Topological Majorana and Dirac zero modes in superconducting vortex cores Rahul Roy Rudolph Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, UK (Dated: 13 January 2010) We provide an argument based on flux insertion to show that certain superconductors with a non-trivial topological invariant have protected zero modes in their vortex cores. This argument has the flavor of a two dimensional index theorem and applies to disordered systems as well. It also provides a new way of understanding the zero modes in the vortex cores of a spinless p$_{x}$ + ip$_{y}$ superconductor. Applying this approach to superconductors with and without time reversal and spin rotational symmetry, we predict the necessary and sufficient conditions for protected zero modes to exist in their vortices. states. Using a flux insertion argument, we first show that a certain class of insulators with π flux inserted through a plaquette have exact zero energy eigenstates which are topologically protected in the sense that their existence is connected to the Hall conductance of the insulator. We then use these results to study superconductors with and without time reversal and spin rotational symmetry and predict precisely which systems have protected zero modes. Fermionic zero energy modes are predicted to exist in defects in a variety of condensed matter systems from one dimensional (1d) chains of polyacetylene to the vortices of certain two dimensional (2d) superconductors [1, 2]. These modes typically occur in systems with a bulk gap which are in a topologically non-trivial phase. The zero energy modes are often topologically protected in the sense that their existence is only tied to the particular topological phase of the system and to some global prop- erties of the system such as the winding number of the phase of the order parameter in a superconductor. Unpaired fermionic zero energy modes which are lo- calized in the vortices of certain superconductors such as a spinless p$_{x}$ + ip$_{y}$ superconductor are of special in- terest. These modes, also called Majorana modes, have attracted a fair bit of recent theoretical attention due to the fact that they can obey non-abelian statistics and have potential applications in decoherence free quantum computation [3–6]. The existence of these modes has tra- ditionally been demonstrated through explicit analytic solutions of the Bogoliubov de-Gennes (BdG) equations [1, 4, 7–9]. In 1d, Jackiw and Rebbi showed that the Dirac equa- tion interacting with a scalar field with a topologically interesting configuration such as a soliton has zero energy solutions whose number was related to an integer which characterizes the non-trivial topology [10]. Similar zero energy states also occur in polyacetylene [2]. Recently many works have attempted to explain the Majorana modes in 2d by reducing the problem to a 1d problem [11–13]. Since the vortex zero energy states occur in 2d sys- tems, a more general and inherently 2d explanation of the Majorana zero modes is desirable. In the current work, this is achieved through a flux insertion argument similar to Laughlin’s gauge argument [14]. The argu- ment is not tied to any particular model and also applies to disordered systems where translational invariance is broken and is particularly useful in the context of recent efforts to find Majorana modes in systems other than a p$_{x}$ + ip$_{y}$ superconductor [15–18]. One of our main re- sults is a prediction about precisely what systems in all the different symmetry classes of the classification intro- duced by Altland and Zirnbauer[19] support zero energy Further, we assume that S acting on single particle po- sition eigenkets in the Hilbert space produces a linear In the first instance, we study systems with neither spin rotational nor time reversal symmetry, though we will analyze systems with these symmetries later on. We further restrict our study to Hamiltonians H which posses a symmetry analogous to that of Bogoliubov de-Gennes Hamiltonians. In other words, we assume the existence of an anti-unitary operator, S , such that The Hamiltonian can be written in the form H = ∑ i,j Ψ † $_{i}$H$_{ij}$ Ψ$_{j}$ where H$_{ij}$ is a 2 s × 2 s dimensional ma- trix and Ψ$_{j}$ = ( ψ$_{j,α}$ ) $^{T}$, Ψ † i = ( ψ † $_{j,β}$) where i, j are the lattice position indices and α, β the orbital and spin in- dices. The Fermi energy is set to zero and is assumed to lie in a gap. The insulator Hamiltonians studied below have the same symmetries as the mean field Bogoliubov-de- Gennes Hamiltonians of superconductors. Though such insulators are possibly of little physical significance by themselves, we study them since their spectra are identi- cal to that of the BdG Hamiltonians of superconductors. Further since they are insulators rather than supercon- ductors, they do not expel flux, making a flux insertion argument possible. Consider a 2d tightbinding insulator on a lattice with an even number of orbitals, 2 s per site. The basis states are written in the form | i, α 〉 where i is an index for the position and α for the orbital. We first study infinite sized systems where the energy of the topo- logically protected vortex states is exactly zero and later consider corrections that would occur in a finite sized system. S H S − 1 = −H . (1) arXiv:1001.2571v1 [cond-mat.supr-con] 14 Jan 2010