Friedel oscillations in a Fermi liquid

The problem of the Friedel oscillations in two- and three-dimensional electron Fermi liquids is examined by means of the many-body local fields theory as aided by the most recent results of accurate numerical studies based on the quantum Monte Carlo method. Within linear response, an exact answer is obtained for the amplitude of the electron density distortion due to both short- and long-ranged impurity potentials. It is discussed how a measurement of the local environment of an impurity embedded in an otherwise homogeneous electron liquid can be used to characterize the systems as a Fermi liquid as well as to extract the magnitude of the static many-body corrections beyond the random phase approximation for wave vector equal to $2k_F$.

Figure 1

The wave vector dependence of the static spin-symmetric many-body local field factor $G_{+}\left(q\right)$ for two- (dashed line) and three- (solid line) dimensional homogeneous, paramagnetic Fermi liquids as obtained from interpolation formulas of accurate quantum Monte Carlo data. Here $r_s=2$.

Figure 2

The linear density modulation due to a short-range impurity potential in a two-dimensional electron gas for various densities. Solid lines: fully interacting; dashed lines: RPA; dotted lines: non interacting. Top panel: $r_s=0.5$. Middle panel: $r_s=1.0$. Bottom panel: $r_s=4.0$.

Figure 3

The linear density modulation due to a short range impurity potential in a three-dimensional electron gas for various densities. Solid lines: fully interacting; dashed lines: RPA. Top panel: $r_s=2.0$. Middle panel: $r_s=5.0$. Bottom panel: $r_s=10.0$.

Figure 4

The linear density modulation due to a long-range impurity potential in a two-dimensional electron gas for various densities $(Z=1)$. Top panel: $r_s=0.5$. Middle panel: $r_s=1.0$. Bottom panel: $r_s=4.0$.

Figure 5

The linear density modulation due to a long-range impurity potential in a three-dimensional electron gas for various densities $(Z=1)$. Solid lines: fully interacting; dashed lines: RPA. Top panel: $r_s=2.0$. Middle panel: $r_s=5.0$. Bottom panel: $r_s=10.0$.

Figure 6

The comparison between the leading term (dashed line) and the exact expression (solid line) of the linear density modulation due to a short-range impurity potential in a fully interacting two- dimensional electron gas with $r_s=1.0$. Top panel: short range impurity potential. Bottom panel: long range impurity potential.

Figure 7

The density response function in a quantizing magnetic field. Solid lines: RPA (upper curve: $\ell ∕a_B=0.5$, lower curve $\ell ∕a_B=4.0$). Dotted line: free electrons.

Figure 8

The linear density modulation due to an isolated impurity in a two-dimensional electron liquid for $\ell ∕a_B=0.5$. Solid line: RPA; dashed line: free electrons. Top panel: short-range impurity potential. Bottom panel: long-range impurity potential.