Title: Supersymmetry in Closed Chains of Coupled Majorana Modes,
Abstract: 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.