  A superconducting wire described by a p-wave pairing and a Kitaev Hamiltonian
exhibits Majorana fermions at its edges and is topologically protected by
symmetry. We consider two Kitaev wires (chains) coupled by a Coulomb type
interaction and study the complete phase diagram using analytical and numerical
techniques. A topological superconducting phase with four Majorana fermions
occurs until moderate interactions between chains. For large interactions, both
repulsive and attractive, by analogy with the Hubbard model, we identify Mott
phases with Ising type magnetic order. For repulsive interactions, the Ising
antiferromagnetic order favors the occurrence of orbital currents spontaneously
breaking time-reversal symmetry. By strongly varying the chemical potentials of
the two chains, quantum phase transitions towards fully polarized (empty or
full) fermionic chains occur. In the Kitaev model, the quantum critical point
separating the topological superconducting phase and the polarized phase
belongs to the universality class of the critical Ising model in two
dimensions. When increasing the Coulomb interaction between chains, then we
identify an additional phase corresponding to two critical Ising theories (or
two chains of Majorana fermions). We confirm the existence of such a phase from
exact mappings and from the concept of bipartite fluctuations. We show the
existence of negative logarithmic corrections in the bipartite fluctuations, as
a reminiscence of the quantum critical point in the Kitaev model. Other
entanglement probes such as bipartite entropy and entanglement spectrum are
also used to characterize the phase diagram. The limit of large interactions
can be reached in an equivalent setup of ultra-cold atoms and Josephson
junctions.
