Page 238 - Principles and Applications of NanoMEMS Physics

228 Appendix B

fashion the procedure of bosonization, and we follow their exposition
closely.
In general, bosonizing a theory involving M species of fermions may be
accomplished when a fix specific set of conditions are met, in particular:
1) The theory can be formulated in terms of a set of fermion creation and
+ +
annihilation operators, c and c , which obey the following canonical
kq kq
anti-commutation relations
+
δ
{ c kη ,c kη } δ= ηη ' kk ' , (B.2)
where the index η = 1 ,... M labels M different species, which might be
−
present, and the index ∈k [ ∞, ∞ ] is a discrete, unbounded wave number
of the form,
π
1
k = 2 § ¨n k − δ b · , ¸ with n k ∈ Z and δ b ∈ [ ) 2,0 , (B.3)
L © 2 ¹

with n are integers, L is a length associated with the size of the system, and
k
δ is a parameter that embodies the nature of the boundary conditions of the
b
fermion fields, i.e., whether they periodic or fixed. According to Delft and
Scholler [139], in typical examples Ș can denote electron spin: Ș = (Ĺ, Ļ), in
which case M = 2, or it distinguishes left-moving from right-moving spinless
electrons, as found in a one-dimensional wire, in which case: Ș = (L,R), and
M = 2. k refers to the momentum index that labels the energy states, E , of a
k
free noninteracting Fermi gas, defined with respect to the Fermi energy, so
that E = E . The discrete and unbounded nature of k are needed in order to
0 F
enable the systematic accounting of the states, on the one hand, and the
proper definition of bosonic operators, oh the other.

2) The fermion fields are defined in terms of the creation and annihilation
operators as follows,

∞
ψ () =x 2 π ¦ e −ikx c , (B.4)
η
L k = −∞ kη
with inverse, L 2 /

ikx
c kη = 1 ³ e ψ η ()dxx , (B.5)
π
2 L − / 2
L