Page 250 - Principles and Applications of NanoMEMS Physics

240 Appendix B

~
introducing L- and R-moving fermion fields ψ , and imposing boundary
L / R
conditions (B.Cs) on these to discretize k. Making k ∈ ( ∞− , ∞ ) entails
defining energies of the form E ≡ E () v+0 (k + k ) in the range
k v , F F
k < − k . These additional “unphysical”states do not alter the low-energy
F
physics of the system, however, a strong perturbation, such as might be due
to an electric field or an impurity, then the procedure would not apply
because of the larger energies involved [139]. Extending the range of k, the
fermionic field Ψ is written in terms of fields representing L- and R-
phys
moving electrons which now possess the unbounded k define above. This
new fermionic field takes the form,
Ψ phys () ex = − ik F x ~ L () ex + + ik F x ~ R () x , (B.52 )
ψ
ψ
where,
∞
~
ψ () =x 2π ¦ e B ikx c . (B. 53)
L / R k , L / R
L k = −∞
Lastly, imposing B.C.s quantizes the fermion fields momentum. If these are
~
~
−
taken as anti-periodic, we have, ψ ( /L ) 2 = ψ ( L− ) 2 / , which
L / R L / R
implies δ = 1. Having defined the prerequisite conditions for bosonization,
b
ˆ
the consequent number operators, Klein factors, and boson operators, N ,
L / R
F , and b are defined in terms of the fermion annihilation operator
L / R qL / R
c . This results in the following,
kL / R
§
~
+
φ L / R () = − ¦ 1 e −aq / 2 [e B iqx b qL / R + e B iqx b qL / R ] ¨ q = 2π n q > 0 · , (B. 54)
¸
x
∈Z + n © L ¹
n q q
2π § ¨ N L / − 1 b δ · x ¸ ~
ˆ
i B
~
φ
ψ () ax ≡ 2 / 1 − F e L © R 2 ¹ e − i L / R () , (B.55)
x
L / R L / R
~
~
~
~ +
ρ () x ≡ + ψ ψ + = ±∂ φ () x + 2π N ˆ , (B.56)
L / R + L / R L / R + x L / R L / R
L
~
~
where the boundary conditions φ ( /L ) 2 = φ ( L− ) 2 / (periodic) on the
L / R L / R
bosons and density fields have been imposed. Notice that, while the density
~
ρ is quadratic in the fermion field, it is only linear in the boson field.
L / R
This is key to the simplification brought about by the bosonization
procedure.