ABSTRACT - DRUM - University of Maryland

where n = (R∆, −I∆) field describes the non-trivial configuration of the superconducting
order parameter. We assume the following boundary condition for n
field:
|n(r)| → const as |r| → ∞. (3.35)
As mentioned above, this model Hamiltonian has particle-hole symmetry, timereversal
symmetry and chiral symmetry which is given by γ 5 . It anticommutes with
the Dirac Hamiltonian {γ 5 , H D } = 0. Therefore, all zero modes Ψ 0 of H D are
eigenstates of γ 5 . Since (γ 5 ) 2 = 1 eigenvalues of γ 5 are ±1. We define ± chirality of
zero modes as γ 5 Ψ ± 0
= ±Ψ ± 0 . The analytical index of H D is defined as
ind H D = n + − n − , (3.36)
where n ± are number of zero modes with ± chirality.
The index theorem for the Hamiltonian H D states that the analytical index is
identical to the winding number of the background scalar field in the two-dimensional
space [84]:
ind H D = − 1 ∫
2π
d i x ɛ abˆn a ∂ iˆn b , (3.37)
where ˆn = n/|n|. According to the index theorem, the number of zero modes is
determined by the topology of order parameter at infinity. The right hand side is
ensured to be an integer by the fact that the homotopy group π 1 (S 1 ) = Z. If we
have a vortex in the system with vorticity l, the right hand side of (3.37) evaluates
exactly to l. Thus the index theorem implies that the Dirac Hamiltonian has at least
l zero modes which agrees with explicit solution obtained by Jackiw and Rossi [82].
Our explicit solutions for a single vortex in the previous section also agrees perfectly.
59