ABSTRACT - DRUM - University of Maryland

Generalization to the case of many vortices is straightforward. Order parameter
with 2N vortices pinned at R i is already given in (3.16). Assuming that they are
well separated from each other, we can find an approximate zero-energy bound state
localized in each vortex core:
⎛
⎞
e −iπ/4 χ ↑ (r i )
e iπ/4 χ ↓ (r i )e iϕ i
Ψ i (r)=e i(l−1)ϕi/2 e iΩ iτ z/2
e −iπ/4 χ ↓ (r i )e −ilϕ i
⎜
⎟
⎝
⎠
−e iπ/4 χ ↑ (r i )e −i(l−1)ϕ i
(3.33)
the construction of N Dirac fermions and 2 N−1 ground state Hilbert space are the
same as the case of spinless p x + ip y superconductors.
3.3 Atiyah-Singer-type index theorem
Index theorem provides an intelligent way of understanding the topological
stability of zero modes. It is well-known that one can relate the analytical index of
an elliptic differential operator (Dirac operator) to the topological index (winding
number) of the background scalar field in 2D [84] through the index theorem. Since
BdG Hamiltonian for TI/SC system at µ = 0 can be presented as a Dirac operator
(see Eq.(3.21)), we give a brief review of this index theorem, see also recent exposition
in Ref. [85]. Specifically, the Hamiltonian for TI/SC heterostructure can be
written as
H D = iγ · ∇ + Γ · n, (3.34)
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