Dynamical susceptibility of a Fermi liquid

We study dynamic response of a Fermi liquid in the spin, charge, and nematic channels beyond the random phase approximation for the dynamically screened Coulomb potential. In all the channels, one-loop order corrections to the irreducible susceptibility result in a nonzero spectral weight of the corresponding fluctuations above the particle-hole continuum boundary. It is shown that the imaginary part of the spin susceptibility, $\mathrm{Im}{\chi }_s(\mathbf{q},\omega )$, falls off as $q^2/\omega$ for frequencies above the continuum boundary ($\omega \gg v_{\mathrm{F}}q$) and below the model-dependent cutoff frequency, whereas the imaginary part of the charge susceptibility, $\mathrm{Im}{\chi }_c(\mathbf{q},\omega )$, falls off as ${(q/k_F)}^2{q}^2/\omega$ for frequencies above the plasma frequency. An extra factor of ${(q/k_F)}^2$ in $\mathrm{Im}{\chi }_c(\mathbf{q},\omega )$ as compared to $\mathrm{Im}{\chi }_s(\mathbf{q},\omega )$ is a direct consequence of Galilean invariance. The imaginary part of the nematic susceptibility increases linearly with $\omega$ up to a peak at the ultraviolet energy scale—the plasma frequency and/or Fermi energy—and then decreases with $\omega$. We also obtain explicit forms of the spin susceptibility from the kinetic equation in the collisionless limit and for the Landau function that contains up to the first three harmonics.

Figure 1

A sketch of the imaginary part of the spin (solid) and charge (dashed) susceptibilities. The Bohr magneton is set to unity, such that the units of ${\chi }_s$ and ${\chi }_c$ are the same. ${\omega }_p\left(q\right)$ denotes the plasmon frequency which may depend on $q$. The particle-hole continuum occupies the range $0\le \omega \le v_{\mathrm{F}}^{*}q$. Outside the continuum, $\mathrm{Im}{\chi }_s(\mathbf{q},\omega )$ falls off a $q^2/\omega$ both in 2D and 3D. For $\omega \gg {\omega }_p\left(q\right)$, the tail of $\mathrm{Im}{\chi }_c(\mathbf{q},\omega )$ is smaller than that of $\mathrm{Im}{\chi }_s(\mathbf{q},\omega )$ by a factor of ${(q/k_{\text{F}})}^2\ll 1$, which reflects Galilean invariance of the system. In the intermediate range, $v_{\mathrm{F}}^{*}q\ll \omega \ll {\omega }_p\left(q\right)$, $\mathrm{Im}{\chi }_c(\mathbf{q},\omega )$ increases with $\omega$ as $q^2{\omega }^3$ and $q^4{\omega }^3$ in 2D and 3D, respectively. The relative magnitude of ${\chi }_s$ and ${\chi }_c$ within the continuum depends on the interaction, and the choice made in the sketch is completely arbitrary.

Figure 2

Imaginary part of the spin susceptibility, ${\chi }_s(\mathbf{q},\omega )$, within the RPA [Eq. (2.2)]. The inset shows $\mathrm{Im}{\chi }_s(\mathbf{q},\omega )$ near the boundary of the particle-hole continuum ($\omega =v_{\mathrm{F}}q$), where it vanishes as $\sqrt{v_{\mathrm{F}}q-\omega }$ and $1/{ln}^2(v_{\mathrm{F}}q-\omega )$ in 2D and 3D, respectively. The dimensionless coupling constant is $U{N}_F/2=0.5$ for both dimensions, where $N_F$ is the density of states at the Fermi level.

Figure 3

Imaginary part of the spin susceptibility (normalized to its static value) for a 2D (a) and 3D (b) Fermi liquid, obtained by solving the kinetic equation for a model form of the Landau function, which contains up to the three first harmonics in the spin channel. The legends specify the harmonics of the Landau function following the $(F_0^a,F_1^a,F_2^a)$ format.

Figure 4

Imaginary part of the spin susceptibility (normalized to its static value) for a model form of the Landau function corresponding to the statically screened Coulomb potential in 2D. The system of equations (2.16) was solved numerically using the first 101 harmonics of the Landau function in Eq. (2.39) with $a=0.5$.

Figure 5

Diagrams for a susceptibility to lowest order in the dynamically screened Coulomb interaction (wavy line). Diagrams 1–3 give a correction to the spin susceptibility. Diagrams 1–5 give a correction to the irreducible charge susceptibility. The full charge susceptibility is obtained by resumming irreducible parts connected by a single interaction line, with the result given by Eq. (2.40).

Figure 6

Semilog scale: Imaginary part of the charge susceptibility in 2D, as given by Eq. (3.31).

Figure 7

A sketch of the imaginary part of the spin-current (nematic) susceptibility. For $\omega \gg v_{\mathrm{F}}q$, the behavior is described by Eqs. (3.37)–(3.40).