imagine that this flux does not leak into the rest of the superconductor and that the flux is inserted in such a way that the low energy configuration where the phase of the superconductor winds by 2 π around the flux tube is attained. 

In the presence of the flux, the single particle Hamil- tonian gets transformed as follows. Let r$_{i}$, θ$_{i}$ be re- spectively the distance from the origin and the polar angle of site i in a coordinate system with the ori- gin located at the position of the flux tube. Then, h$_{ij}$ → h ′ ij = h$_{ij}$e i ( e/$_{ℏ}$ ) ∫ dr . A = h$_{ij}$e i ( θ$_{i}$ − θ$_{j}$ ) / $^{2}$, where A is chosen to be ℏ 2 $_{e}$∇ ( θ ). The gap matrix transforms as: Δ$_{ij}$ → Δ ′ ij = Δ e i ( θ$_{i}$ + θ$_{j}$ ) / 2 in the presence of the flux tube. 

The BdG eigenvalue equation in the presence of the vortex is thus: H $^{′}$ψ = Eψ , where ψ = ( u, v ) T and H ′ has the form of Eq. (6) with M$_{11}$ = h $^{′}$, M$_{12}$ = Δ ′ and M$_{22}$ = − ( h $^{′}$) $^{T}$. Let ˜ u$_{i}$ = u$_{i}$, ˜ v$_{i}$ = e $^{iθ$_{i}$}$v$_{i}$ Then ˜ ψ = (˜ u, ˜ v ) T satisfies the eigenvalue equation : H ′′ ˜ ψ = E ˜ ψ , where H ′′ can be written in the form of Eq. (6) with M$_{11}$ = h $^{′}$, M$_{12}$ = Δ $^{′′}$, M$_{22}$ = − ( h $^{′′}$) , h ′′ ij = h$_{ji}$e i ( θ$_{i}$ − θ$_{j}$ ) / 2 and Δ ′′ ij = Δ$_{ij}$ e i ( θ$_{i}$ − θ$_{j}$ ) / $^{2}$. 

We now replace the superconductor with the associ- ated insulator, H$_{I}$ with the same flux configuration, i.e., half a quantum of flux inserted at the origin, which lies at the center of a plaquette. It is easy to verify that H$_{I}$ ( π ) can be written as ∑ ij Ψ † $_{i}$H ′′ $_{ij}$Ψ$_{j}$ with H ′′ as given above. Further, H$_{I}$ satisfies the symmetry in Eq. (1) since it is derived from a BdG Hamiltonian and the analysis for the zero modes for insulators made previously can therefore be used. 

The necessary and sufficient condition for the existence of an exact zero-energy mode which is localized around the flux tube is therefore that the Hall conductance of H$_{I}$ is pe $^{2}$/ 2 π$_{ℏ}$ where p is odd. If this condition is met, it follows that there is a zero energy mode localized around the plaquette containing the tube. Since the Hall con- ductance is a robust topological invariant, the existence of the zero mode for the superconductor is also topolog- ically protected. 

A more realistic description of a vortex would include a finite region larger than a single plaquette where the magnetic field is non-zero rather than the situation con- sidered above where the flux is confined to a single pla- quette. In the general case, the gap parameter and the single particle Hamiltonians may be modeled as: Δ ′ ij = Δ$_{ij}$ f (( r$_{i}$ + r$_{j}$ ) / 2) e i ( θ$_{i}$ + θ$_{j}$ ) / $^{2}$; h ′ ij = h$_{ij}$e $^{iφ$_{ij}$}$. Here, f ( r ) is some function which is zero at the origin and goes to one for distances larger than the coherence length and φ$_{ij}$ = $^{∫}$( e/$_{ℏ}$ ) dr.A where A is the vector potential which corresponds to a smeared out flux and which approaches the value ($_{ℏ}$ /e ) ∇ ( θ ) / 2 for distances much larger than the penetration depth. 

The corresponding BdG Hamiltonian may therefore be written as H = H$_{0}$ + V where H$_{0}$ is the Hamiltonian which corresponds to the idealized vortex discussed above and V is a local perturbation in the sense that V$_{ij}$ = 0 

sufficiently far from the vortex. If there is an odd number of Majorana zero energy states, then the Hilbert space of the idealized vortex has one unpaired fermionic mode, while every other eigenstate of energy, E , can be paired with a state of energy, − E . Any local perturbation can- not alter the Hilbert space structure. It follows that the perturbation, V , does not alter the existence of a zero en- ergy mode and that the zero mode persists in the more realistic configuration. 

We now intend to study superconductors with time re- versal symmetry using the same technique and therefore first study the corresponding insulators. Insulators with time reversal symmetry are classified by a Z$_{2}$ invariant. Any Hamiltonian, H , of this class can be continuously deformed, without closing the gap or breaking time re- versal symmetry, to a Hamiltonian H ′ which is diagonal in the basis of the z-component of spin and can thus be written in the form H ′ = H ′ ↑ + H ′ $_{↓}$. The Hamiltonians H ′ ↑ and H ′ ↓ can be analyzed separately for studying their zero modes. The sum of the Hall conductance of H ′ ↑ and H ′ ↓ must be zero. If | σ$_{xy}$ 2 π$_{ℏ}$/e $^{2}$| of H ′ ↑ is an odd number, then the insulator belongs to the non-trivial class. By the previous analysis, in this case, when a flux of π is inserted through the central plaquette the system has an odd number of Kramers pairs at zero energy, one mem- ber of each pair associated with H ′ ↑ and the other with H ′ $_{↓}$. Kramers theorem then prevents a gap from opening up at π flux when the Hamiltonian is continuously de- formed without breaking time reversal symmetry. This implies that the original Hamiltonian, H , must also have an odd number of pairs of zero modes at π flux. When σ$_{xy}$ ( H ′ $_{↑}$) is an even multiple of e $^{2}$/ 2 π$_{ℏ}$ on the other hand, the number of pairs of zero modes must be even and is therefore not protected. 

Superconductors with time reversal symmetry fall in the class DIII of the classification scheme [19]. These superconductors are classified by a Z$_{2}$ invariant [21] and can be mapped onto insulators with time reversal symme- try which satisfy Eq.(1) and which have the same spec- trum, exactly as in the case of superconductors with- out time reversal or spin rotational symmetry. We have shown in the previous paragraph that when flux is in- serted through a plaquette in these associated insulators, they have an odd number of Kramers pairs of zero en- ergy modes if and only if they belong to the non-trivial topological class. The zero eigenmodes of the equation that determines the spectra of the insulator with half a quantum of flux inserted through a plaquette may be re- lated to the zero eigenmodes of the superconductor with a vortex following the discussion after Eq. (6). Thus, superconductors with time reversal symmetry have an odd number of Kramers pairs of zero energy modes in their vortex cores whenever the superconductors are in the non-trivial topological class. A pair of zero-energy modes may be regarded as a single Dirac mode. 

Hamiltonians of superconductors with spin-rotational symmetry but not time reversal symmetry fall in the class C of the Altland-Zirnbauer symmetry classes 

