Page 116 - Principles and Applications of NanoMEMS Physics

104 Chapter 3

simple picture for visualizing spin-charge separation is shown in Figure 3-
11.

ho l o n n
ho l o s pi n o
s pi n o n n
Figure 3-11. Illustration of spin charge separation. If a photon impinges on an
antiferromagnetic Mott insulator an removes an electron, the disruption left behind changes
both the spin and charge order. Electron motion into the vacant site results in spin and charge
separation, giving rise to two distinct particles, namely, a holon and a spinon. (After [134].)

Qualitatively, the pertinent physics of the Lüttinger liquid follow from the
dispersion relation and may be surmised from Fig. 3-11 [134]. An
examination of this figure indicates that, due to the linear dispersion relation,
changes in momentum determine energy changes. In particular, a momentun
G
excitation q imposed on the 1D electron system, will cause a compression
G
and rarefaction of the electron density with a wavelength 2π q . The
degrees of compression and rarefaction embody a density wave, and has two
G
consequences. First, because q determines the kinetic energy E in a unique
way, the density wave has a well-defined kinetic energy. Second, the
concomitant density will depend on both the spin interaction and the
Coulomb interaction amongst electrons which, being functions of distance,
embody the potential energy of the system. Therefore, the total energy of the
system may be specified by the properties of a density wave. This density
wave, in turn, contains a spin density and a charge density. This spin-charge
separation and coexistence is the hallmark of the Lüttinger liquid.

En e r g y y
En e r g

δ E E
0 0 k k F F δ k k
δ
δ q q

Figure 3-12. Excitation of electron-hole pairs in one-dimensional structure. After [134].