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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