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Fermi point in graphene as a monopole in momentum space

We consider the effective field theory of graphene monolayer with the Coulomb interaction between fermions taken into account. The gauge field in momentum space is introduced. The position of the Fermi point coincides with the position of the corresponding monopole. The procedure of extracting such monopoles during lattice simulations is suggested.


INTRODUCTION
Graphene is a unique 2 + 1 -dimensional nonrelativistic system that shares common properties with relativistic quantum field theory. In particular, in the effective field theory of graphene the massless Dirac spinors appear [1- 3,12]. When the Coulomb interaction is taken into account, the effective Lorentz symmetry is broken. The phase structure of the model may be changed when the external conditions are changed (that may lead, say, to the change of Fermi velocity v F ) [9-11, 13, 15]. Change of the phase structure of the model must be accompanied with the deformation of the momentum space topology [4,5]. Therefore, it is important to investigate various topological invariants in momentum space of the effective field model.
In general in 3D the Fermi points are not topologically stable [4]. This is because π 2 (GL(N, C)) = 0 for N ≥ 2. The N × N Green function in momentum space belongs to GL(N, C). That's why topological triviality of mapping S 2 → GL(N, C) does not allow topological stability of the Green function's poles in general case. However, if a certain symmetry is present that reduces the size of the space of the Green functions, the topological stability becomes possible [5,15]. In particular, the 3D model of graphene monolayer has such a symmetry that effectively reduces space of the Green functions considerably. As a result, the topological invariant N 2 appears. This invariant is expressed through Green function at zero frequency ω = 0 and is an integral over the closed contour C around Fermi point in the plane of 2D momenta p (see, for example, [5]). The important advantage of the existence of this invariant is that the pole of the Green function cannot disappear without a phase transition. It is worth mentioning, however, that this construction is less natural than that of the invariant N 3 of 4D theory [4]. This is because only the structure of ω = 0 plane in momentum space is reflected by N 2 . Moreover, the given construction was introduced when Coloumb interaction between quasiparticles is neglected, and in the case, when these interactions are present, it requires an additional investigation.
In this paper we extend the construction of N 2 to the effective field model of graphene in such a way that the topological invariant is written as an integral over the surface in 3D momentum space (ω, p). In the form presented here this invariant works also for the case, when the Coulomb interaction is present. We show that when the Green function is smooth enough, our construction can be reduced to the original construction of N 2 . In addition, we present the definition of the gauge field in momentum space such that the positions of the corresponding monopoles coincide with the positions of the poles or zeros of the Green function. This construction is intended mainly to be used during lattice simulations. We also suggest the procedure of extraction the monopoles in momentum space for the lattice discretization with staggered fermions.

THE FIELD THEORETICAL EFFECTIVE MODEL FOR GRAPHENE
The low energy effective model of graphene may be derived [1-3] starting from the simple non -relativistic Hamiltonian that describes the interactions of electrons that belong to neighbor Carbon atoms. The carbon atoms of graphene form a honeycomb lattice with two sublattices A and B (or the triangular form). Further we denote the lattice spacing by a.
Let us introduce vectors that connect a vertex of the sublattice A to its neighbors (that belong to the sublattice B): l 1 = (−a, 0), The Hamiltonian has the form Here t is the hopping parameter, operator ψ † creates electrons at the points of the lattice.
Let us define two electron fields in momentum space that correspond to two sublattices: Here V is the number of points in the sublattice A. The Brillouin zone is a hexagon with opposite sides identified. There are two different vertices of the hexagon that are denoted K + , K − . Quasiparticle energy vanishes at these points. We expand ψ around K + , K − , denote At low energy the effective field theory appears. Taking the Fourier transform from q to the coordinate space we come to the field -theoretic formulation of the model: whereD has the form of the usual Dirac operator taken on the 2D hypersurface x 3 = 0: Here γ are the gamma -matrices in the representation to be specified below. Let us remind that we started from the nonrelativistic Hamiltonian and completely disregarded spin degrees of freedom. Now we take them into account adding a new index to the field ψ. We assume it has two spin components. In hamiltonian (3) gamma -matrices act on the pseudospin index while the true spin operator does not enter the Hamiltonian.
We consider the interaction between quasiparticles due to the photon exchange (A is the 3 + 1 electromagnetic field). Let us perform the Wick rotation, the rescaling of time, and gauge fields: Further we denote g = e/ √ v F . Therefore, the analogue of the fine structure constant We also introduce Euclidean Dirac matrices that satisfy We introduce finite temperature T via taking an integral over x 4 within the interval [0, v F T ] and adopting the periodic in x 4 boundary conditions. The chemical potential is assumed to be equal to zero. Due to v F ∼ 1/300 << 1 the fluctuations of A a are suppressed and we neglect them in the functional integral. We arrive at the partition function: Here index A = 1, 2 belongs to the spin degrees of freedom.

GREEN FUNCTIONS
It is worth mentioning that the Green function has to be considered in a certain gauge. The gauge freedom of the system corresponds to the transformation A 4 → Fermion Green function has the form: In order to reveal the 3D nature of the system let us consider the following representation of the spinor field: In terms of χ + and χ − the Green functions are: We have, obviously, We also imply that the Green function is diagonal in spin index. That's why G can be understood as the 2 × 2 matrix. On the language of χ ± different Γ 3 chiralities correspond to the states Here vectors p, x are two -component.

Direct calculation gives
Operator is Hermitian for any real A 4 . That's why we come to the conclusion that the operator iGσ 3 is also Hermitian. This means that the functions g a (ω, p), a = 0, 1, 2, 3 are real. As a results −iGσ 3 belongs to u(2). Considering symmetries of the Green function, we come to the following form of G (see Appendix): That's why iGσ 3 ∈ su (2). If in addition, the scale invariance is not broken (in particular, T = 0), and the functionsf ,h are smooth enough, we have the further simplification: In some publications (see, for example, [5]) the following expression has been considered that is shown to be a topological invariant both with and without external magnetic field when the interaction with A 4 is switched off: Here However, using the above mentioned symmetry considerations we rewrite this function in the absence of external fields as follows: Tr σ 3 C (dg 3 σ 3 + (dg, σ))(g 3 σ 3 + (g, σ)) g 2 3 + g 2 where n 1 = p 1 / [p 1 ] 2 + [p 2 ] 2 , n 2 = p 2 / [p 1 ] 2 + [p 2 ] 2 , and it is implied thath(0, p 2 ) = 0.
As it was mentioned above, when the functions g a are smooth enough, we have g 3 (0, p) = 0 and, therefore, again N 2 = 1. This means that the pole of the Green function (the Fermi point) is topologically stable if the symmetries considered above take place.

TOPOLOGICAL INVARIANT IN SPACE (ω, p)
Below we generalize the construction of the topological invariant N 2 considered above.
The resulting construction uses the Green function defined on the surface that encloses the Fermi point in ω − p space. The considered construction also works for nonzero g 3 (0, p).
Let us define the function in momentum space We can express H through the functions g a mentioned above: H = n a σ a , n a = g a /|g|, |g| = √ g a g a , a = 1, 2, 3. Now let us consider the following integral over closed surface Σ in momentum space such that G does not have poles on Σ: The given expression (16) for the invariant N 2 is, obviously, reduced to (13) in the case, when g 3 (0, p) = 0. Without interactions and without external magnetic field G has the pole at p = ω = 0 that corresponds to the Fermi point. In this case n 3 (ω, p) = ω √ ω 2 +p 2 , n(ω, p) = p √ ω 2 +p 2 , and N 2 = 1 for any surface that encloses the pole. When the interaction with the electromagnetic field is turned on, the value of N 2 for the surface that encloses this pole remains equal to unity until a phase transition is encountered. The important advantage of the given formulation is that we already do not need the condition g 3 (0, p) = 0 to be satisfied. We only need g 0 = 0. The situation, when g 0 = 0 and g 3 (0, p) = 0 may appear in the other 2+1 systems or even in the effective field model of graphene for the inhomogenious gauge or when some of the symmetries are broken dynamically.

FERMI POINT AS A MONOPOLE
As it was explained above,G = −iGσ 3 ∈ su(2) out of the region, where G has poles. We can diagonalizeG via SU(2)/U(1) transformations: V is defined up to the U(1) transformation V → e ασ 3 V . That's why here V ∈ SU(2)/U(1) ∼ S 2 . We can choose V to be smooth on the surface Σ except for a small vicinity Ω of a certain point. We have π 2 (SU(2)/U(1)) = Z. Actually the invariant N 2 is equal to the degree of the mapping S 2 → SU(2)/U(1): In 4D notations Green function (6) has the form: Again, we define the function in momentum space: We can express H through three real functions g a mentioned above: H = −n 1 Γ 1 − n 2 Γ 2 + n 3 Γ 4 , n a = g a /|g|, |g| = √ g a g a , a = 1, 2, 3. Now the topological invariant can be expressed as We denoteG = −iGΓ 4 .G can be diagonalized via the SO(4)/(SU(2) ⊗ U(1)) transfor- spinors. This regularization has been used in practical numerical simulations of the considered model [11,16]. However, the additional doublers ever appear in lattice propagator as it will be explained below. Staggered fermion variables Ψ are obtained via the spin diagonal- Here always x 3 = 0. In terms of Ψ the free fermion action has the form: We keep the only component of Ψ. As a result the doublers play the role of the components of the two original spinors. In order to reconstruct the original spinor and flavor indices of the fermions we consider the lattice with even number of lattice spacings in each direction.
Let us subdivide the lattice into the blocks consisted of elementary cubes. We denote the coordinates of the blocks by y i . Therefore, the coordinates of the lattice sites are x i = 2y i + η i , η i = 0, 1. We define the new fields [17]: Here T i = Γ T i acts on the flavor indices while Γ matrices act on the Dirac indices. Momenta k are k 1 = 2πK 1 Nx/2 k 2 = 2πK 2 Ny/2 k 4 = 2πK 4 +π Nt/2 , K 1 , K 2 , K 4 ∈ Z. In this regularization the mass term is necessarily added. At the end of the calculation one must set m = 0. This Green function turns to the form (19) with G in the form (9) in the continuum limit at m = 0. For m = 0 the only pole of the Green function at p = 0 appears. The fermion doublers do not have such poles. However, zeros of the functions g a , a = 1, 2, 4 appear at p a = πk a , k a ∈ Z. At any value of m vector n mentioned above has the following components: n a = g a / √ g a g a , g 1 = iTr G Γ 4 Γ 1 /4 g 2 = iTr G Γ 4 Γ 2 /4 g 3 = −iTr G/4. formulation also gives monopoles that correspond to the unphysical doublers.
When the interaction is switched on the practical prescription for the calculation of the vector n is n a (k) = g a (k)/ g a (k)g a (k), a = 1, 2, 4 with Here G(2y +η, 2z +η ′ ) is the staggered fermion one -component propagator in the external field averaged over the configurations of the U(1) gauge field A 4 and over the pseudofermion configurations (the latter give the fermion determinant in the averaging over gauge fields).
Using expression (22) we may calculate the value of V ∈ SU(2)/U(1) at any point of the momentum space lattice. Next, we may define the U(1) gauge field B at any link of this momentum lattice via the following equation: The position of the monopole is given by j = 1 2π * d[dB mod 2π]. We expect that the pattern described above with 4 monopole -antimonopole pairs in momentum space will remain until a phase transition is encountered.

CONCLUSIONS AND DISCUSSION
In this paper we extend the construction of the topological invariant N 2 to ω − p space.
The suggested construction works for the case when the Coulomb interaction between the quasiparticles is present. We also construct the gauge field in momentum space that has vanishing field strength everywhere except for the poles and zeros of the Green function (and the strings that connect them). The positions of poles and zeros themselves coincide with the positions of monopoles extracted from the given gauge field.
The 8 × 8 Green functions of the fermion quasiparticles are reduced to 2 × 2 matrices, and even further, to the elements of su (2). These matrices can be represented as iGσ 3 = g 3 σ 3 + g a σ a with real g 3 , g a . The constructed topological invariant catches zeros and poles of G. If interactions are absent, N 2 = 1. When the Coulomb interaction is turned on, the equation N 2 = 1 holds until the phase transition is encountered. This means that the pole in G cannot disappear until the phase transition occurs.
The system may be transferred to various phases, where different symmetries of the initial system are broken. There may appear different fermion condensates [9-11, 13, 15]. The phase structure of the effective field model of graphene is still unknown. Topology of momentum space must have the relation to this phase structure. The constructed invariant N 2 and the monopoles in momentum space have direct connection only to the phase that includes the noninteracting system. The transition to the other phas(es) may be accompanied with the change of N 2 . The transition between the new phases may have connection to the other topological invariants. In particular, the topological invariants for the 2 + 1 gapped systems enter the expression for the quantized Hall conductivity [4,7,8,14]. Also it is worth mentioning that in the presence of the finite chemical potential the Fermi surface appears that is related to the invariant N 1 [4].
The construction presented here is intended for the use mainly in lattice simulation of the effective field theory of graphene at vanishing chemical potential and in the absence of external fields (for the review of recent numerical investigations of the model see [10,11,16] and references therein). We expect that the phase transition(s) may take place to the phase(s), where chiral symmetry of the noninteracting system is broken [15] in a certain way.
The transition to the new phase must lead to the change of the momentum space topology.
The behavior of the monopoles in momentum space constructed here are expected to be related intimately to the mechanism of the phase transition(s) and to the nature of the new phase(s). Their investigation may also be important for the understanding of the role of doublers in various lattice discretizations of the Fermion systems.