Dynamical gap and cupratelike physics from holography

We study the properties of fermion correlators in a boundary theory dual to the Reissner-Nordström ${\mathrm{AdS}}_{d+1}$ background in the presence of a bulk dipole (Pauli) interaction term with strength $p$. We show that by simply changing the value of the parameter $p$ we can tune continuously from a Fermi liquid (small $p$), to a marginal-Fermi liquid behavior at a critical value of $p$, to a generic non-Fermi liquid at intermediate values of $p$, and finally to a Mott insulator at large values of the bulk Pauli coupling. As all of these phases are seen in the cuprate phase diagram, the holographic model we study has the key elements of the strong-coupling physics typified by Mott systems. In addition, we extend our analysis to finite temperature and show that the Mott gap closes. Of particular interest is that it closes when the ratio of the gap to the critical temperature is of the order of 10. This behavior is very much similar to that observed in the classic Mott insulator ${\mathrm{VO}}_2$. We then analyze the nonanalyticities of the boundary theory fermion correlators for generic values of frequency and momentum by calculating the quasinormal modes of the bulk fermions. Not surprisingly, we find no evidence for the dipole interaction inducing an instability in the boundary theory. Finally, we briefly consider the introduction of superconducting condensates and find that, in that case, the fermion gap is driven by scalar-fermion couplings rather than by the Pauli coupling.

Figure 1

A cartoon of the phase diagram of the boundary theory considered here. MI indicates a Mott insulator, a phase with a gap in the absence of symmetry breaking. NFL denotes non-Fermi-liquid behavior which is distinct from the gapped spectrum of a Mott insulator. MFL indicates marginal-Fermi liquid behavior in which the electron self-energy scales as $\omega \log \omega$ at $T=0$, and FL (Fermi liquid) a regime in which the dispersion is linear in frequency. The tuning parameter in this model is the Pauli (dipole) coupling. Similar behavior is obtained in the nonsuperconducting features of the cuprate materials by tuning the hole-doping level $x$.

Figure 2

Plots of ${\mathcal{I}}_{\pm }$ versus $p$ (for $d=3$, ${\Delta }_{\psi }=3/2$, and $q=1$). The red band depicting ${\mathcal{I}}_{-}$ is where $\mathrm{Im}{G}_{-}(\omega ,k)$ becomes oscillatory at small $\omega$. The blue band (${\mathcal{I}}_{+}$) shows the region where $\mathrm{Im}{G}_{+}(\omega ,k)$ is oscillatory (at small $\omega$).

Figure 3

Plots of $\mathrm{Im}{G}_{-}(0,k)$ (solid line) and $\mathrm{Im}{G}_{+}(0,k)$ (dashed line) for (a) $p=0$, (b) $p=0.1$, and (c) $p=1.8$. We set $d=3$, ${\Delta }_{\psi }=3/2$, and $q=1$. Similar plots can be obtained for negative values of $p$ by switching the solid lines with the dashed lines. Focusing on positive $p$, we see that the maximum value of $\mathrm{Im}{G}_{\pm }(0,k)$ increases as $p\to 1/\sqrt{6}$, after which (namely, for $p>1/\sqrt{6}$) it rapidly decreases.

Figure 4

$k_{\mathrm{F}}$’s (shown by blue dots) versus $p$. For $p\le 1/\sqrt{6}$ there is a single Fermi surface for each $p$. For $p>1/\sqrt{6}$ we do not find Fermi surfaces. The orange band shows the oscillatory region ${\mathcal{I}}_{-}$.

Figure 5

Plots of ${\alpha }_{-}(u=\infty ,k)$ (black curves) and ${\beta }_{-}(u=\infty ,k)$ (red curves) versus $k$ for (a) $p=-0.4$, (b) $p=0$, (c) $p=0.2$, and (d) $p=1$. The orange strip in each plot shows the oscillatory region ${\mathcal{I}}_{-}$. The plots are generated for $d=3$, $m=0$, and $q=1$. By $k\to -k$, similar plots could be obtained for ${\alpha }_{+}(u=\infty ,k)$ and ${\beta }_{+}(u=\infty ,k)$.

Figure 6

(a) $\mathrm{Im}{G}_{-}(\omega ,k)$ as a function of $\omega$ for $k=2$. (b) The quasinormal frequencies of ${\alpha }_{-}$ for $k=2$. $d=3$, $p=5$, $q=1$, and $m=0$ in both plots. Also, $M=250$.

Figure 7

The top two plots show the dependence on $k$ (dispersion relation) of the real and imaginary parts of the first five quasinormal modes [depicted in Fig. 6b] on the left-hand side of the negative imaginary axis. The bottom two plots show the dispersion relation of the real and imaginary parts of the first five quasinormal modes of Fig. 6b which are on the right-hand side of the negative imaginary axis. The plots are generated for $d=3$, $p=5$, $m=0$, $q=1$, and $M=250$. The red data correspond to the mode closest to the real axis in the complex $\omega$ plane.

Figure 8

$|\mathrm{Res}{G}_{-}({\omega }_{*},k)|$ as a function of $k$ for the leading negative-frequency pole in Fig. 6b which is closest to the real axis and located to the left of the negative imaginary axis. We set $d=3$, $p=5$, $q=1$, and $m=0$.

Figure 9

A close-up of the real (left plot) and imaginary (right plot) parts of the dispersion relation of the leading pole shown in Fig. 6b which is closest to the real axis and located to the left of the negative imaginary axis.

Figure 10

A close-up of the density plots of $\mathrm{Im}{G}_{-}(\omega ,k)$ for $p=6$ and $T/\mu \simeq 5.15\times {10}^{-3}$ (left) and $T/\mu \simeq 3.98\times {10}^{-2}$ (right). A gap is still seen in the plot on the left while it is closed in the plot on the right.

Figure 11

A close-up of the density of states $A(\omega )$ at $p=6$ for $T/\mu \simeq 0.44$ (dotted line), 0.16 (dashed line), and $5.15\times {10}^{-3}$ (solid line).

Figure 12

The plots in (a) and (b) show, for $k=2$ and $T/\mu =0.16$, $\mathrm{Im}{G}_{-}(\omega ,k)$ as a function of $\omega$ and the quasinormal frequencies of ${\alpha }_{-}$, respectively. Here, $d=3$, $p=5$, $q=1$, $m=0$, and $M=250$.

Figure 13

A cartoon of the zero-temperature “phase diagram” in the $m,q,p$ parameter space. Different regions of the phase diagram correspond to each of the principal structures in the cuprate phase diagram (compare to Fig. 1).

Figure 14

Density plot of the boundary theory fermion spectral function for (a) $p=0$ and (b) $p=3$. Here, $q_{\varphi }=1.5$, $L=1$, and $\mu =2\sqrt{3}$. The black lines depict the IR lightlike region and the red curves represent the bound states.

Figure 15

$k_F$ as a function of $p$. Here, $q_{\varphi }=1.5$, $L=1$, and $\mu =2\sqrt{3}$.