Wednesday, May 26, 2010

One dimension is different

This weeks reading from Phillips, Advanced Solid State Physics, is Section 8.4, on the dielectric response function. This is calculated at the level of the Random-Phase-Approximation (RPA) for a Fermi liquid (weakly interacting fermion gas). One finds the density-density response function. The imaginary part is related to the structure factor (via a fluctuation-dissipation relation). This can be thought of as an effective density of states for particle-hole excitations.
In three-dimensions these excitations are gapless for all wavevectors. However, one dimension is different. The shaded area in Figure (b) above shows the relevant excitations for on one-dimensional fermion gas. This Figure is taken from a seminal paper by Haldane, who emphasized the distinct difference from higher dimensions.

The fact that there is a well defined dispersion for low momenta, shown above means that density fluctuations are well-defined quasi-particles in the one dimensions. This is the basis of bosonisation and the Luttinger liquid, discussed in Chapter 9.

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