ABSTRACT - DRUM - University of Maryland

Majorana zero modes can survive under strong quantum fluctuations, especially in
one dimension. We first considered a continuum field theory of spinless fermions
on a two-leg ladder with pair tunneling, in the presence of quasi-long-range superconducting
order. Using bosonization technique we analyze non-perturbatively the
strong-coupling phase and found interesting degeneracies of low-energy states that
can be interpreted as Majorana zero-energy edge states. We discussed the stability
of these degeneracies under various perturbations.
Then we proposed a possible
lattice realization of this field theory.
We now discuss possible future research directions. It is interesting to explore
the possible vortex lattice phase in a topological superconductor where low-energy
physics can be described by Majorana fermions hopping on the lattice with hopping
amplitudes determined by the energy splitting calculated in Chapter 3. More work
needs to be done to fully understand the robustness of topological qubits, including
the effect of disorder and possible low-energy impurity bound states, and how they
affect the braiding and the read out schemes. The effect of quantum fluctuations on
Majorana zero modes in higher dimensions remains a open problem although the
one-dimensional case has been rather well understood. It would be very interesting
to generalize the bosonization approach to higher dimensions.
144