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A one-dimensional Ising model in a transverse field can be mapped onto asystem of spinless fermions with p-wave superconductivity. In the weak-couplingBCS regime, it exhibits a zero energy Majorana mode at each end of the chain.Here, we consider a variation of the model, which represents a superconductorwith longer ranged kinetic energy and pairing amplitudes, as is likely to occurin more realistic systems. It possesses a richer zero temperature phase diagramand has several quantum phase transitions. From an exact solution of the modelthese phases can be classified according to the number of Majorana zero modesof an open chain: 0, 1, or 2 at each end. The model posseses a multicriticalpoint where phases with 0, 1, and 2 Majorana end modes meet. The number ofMajorana modes at each end of the chain is identical to the topological windingnumber of the Anderson's pseudospin vector that describes the BCS Hamiltonian.The topological classification of the phases requires a unitary time-reversalsymmetry to be present. When this symmetry is broken, only the number ofMajorana end modes modulo 2 can be used to distinguish two phases. In one ofthe regimes, the wave functions of the two phase shifted Majorana zero modesdecays exponentially in space but but in an oscillatory manner. The wavelengthof oscillation is identical to the asymptotic connected spin-spin correlationof the XY-model in a transverse field to which our model is dual.

Majorana zero modes in a quantum Ising chain with longer-ranged interactions