and each of them are doubly degenerate (signs are not correlated). Note that the spectrum is fully gapped as long as Δ$_{0}$ = 0. The computation of its topological charge is lengthy but straightforward. 3 In order to eluci- for arbitrary m , which means that the system is topolog- ically trivial. In the following subsection, we will see how these observations in the free space are reflected in the spectrum of fermions localized around a vortex line. When the fermion mass is nonzero m = 0, the right- handed and left-handed sectors are coupled and thus the only total topological charge N is well defined. Because the spectrum is fully gapped for Δ$_{0}$ = 0, the inclusion of the fermion mass can not change the topological charge. Therefore we find When the pairing gap is a constant, we can choose it to be real; Δ( x ) = Δ $^{∗}$( x ) = Δ$_{0}$. From Eq. (3), the free space Hamiltonian in the momentum space is given by Its energy eigenvalues have the usual form H p = ( α · p + βm − µ Δ$_{0}$ Δ$_{0}$ − α · p − βm + µ ) . (12) N = 0 (17) 5 The half-integer value of N is common to relativistic fermions because q$_{p}$ is noncompact at | p | → ∞ . See, e.g., Ref. [20]. 4 The topological charge for the full Hamiltonian (1) can be ob- tained by simply summing contributions from all gapped sectors. 3 Note that the topological charge is invariant as long as the spec- trum is fully gapped. Therefore one can set µ → 0 to reduce the computational complication. 2. Nonzero fermion mass m = 0 A. Topological charge of a free space Hamiltonian III. EVEN PARITY PAIRING We find that each sector is topologically nontrivial having the nonzero topological charge. 5 This implies the exis- tence of a localized zero energy state for each sector in the presence of a vortex. However, their signs are oppo- site and the total topological charge of the Hamiltonian (12) is vanishing; N = N$_{R}$ + N$_{L}$ = 0. We first consider the case where the pairing takes place in the even parity channel, which is relevant to the color superconductivity of quarks [29, 30]. and and N$_{L}$ ≡ 1 24 π 2 ∫ d p ϵ ijk Tr [ ( q − 1 L $_{p}$∂$_{i}$ q$_{L}$$_{p}$ )( q − 1 L $_{p}$∂$_{j}$ q$_{L}$$_{p}$ )( q − 1 L $_{p}$∂$_{k}$ q$_{L}$$_{p}$ ) ] = − Δ$_{0}$ 2 | Δ$_{0}$ | . (16) γ 5 = ( 1 0 0 − 1 ) . (11) α = γ $^{0}$γ = ( σ 0 0 − σ ) , β = γ 0 = ( 0 1 1 0 ) , (10) N$_{R}$ ≡ 1 24 π 2 ∫ d p ϵ ijk Tr [ ( q − 1 R $_{p}$∂$_{i}$ q$_{R}$$_{p}$ )( q − 1 R $_{p}$∂$_{j}$ q$_{R}$$_{p}$ )( q − 1 R $_{p}$∂$_{k}$ q$_{R}$$_{p}$ ) ] = Δ$_{0}$ 2 | Δ$_{0}$ | (15) q$_{p}$ = ( q$_{R}$$_{p}$ 0 0 q$_{L}$$_{p}$ ) . (14) N ≡ 1 24 π 2 ∫ d p ϵ ijk Tr [ ( q − 1 p ∂$_{i}$q$_{p}$ )( q − 1 p ∂$_{j}$q$_{p}$ )( q − 1 p ∂$_{k}$q$_{p}$ ) ] . (9) When a given Hamiltonian has a nonzero topological charge, such a system is said to be topological . Because Hamiltonians having different topological charges can not be continuously deformed into each other without clos- ing energy gaps in their spectrum, the topological charge defined in Eq. (9) classifies 3D Hamiltonians belonging to the symmetry class DIII [20]. It is known in the case of the p$_{x}$ + ip$_{y}$ superconductor in 2D that a nontrivial topological charge of the free space Hamiltonian has a close connection to the existence of a localized zero energy state in the presence of a vortex [16, 28]. We will see in the subsequent sections that the same correspondence is true in our Hamiltonian (3) describing superconducting Dirac fermions in 3D. We shall work in the chiral representation: Accordingly, we can define the topological charges for the right-handed sector and for the left-handed sector independently. The results are 4 In the chiral limit m = 0, because the right-handed sector and the left-handed sector of the Hamiltonian (12) are decoupled, the unitary matrix q$_{p}$ in Eq. (8) has the block diagonal form: date the effect of the fermion mass, we shall present the results for the m = 0 case and the m = 0 case separately. the winding number of q$_{p}$ associated with the homotopy group π$_{3}$ [ U ( n ≥ 2)] = Z : 3 1. Chiral limit m = 0