Page 249 - Principles and Applications of NanoMEMS Physics

B. BOSONIZATION 239

+
H 0 η b q + η E = ( + qE )b , (B.4 )
8
η
q
G
+
and implies that the b acting on the ground state N may generate the
η
q
G 0
complete N -particle Hilbert space. The bosonic variables then may be
employed to represent H , including both the ground and excited states.
0 η
This is accomplished when it takes the bosonized form,
=
ˆ
ˆ
H η ¦ qb + b + 2 π 1 N (N +1 − δ ). (B.49 )
0 qη qη η η b
q>0 L 2
+
Since H does not change the particle number, no Klein factors, F ,
0 η η
appear.
B.5 Bosonization Treatment of Spinless Electrons in One-
Dimensional Wire

The one-dimensional wire is the prototypical system of a Lüttinger liquid.
It is described as a one-dimensional conductor of length L with free spinless
left- and right-moving electrons. The electrons possess momentum
p ∈ ( ∞− , ∞ ), and propagate according to a dispersion relation given by
2
E () (pp = 2 − p 2 ) m . Since electrons are confined to move either to the
F
left or to the right in a 1D conductor, the usual fermion field,
∞
Ψ ()≡x 2 π ¦ e ipx c , (B. 0 5 )
phys p
L p = −∞
is expressed as,
∞
Ψ () =x 2 π ¦ (e − ki ( F +k )x c + e i ( F +k )x c ) , (B. 51)
k
phys
L k = − k −k F −k k F +k
F
where the momentum p is written as p = B (k + k ), with ∈k [− k ∞ ),
F F ,
and p < 0 corresponds to the left(L)-moving electrons, and p > 0 to the
right(R)-moving electrons. In the context of our species definitions, the
index v = ( RL, ) plays the analogous role to η .
We now effect the bosonization procedure described previously. First, one
must make k ∈ [− k F , ∞ ) unbounded from below and discrete. This is
accomplished by artificially extending the range of k to be unbounded,