ABSTRACT - DRUM - University of Maryland

etween these Majorana fermions, which is equivalent to the process of tunneling of a
Majorana quasiparticle from one place to another, split the topological degeneracy.
The energy splitting determines the fusion channel of two non-Abelian vortices.
However, such tunneling process has to overcome the bulk gap, very similar to
tunneling of a quantum-mechanical particle through a potential barrier that is higher
than its kinetic energy. So the splitting is exponentially suppressed in the topological
phase as e −R/ξ where R is the separation between anyons and ξ is the correlation
length (the coherence length in a superconductor).
Another possibility is thermal excitations of non-Majorana fermionic modes.
The process (1.33) has non-vanishing probability to occur at finite temperature and
as such, represents a thermal decoherence of Majorana qubits. This issue is particularly
pronounced when there are low-energy (but not zero) bound states present
together with the Majorana zero-energy states which is the case in superconducting
vortices.
At a more fundamental level, the BCS theory which all our discussions are
based on, is a mean-field theory neglecting all quantum and thermal superconducting
fluctuations. In three-dimensional electronic superconductors the fluctuations
are gapped due to the famous Anderson-Higgs mechanism [69] and the relevant
energy scale is the plasmon frequency. However, since non-Abelian topological superconductors
all exist in dimensions smaller than three, the fluctuation effect needs
to be reconsidered. For example, in quasi-two-dimensional superconductors the London
penetration length is inversely proportional to the thickness of the system. In
the limit of vanishing thickness, the superconducting fluctuations are essentially
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