Current state of the art devices for detecting and manipulating Majorana fermions commonly consist of networks of Majorana wires and tunnel junctions. We study a key ingredient of these networks -a topological Josephson junction with charging energy -and pinpoint crucial features for device implementation. The phase dependent tunneling term contains both the usual 2π-periodic Josephson term and a 4π-periodic Majorana tunneling term representing the coupling between Majoranas on both sides of the junction. In non-topological junctions when the charging energy is small compared to the Josephson tunneling scale the low energy physics is described by 2π phase slips. By contrast, in a topological junction, due to the 4π periodicity of the tunneling term it is usually expected that only 4π phase slips are possible while 2π phase slips are suppressed. However, we find that if the ratio between the strengths of the Majorana assisted tunneling and the Josephson tunneling is small, as is likely to be the case for many setups, 2π phase slips occur and may even dominate the low energy physics. In this limit one can view the 4π phase slips as a pair of 2π phase slips with arbitrarily large separation. We provide an effective descriptions of the system in terms of 2π and 4π phase slips valid for all values of the tunneling ratio. Comparing the spectrum of the effective models with numerical simulations we determine the cross-over between the 4π phase slip regime to 2π phase slip dominated regime. We also discuss the role of the charging energy as well as the implications of our results on the dissipative phase transitions expected in such a system. arXiv:1805.10601v1 [cond-mat.supr-con]

The properties of Josephson tunneling between a single band s-wave superconductor and a two band s± superconductor are studied, in relation to recent experiments involving iron-based superconductors. We study both a single junction and a loop consisting of two junctions. In both cases, the relative phase between the order parameters of the two superconductors is tuned and the energy of the system is calculated. In a single junction, we find four types of behaviors characterized by the location of minima in the energy/phase relations. These phases include a newly found double minimum junction which appears only when the order parameters are treated self-consistently. We analyze the loop geometry setup in light of our results for a single junction, where the phase difference in the junctions is controlled by a threaded flux. We find four types of energy/flux relations. These include states for which the energy is minimized when the threaded flux is an integer or half integer number of flux quanta, a time reversal broken state and a meta-stable state.

Unequivocal signatures of Majorana zero energy modes in condensed matter systems and manipulation of the associated electron parity states are highly sought after for fundamental reasons as well as for the prospect of topological quantum computing. In this paper, we demonstrate that a ring of Josephson coupled topological superconducting islands threaded by magnetic flux and attached to a quantum dot acts as an excellent parity-controlled probe of Majorana mode physics. As a function of flux threading through the ring, standard Josephson coupling yields a Φ 0 = h/(2e) periodic features corresponding to 2π phase difference periodicity. In contrast, Majorana mode assisted tunneling provides additional features with 2Φ 0 (4π phase difference) periodicity, associated with single electron processes. We find that increasing the number of islands in the ring enhances the visibility of the desired 4π periodic components in the groundstate energy. Moreover as a unique characterization tool, tuning the occupation energy of the quantum dot allows controlled groundstate parity changes in the ring, enabling a toggling between Φ 0 and 2Φ 0 periodicity.

We study the molecular dynamics of two discs undergoing Newtonian ("inertial") dynamics, with elastic collisions in a rectangular box. Using a mapping to a billiard model and a key result from ergodic theory, we obtain exact, analytical expressions for the mean times between the following events: hops, i.e. horizontal or vertical interchanges of the particles; wall collisions; and disc collisions. To do so, we calculate volumes and cross-sectional areas in the four-dimensional configuration space. We compare the analytical results against Monte Carlo and molecular dynamics simulations, with excellent agreement.