We discuss the emergence of zero-energy Majorana modes in a disordered finite-length p-wave one-dimensional superconducting ring, pierced by a magnetic flux Φ tuned at an appropriate value Φ = Φ * . In the absence of fermion parity conservation, we evidence the emergence of the Majorana modes by looking at the discontinuities in the persistent current I[Φ] at Φ = Φ * . By monitoring the discontinuities in I[Φ], we map out the region in parameter space characterized by the emergence of Majorana modes in the disordered ring.PACS numbers: 73.23. Ra, 74.81.g, 74.45.+c, I. INTRODUCTIONMajorana fermions, particles coinciding with their own antiparticles, were proposed by Majorana in 1937 1 . While, so far, they have never been detected in a particle physics experiment in the last years, after Kitaev's proposal that Majorana fermions may appear as zero-energy real fermionic modes ["Majorana modes" -MMs] localized at the interface between a p-wave one-dimensional superconductor and a normal metal 2 , the search for Majorana fermions in such systems has become one of the most relevant and promising areas in condensed matter physics 3 .Besides Kitaev's proposal, the emergence of MMs in condensed matter systems has been predicted in a number of systems such as superconductor-topological insulator interfaces 4-7 , in proximity-induced superconducting quantum wires with strong spin-orbit interaction 8-11 , in helical magnets 12 , in ferromagnetic atoms in proximity to superconductors 13,14 . In this context, interesting phases with unconventional properties have been predicted at junctions between topological superconductors, hosting MMs at their endpoints, and interacting one-dimensional electronic systems (Luttinger liquids) 15-17 . In addition, MMs emerging at junctions of one-dimensional interacting quantum wires 18-21 , or of systems formally described as interacting electronic systems, such as quantum Ising spin chains 22-25 , one-dimensional XX-26 , or XY-27 models, or pertinently designed Josephson junction networks 28 , have been predicted to give rise to the so-called "Topological Kondo Effect", a remarkable realization of Kondo Effect in which the impurity spin, determined by the MMs, is nonlocal in the wire indices and, thus, cannot be expressed as a functional of local operators 18,19 . Finally, it is worth mentioning that, besides being of remarkable interest for fundamental physics, MMs are also of great interest for quantum computation since, due to their nonabelian statistics 29 , they appear to be among the most natural candidates to work as robust qubits 30 .The proliferation of theoretical literature about Majorana fermions in condensed matter systems has triggered a number of experimental attempts to probe MMs in pertinently designed devices. The main route followed in the experiments consists in measuring the effects in the transport across junctions between topological superconductors and normal metals possibly due to the presence of localized MMs at the interfaces 31-33 . Unfortunately, despite the ex...

Using the properties of the transfer matrix of one-dimensional quantum mechanical systems, we derive an exact formula for the persistent current across a quantum mechanical ring pierced by a magnetic flux Φ as a single integral of a known function of the system's parameters. Our approach provides exact results at zero temperature, which can be readily extended to a finite temperature T . We apply our technique to exactly compute the persistent current through p-wave and s-wave superconducting-normal hybrid rings, deriving full plots of the current as a function of the applied flux at various system's scales. Doing so, we recover at once a number of effects such as the crossover in the current periodicity on increasing the size of the ring and the signature of the topological phase transition in the p-wave case. In the limit of a large ring size, resorting to a systematic expansion in inverse powers of the ring length, we derive exact analytic closed-form formulas, applicable to a number of cases of physical interest.

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