Exact zero modes in interacting Majorana X- and Y-junctions
We report the emergence of exact zero modes in junctions of two, three and four short interacting Majorana wires, equivalent to a chain with an impurity bond, Y- and X- junctions respectively. These exact zero modes arise from incommensurate short-range correlations induced by interacting Majorana fermions and manifest as exact level crossings between in-gap states upon continuously tuning the interaction strength. In a junction of only two chains we report exact zero modes and parity switching as soon as the coupling between the chains across a junction is positive. Remarkably, for junctions with multiple chains the in-gap states group up into sets of parity pairs -- pairs of states with opposite parity and similar energies. We demonstrate that the formation of these parity pairs are always due to the effective interaction of the outer edges of the junction. The behavior within each pair can be efficiently described by two coupled chains. In the Y-junction, we detect four in-gap states (two parity pairs) that show exact zero modes not only within each pair but also between them. This is attributed to an additional Majorana fermion localized at the center of junction that is protected by symmetry. Therefore, coupling between the Majorana fermions at the outer edges of the junction is mediated by that in the center. We argue that this is a generic feature of junctions with an odd number of arms. In the X-junction we detect eight in-gap states (four parity pairs) that are the result of two Majorana degrees of freedom localized at the center of the junction. However, we demonstrate that, by contrast to the Y-junction, the appearance of Majorana fermions at the center of the X-junction is not protected and the interaction across the junction can be tuned to the point where there are only Majorana fermions localized at the four outer edges of the junction, forming four in-gap states.