ABSTRACT - DRUM - University of Maryland

of (7.2) has a product of the Klein factors ˆF † R2 ˆF † L2 ˆF L1 ˆFR1 in it. Since this term is
to be refermionized as ∼ ˆχ L ˆχ R , we are naturally led to define new Klein factors
ˆF r = ˆF † r2 ˆF r1 for ˆχ r . Notice that so-defined Klein factors satisfy { ˆP a , ˆF r } = 0, i.e.
ˆF r change single-chain fermion parity. Then the fermionic fields that refermionize
the sine-Gordon theory at the Luther-Emery point should take the form
ˆχ r =
1
√ 2πa0
ˆξr ˆFr e ir√ 4π ˜ϕ r−
. (7.14)
Thus one can identify that ˆχ r corresponds to inter-chain single-particle tunneling.
The ground state |G〉 of the Hamiltonian (7.9) can be schematically expressed as
[∫
|G〉 = exp
]
dx 1 dx 2 ˆχ † (x 1 )g(x 1 , x 2 )ˆχ † (x 2 ) |vac〉, (7.15)
where g(x 1 , x 2 ) is the Cooper-pair wave function of the spinless p-wave superconductor
and |vac〉 is the vacuum state of ˆχ fermion. With the definition (7.14), it is
easy to check that |G〉 is a coherent superposition of Fock states having the same
single-chain fermion parity, thus an eigenstate of ˆP a . On the other hand, the Majorana
fermion ˆγ, being a superposition of ˆχ and ˆχ † , changes the single-chain fermion
parity: {ˆγ, ˆP a } = 0. As a result, the two degenerate ground states |G〉 and ĉ † |G〉
have different single-chain fermion parity which is the essence of the Majorana edge
states. If the total number of fermions N is even, then the two (nearly) degenerate
ground states correspond to even and odd number of fermions on each chain,
respectively.
So far all the conclusions are drawn at the Luther-Emery point K − = 2. Once
we move away from the Luther-Emery point, the theory is no longer equivalent to
free massive fermions. An intuitive way to think about the situation is that if we
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