Supersymmetry in Closed Chains of Coupled Majorana Modes

We consider a closed chain of even number of Majorana zero modes with nearest-neighbour couplings which are different site by site generically, thus no any crystal symmetry. Instead, we demonstrate the possibility of an emergent supersymmetry (SUSY), which is accompanied by gapless Fermionic excitations. In particular, the condition can be easily satisfied by tuning only one coupling, regardless of how many other couplings are there. Such a system can be realized by four Majorana modes on two parallel Majorana nanowires with their ends connected by Josephson junctions and bodies connected by an external superconducting ring. By tuning the Josephson couplings with a magnetic flux $\Phi$ through the ring, we get the gapless excitations at $\Phi_{SUSY}=\pm f\Phi_0$ with $\Phi_0= hc/2e$, which is signaled by a zero-bias conductance peak in tunneling conductance. We find this $f$ generally a fractional number and oscillating with increasing Zeeman fields that parallel to the nanowires, which provide a unique experimental signature for the existence of Majorana modes.