Page 115 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 103

which captures scattering events in which (k , s;− k F ) t , → ( k ,− F s; k F ) t ,
F
t
for s ≠ . The forward scattering Hamiltonian is given by,

H = 1 ¦ (g () q ρ s , α () q ρ α ( )+− q g () q ρ s , α () q ρ t , α ( )). (72)
− q
2
int_
L 2 − , t 4
The full bosonized Hamiltonian has been shown by Schulz [133] to take the
form,

H = H kin + H int_ 1 + H int_ 2

2g )
= H ρ + H σ + 1 ³ dx cos ( φ8 σ , (73)
( π2 a ) 2

where a is a short-distance cutoff, and for ν = ρ, σ ,

§ π u ν K ν 2 u ν 2 ·
H ν = ³ dx ¨ Π ν + (∂ φ ν ) ¸ , (74)
¸
¨
x
© 2 2π K ν ¹
with,

2 2
§ g , 4 ν · § g · 2 πv F + 2 g ν , 4 + g ν
u ν = ¨ ¨ v + ¸ − ¨ ν ¸ , K = , (75)
ν
¸
F
2 ¹
© π ¹ © π 2 πv F + 2 g ν , 4 − g ν
and g = g − 2g , g = g , g , 4 σ = 0.
ρ
σ
1
2
1
Schulz [133] has exposed a number of situations by examining (75). For
instance, he points out that a noninteracting system, for which u = v and,
ν
F
thus exhibits equal charge and spin velocities, is obtained by setting K ν = 1.
That if g = 0 , then there is no backscattering and (75) describes uncoupled
1
charge and spin density oscillations with a dispersion relation () ukω ν = ν k
and the system is conducting.
The Hamiltonian (75) offers, as one of its consequences, the possibility of
complete separation in the dynamics of spin and charge. In particular, if
u ≠ u , then spin and charge waves propagate with different velocities.
ρ
σ
The electron, in this case, is said to dissolve into two particles, namely, a
spin particle, called a spinon, and a charge particle, called a holon [134]. A