Density response and collective modes of semiholographic non-Fermi liquids

Semiholographic models of non-Fermi liquids have been shown to have generically stable generalized quasiparticles on the Fermi surface. Although these excitations are broad and exhibit particle-hole asymmetry, they were argued to be stable from interactions at the Fermi surface. In this work, we use this observation to compute the density response and collective behaviour in these systems. Compared to the Fermi liquid case, we find that the boundaries of the particle-hole continuum are blurred by incoherent contributions. However, there is a region inside this continuum, that we call inner core, within which salient features of the Fermi liquid case are preserved. A particularly striking prediction of our work is that these systems support a plasmonic collective excitation which is well-defined at large momenta, has an approximately linear dispersion relation and is located in the low-energy tail of the particle-hole continuum. Furthermore, the dynamic screening potential shows deep attractive regions as a function of the distance at higher frequencies which might lead to long-lived pair formation depending on the behavior of the pair susceptibility. We also find that Friedel oscillations are present in these systems but are highly suppressed.

Figure 1

The kinematic region in $\mathrm{\Omega }\text{−}q$ plane where $\mathrm{Im}\mathcal{L}(\mathrm{\Omega },q)$ is nonzero for the Fermi liquid is the combined green and red regions in the plot shown above. For convenience, we have chosen $k_F=1$ and $m=0.5$. This region is essentially the particle-hole continuum. The boundaries of this region are given by ${\mathrm{\Omega }}_{\max }(q)$ and ${\mathrm{\Omega }}_{\min }(q)$. The red region is what we will call as the inner core of the particle-hole continuum and is bounded by the curve ${\mathrm{\Omega }}_{\mathrm{int}}(q)$.

Figure 2

The allowed values of $k_x/k_F$ and $k_y/k_F$ when $q=0.6k_F<2k_F$ is shown above in dark blue. This dark blue region is simply the intersection of the complement of the region outside the circle of unit radius centred at $\{-0.3k_F,0\}$ (representing the on-shell particle in the pair) and the region bounded by the circle of unit radius centred at $\{0.3k_F,0\}$ (representing the on-shell hole in the pair). The allowed values of $\mathrm{\Omega }$ are simply those for which $\mathrm{\Omega }=q{k}_x/m$ with $k_x$ lying within in the dark blue region. Clearly, the smallest possible value of $\mathrm{\Omega }$ is 0 which is realized when $k_x=0$ (the brown line) and the largest possible value of $\mathrm{\Omega }$ is ${\mathrm{\Omega }}_{\max }$ given by Eq. (28) which is realized when $k_x=k_F+q/2$. When $k_F+q/2\ge k_x\ge k_F-q/2$, i.e. ${\mathrm{\Omega }}_{\max }\ge \mathrm{\Omega }\ge {\mathrm{\Omega }}_{\mathrm{int}}$ with ${\mathrm{\Omega }}_{\mathrm{int}}$ latter given by Eq. (30), the allowed values of $k_y$ forms a continuous line instead of disconnected segments. The red vertical line represents $k_x=k_F-q/2$ for which $\mathrm{\Omega }={\mathrm{\Omega }}_{\mathrm{int}}$.

Figure 3

The allowed values of $k_x/k_F$ and $k_y/k_F$ when $q=2.4k_F>2k_F$ is shown above in dark blue. The region bounded by the unit circle centred at $\{1.2k_F,0\}$ (representing the hole-like excitation) now lies entirely in the complement of the unit circle centred at $\{-1.2k_F,0\}$ (representing the particle-like excitation)—the dark blue region simply then coincides with the first region representing the hole excitations. The allowed values of $\mathrm{\Omega }$ are simply those for which $\mathrm{\Omega }=q{k}_x/m$ with $k_x$ lying within the dark blue region. The smallest allowed value of $\mathrm{\Omega }={\mathrm{\Omega }}_{\min }$ is given by Eq. (31) for which $k_x=q/2-k_F$ (the brown vertical line) and the largest allowed value of $\mathrm{\Omega }={\mathrm{\Omega }}_{\max }$ is given by Eq. (28) for which $k_x=q/2+k_F$ (the blue vertical line). There is no analogue of ${\mathrm{\Omega }}_{\mathrm{int}}$ because the allowed values of $k_y$ for fixed $k_x$ (i.e. $\mathrm{\Omega }$) and $q$ always form a continuous line segment.

Figure 4

Plots of $\mathrm{Im}{\mathcal{L}}^{\mathrm{FL}}(q)$ for various fixed values of $\mathrm{\Omega }$ are shown above. Note ${\mathcal{L}}^{\mathrm{FL}}(q,\mathrm{\Omega })=(m/\pi )f(q/k_F,\mathrm{\Omega }/{\epsilon }_F)$ with ${\epsilon }_F=k_F^2/2m$. Therefore, we have used the dimensionless variables $q/k_F$, $\mathrm{\Omega }/{\epsilon }_F$ and $(\pi /m)\mathrm{Im}\mathcal{L}$ in the plots above. Note that the intermediate plateau and the two intermediate kinks where $\partial \mathrm{Im}{\mathcal{L}}^{\mathrm{FL}}(q)/\partial q$ is discontinuous appear for $\mathrm{\Omega }/{\epsilon }_F<1$ and disappears when $\mathrm{\Omega }/{\epsilon }_F\ge 1$.

Figure 5

Plots of $\mathrm{Im}{\mathcal{L}}^{\mathrm{FL}}(\mathrm{\Omega })$ for various fixed values of $q$ are shown above. We have used dimensionless variables for plotting as in Fig. 4. Note the behavior for $q<2k_F$ is different from that for $q>2k_F$. In particular, the minimum value of $\mathrm{\Omega }$ for the latter case for which $\mathrm{Im}{\mathcal{L}}^{\mathrm{FL}}(\mathrm{\Omega })$ is nonvanishing is shifted from the origin. Also the intermediate kink where $\partial \mathrm{Im}{\mathcal{L}}^{\mathrm{FL}}(q)/\partial \mathrm{\Omega }$ is discontinuous appears only for $q<2k_F$.

Figure 6

Plots of $\mathrm{Im}\mathcal{L}(q)$ of the semiholographic non-Fermi liquid for $\nu =2/3$ and $\varphi =\pi /4$ for various fixed values of $\mathrm{\Omega }$. In order to compare with plots of $\mathrm{Im}{\mathcal{L}}^{\mathrm{FL}}(q)$ shown in Fig. 4, we need to take into account that for the above plots we have chosen $k_F=0.4$, $m=0.5$ (i.e. ${\epsilon }_F=0.16$) and $|\zeta |=1$.

Figure 7

Plots of $\mathrm{Im}\mathcal{L}(\mathrm{\Omega })$ of the semiholographic non-Fermi liquid for $\nu =2/3$ and $\varphi =\pi /4$ for various fixed values of $q$. In order to compare with plots of $\mathrm{Im}{\mathcal{L}}^{\mathrm{FL}}(\mathrm{\Omega })$ shown in Fig. 5, we need to take into account that for the above plots we have chosen $k_F=0.4$, $m=0.5$ (i.e. ${\epsilon }_F=0.16$) and $|\zeta |=1$.

Figure 8

Plots of $\mathrm{Im}\mathcal{L}(q)$ for $\mathrm{\Omega }=0.01$ for $\nu =n/(n+1)$ and $\varphi =\pi /(n+2)$ with $n=4$, 5, 6 and 7. The other parameters are kept the same as in previous plots, i.e. $k_F=0.4$, $m=0.5$ and $|\zeta |=1$. We see that as we increase $\nu$ keeping $\varphi$ near-extremal, $\mathrm{Im}\mathcal{L}(q)$ becomes closer to the Fermi liquid. The inlay plots demonstrate the behavior near $q*$ for which $\mathrm{\Omega }={\mathrm{\Omega }}_{\min }(q*)=0.01$ (corresponding to the second boundary of the particle-hole continuum) where ${\partial }^2\mathrm{Im}\mathcal{L}(q)\partial q^2$ varies very rapidly. It is clear that with increasing $\nu$ the decay of $\mathrm{Im}\mathcal{L}(q)$ for $q>q*$ occurs faster.

Figure 9

Plots of $\mathrm{Re}{\mathcal{L}}^{\mathrm{FL}}(\mathrm{\Omega })$ for various fixed values of $q$ are shown above.

Figure 10

Plots of $\mathrm{Re}\mathcal{L}(\mathrm{\Omega })$ for various fixed values of $q$ are shown above. We have made the same choices of parameters as for $\mathrm{Im}\mathcal{L}(q,\mathrm{\Omega })$ in Fig. 7, i.e. for $\nu =2/3$, $\varphi =\pi /4$, $k_F=0.4$, $m=0.5$ and $|\zeta |=1$.

Figure 11

Plots of $\mathrm{Re}{\mathcal{L}}^{\mathrm{FL}}(q)$ for various fixed values of $\mathrm{\Omega }$ are shown above.

Figure 12

Plots of $\mathrm{Re}\mathcal{L}(q)$ for various fixed values of $\mathrm{\Omega }$ are shown above. We have made the same choices of parameters as for $\mathrm{Im}\mathcal{L}(q,\mathrm{\Omega })$ in Fig. 6, i.e. for $\nu =2/3$, $\varphi =\pi /4$, $k_F=0.4$, $m=0.5$ and $|\zeta |=1$.

Figure 13

Plots of $\mathrm{Re}\mathcal{L}(q)$ for $\mathrm{\Omega }=0.01$ for $\nu =n/(n+1)$ and $\varphi =\pi /(n+2)$ with $n=4$, 5, 6 and 7. The other parameters are kept the same as in previous plots, i.e. $k_F=0.4$, $m=0.5$ and $|\zeta |=1$.

Figure 14

The ring diagrams summing which we obtain ${\mathcal{L}}^{\mathrm{imp}}(q,\mathrm{\Omega })$.

Figure 15

The ring diagrams summing which we obtain $V_s(q,\mathrm{\Omega })$.

Figure 16

We see that in the case of the Fermi liquid, $q-\mathrm{Re}{\mathcal{L}}^{\mathrm{imp}}(\mathrm{\Omega },q)$ has two zeroes for each value of $q<q_{\mathrm{crit}}\approx 0.25k_F$.

Figure 17

$\mathrm{Re}{\mathcal{L}}^{\mathrm{imp}}(\mathrm{\Omega },q)$ for the Fermi liquid as a function of $\mathrm{\Omega }$ for various values of $q$.

Figure 18

$-\mathrm{Im}{\mathcal{L}}^{\mathrm{imp}}(\mathrm{\Omega },q)$ for the Fermi liquid as a function of $\mathrm{\Omega }$ for various values of $q$. Note that $-\mathrm{Im}{\mathcal{L}}^{\mathrm{imp}}(\mathrm{\Omega },q)$ never diverges in the continuum. The delta function peaks outside the continuum corresponding to the plasmonic poles where $\mathrm{Re}{\mathcal{L}}^{\mathrm{imp}}(\mathrm{\Omega },q)$ also diverges are not shown in the above plots due to numerical limitations.

Figure 19

Three-dimensional plots of real and imaginary parts of ${\mathcal{L}}^{\mathrm{imp}}(q,\mathrm{\Omega })$ for $e^2/(2{\epsilon }_b)=1$, $\nu =2/3$, $k_F=0.4$, $\arg (\zeta )=\pi /4$ and all other parameters as in the previous section.

Figure 20

A contour plot of $\mathrm{Im}{\mathcal{L}}^{\mathrm{imp}}(q,\mathrm{\Omega })$ is shown with the black dots indicating the maxima.

Figure 21

The response of the induced $\delta \rho (t)$ at $\mathbf{x}=0$ for the semi-holographic non-Fermi liquid.

Figure 22

$\mathrm{Re}{V}_s(\mathrm{\Omega },r)$ shown as a function of $r$ for various fixed values of $\mathrm{\Omega }$.

Figure 23

$\mathrm{Im}{V}_s(\mathrm{\Omega },r)$ shown as a function of $r$ for various fixed values of $\mathrm{\Omega }$.

Figure 24

The induced charge density as a result of the introduction of static charge shows suppressed Friedel oscillation.