ABSTRACT - DRUM - University of Maryland

the bulk superconducting gap and the plasma frequency, the only degrees of freedom
of this system are the superconducting phase φ and the midgap fermions. We also
assume that the phase varies slowly so the fermionic part of the system follows the
BCS mean-field Hamiltonian with superconducting phase φ.
We want to know the geometric phase associated with vortex tunneling in
the presence of midgap fermions. It can be derived by calculating the transition
amplitude A fi associated with a time-depedent phase φ = φ(t):
A fi = 〈φ f | ˆQ f Û(t f , t i ) ˆQ † i |φ i〉, (5.21)
where |φ〉 denotes the BCS ground state with superconducting phase φ and φ f −φ i =
2wπ. ˆQ† = ∏ m ( ˆd † m) nm denote the occupation of the midgap fermionic states with
n m = 0, 1.
The midgap fermionic operators ˆd † m are explicity expressed in terms of Bogoliubov
wavefunctions u m and v m :
ˆd † m(t) = e −iεmt ∫
dr [ u m (r) ˆψ † (r)e iφ/2 + v m (r) ˆψ(r)e −iφ/2] . (5.22)
Therefore,
Û(t f , t i ) ˆd † m(t i )Û † (t f , t i ) = ˆd † m(t f )e iπwnm . (5.23)
So the transition amplitude is evaluated as
A fi = 〈φ f | ˆQ f Û(t f , t i ) ˆQ † iÛ† (t f , t i )Û(t f, t i )|φ i 〉
= e iπwn e −i ∑ m nmεm(t f −t i ) 〈φ f | ˆQ f ˆQ† fÛ(t f, t i )|φ i 〉
(5.24)
= e iπwn e −i ∑ m nmεm(t f −t i ) 〈φ f |Û(t f, t i )|φ i 〉
90