content_image/413042.jpg content_image/413021.jpg content_image/413026.jpg content_image/413027.jpg content_image/413038.jpg content_image/413018.jpg content_image/413019.jpg content_image/413020.jpg content_image/413039.jpg content_image/413022.jpg content_image/413023.jpg content_image/413041.jpg
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Refs. 12,16). 

A hole-like excitation Ψ ( h ) K in the valley associated with the point K is the complex conjugated wave function of a particle-like excitation in the valley − K , i.e., Ψ ( h ) K = Ψ ∗ − $_{K}$. In what follows we denote particle-like states by u while hole-like states by v . The Bogoliubov–de Gennes equations have the form 7,12 

<img src='content_image/413021.jpg'>

The two-component wave functions are in a form of pseudo-spinors 

<img src='content_image/413026.jpg'>

where the two components are the wave functions of elec- trons and holes on two sublattices 1 and 2 in the honey- comb lattice, Fig. 1(a); ˆ σ = (ˆ σ$_{x}$ , ˆ σ$_{y}$ ) are Pauli matrices in the pseudo-spin space: 

Equations for the valley at the point K ′ can be obtained with the replacement u$_{1}$ → u$_{2}$ , v$_{1}$ → v$_{2}$ . Pairing of a particle u in the valley K in the Brillouin 

zone occurs with a hole v at K , i.e., with a particle in the valley − K . Since the points − K and K ′ are equivalent, K + K ′ = b$_{1}$ + b$_{2}$ where b$_{1}$ and b$_{2}$ are the vectors of the reciprocal lattice, Fig. 1(b), one may also say that pairing is between particles from the valleys K and K $^{′}$. The model assumes that the order parameter is the same for both sublattices, 

<img src='content_image/413027.jpg'>

<img src='content_image/413038.jpg'>

where α labels four independent solutions of the Bogoliubov–de Gennes equations with the momentum p (see below) and f$_{p}$$_{,α}$ is the Fermi occupation number in the state p , α . The sum runs over all states within the Brillouin zone. We do not concentrate here on the specific nature of the pairing interaction assuming that the pairing potential may be either due to some intrinsic mechanism or due to an interaction induced by a prox- imity to a usual superconductor. 

The particle density is 

<img src='content_image/413018.jpg'>

Factor 2 accounts for the true spin of electrons. The statistical average of the current operator is 

<img src='content_image/413019.jpg'>

Sometimes the currents j$_{e}$ and j$_{p}$ are defined, 

<img src='content_image/413020.jpg'>

such that j = j$_{e}$ + j$_{p}$ . The current j$_{p}$ is the quasiparticle flux, which vanishes in our spatially uniform case (see below). The current j$_{e}$ is the sum of currents in each state, which may be not conserved separately in some spatially inhomogeneous or non-equilibrium situations, but the total current, however, is conserved (div j$_{e}$ = 0) taking into account the self-consistency equation $^{17}$. 

We will consider the case of zero magnetic field and look for the solution in the form of plane waves 

assuming that the order parameter Δ = | Δ | e i k · r corre- sponds to a moving condensate of Cooper pairs. Equa- tions (1) and (2) give 

Let us consider the ground state with zero current ( k = 0). Equations (8), (9) define four linearly independent solutions. Let us introduce the spinors 

<img src='content_image/413039.jpg'>

which satisfy 

<img src='content_image/413022.jpg'>

<img src='content_image/413023.jpg'>

## A. Ground state 

<img src='content_image/413041.jpg'>

