Page 245 - Principles and Applications of NanoMEMS Physics

B. BOSONIZATION 235

where λ η is a phase operator, and F effects the η -particle removal
ˆ
η
normally effected by ψ η () x . Inserting (C.27) into (B.37), the following
expression is obtained,
G + () G
ψ () Nx = e − iϕ η x F λ () Nx . (B.38)
ˆ
η
0 η η 0
B
B
Use of the operator identity, e − B Ae = A + C or [ eA, B ]Ce , and
identification of A = b , B = i − ϕ + () x , and C = δ α () x , secures the
q ' η η ηη' q
compliance of (B.38) with (B.36).
The crux of the bozonization identity lies on (B.38). According to Delft
and Schoeller [139], this expression embodies the fact that acting with the
fermionic field ψ η () x on N 0 may effect the removal of one η -particle

from the ground state in two ways. First, via the interpretation of ψ η () x as

2 π ∞
¦ e −ikx c , it creates an infinite linear combination of single-hole
L k = −∞ kη
states caused by each application of the fermion annihilation operator c ,
k η
see Fig. B-1.

∞ ∞
¦ ¦ y y n n c c − − n n 0 0 0 0 = = + + y y + + y y 2 2 +…
+…
n = = 0 0
n

Figure B-1. Effect of acting with ψ η () x on the ground state. We have expressed

∞
ψ () ~x ¦ y n c −n , with y = e i2π x / L . (After [139].)
n =0

On the other hand, observing the right-hand side of (B.38), this η -particle
removal may also be effected by removing the highest η -electron from
G
N , which yields a different ground state, namely, c N , and then
0 N η 0
η
creation of a linear combination of hole states through the action of the
+ i − ϕ η + ()
x
boson creation operators b present in e . The effect of first operating
q η
with the Klein factor is shown in Fig. B-2, and that of operating with the
field operator is shown in Fig. B-3..