero.
Therefore, taking into account the eigenstates of both and we get, for quasi-particles and for holes, an almost doubly degenerate spectrum, except for one state of quasi-particle and one state of hole with energies near zero. For positive energy states we have
where the gap between contiguous states is
6. The Hall conductivity
As discussed in Appendix A, the Hall conductivity has a topological character and can be calculated from the weak field and gradient expansion of the effective action of the system. However, since we know the exact eigenvalues of energy of our model, we will employ a more direct evaluation method, based on the relation established between the conserved current and the external electromagnetic field, Ecs. (A.13) and (A.14).
For our purposes, it will be sufficient to consider the mean value of the density . Then, as in the previous Sections, we will take and perpendicular to the plane of the system.
Let us start by writing down the partition function of the particles around one Fermi point at inverse temperature and chemical potential . It can be obtained through a Wick rotation of the generating functional (see Eq. (A.8)) to the 2+1 Euclidean space by means of the replacements , , maintaining unaltered the other coordinates and components of the gauge field.
where is (or ) and the functional integral is performed on the set of configurations of the fermionic field which satisfy anti-periodic boundary conditions on . Then,
where use has been made of Eq. (A.14).
Our goal is now to evaluate the partition function as the functional determinant
where is a differential operator defined on a domain of anti-periodic functions of . Even though is not symmetric, since does not depend on this operator has a complete set of orthogonal generalized eigenfunctions constructed as
with , , , , and where the Matsubara frequencies
Then, the eigenvalues of are given by
where and are the eigenvalues of , studied in the previous Sections.
Notice that and are independent of . Therefore, in evaluating we will forget about the index and, at the end, take into account the degeneracy it introduces per unit area, given by the number of flux quanta per unit area [40, 41], (or in full units).
where is an arbitrary mass scale, the trace is evaluated for sufficiently large and stands for the analytic continuation of its derivative to a neighborhood of . In our case,
Let us first consider the contribution of the second term on the right hand side and evaluate
where use has been made of the properties of the Matsubara frequencies stated in Eq. (6.5). We can also write
These series can be expressed in terms of the Hurwitz -function [44, 45]. Taking into account that this function has branch cut discontinuities in the complex plane of its second argument running from 0 to , we get
The expression in braces has a vanishing analytic continuation at . As a consequence, the derivative of the right hand side at does not depend on the arbitrary scale . Then, the contribution of this Landau level to per unit area and at low temperature is given by [44, 45]
Taking into account that we are interested in the mean number of particles with respect to the neutral material (i.e., with all the Landau levels with negative energy filled, which corresponds to ), from Eq. (6.2), we finally get for the contribution of the Landau level with energy to the Hall conductivity at zero temperature
Therefore, for positive , we get a contribution to if and zero otherwise. On the other hand, for negative we get a contribution if and zero otherwise.
At this point, it is clear that a similar result is obtained for any other Landau level. Indeed, the same considerations can be done for any other energy eigenvalue . Then, for a given chemical potential , there is only a finite number of nonvanishing contributions and the Hall conductivity reduces to
where the sums run over the energy levels. Indeed, this is a rather general result for the Landau problem of a fermionic system.
For the model of interest in the present paper, the spectrum to be considered is the union of those of and , , which takes into account the states around both Fermi points. But they must be taken with an additional degeneracy corresponding to the two polarizations of the electron spin, which has played no role up to now in this analysis.
In full units we get
for small (but ). This is the characteristic of the anomalous integer quantum Hall effect found in graphene [15, 17, 16, 18], which shows a nonvanishing Hall conductivity for small (positive or negative) Fermi level.
In Figure 3, the Hall conductivity of this model as a function of (equal to the Fermi level at zero temperature) has been plotted for . Notice that it is not possible to appreciate the structure of each step as a double step (due to the small degeneracy breaking remarked in Eqs. (5.21) and (5.22)) with such a realistic value of (See Figure 4). Also notice that the Hall conductivity of our model vanishes in a small neighborhood of the origin, for .
7. Perturbation with the quadratic nearest neighbor interaction term
Notice that the quadratic term in the nearest neighbor dispersion relation in Eq. (5.8) can be obtained from the solutions of the free problem, Eqs. (3.2) and (3.8), through the (rotational symmetry breaking) perturbation
We will assume that, in the presence of an electromagnetic field , derivatives change into covariant derivatives and the perturbation just turns into the Hermitian expression
Notice that there is an order indeterminacy in this definition. We will come back later on this point.
We now consider a constant magnetic field as in Eq. (4.1). Then,