Fermionic response in a zero entropy state of $\mathcal{N}=4$ super-Yang-Mills

We analyze fermionic response in the geometry holographically dual to zero-temperature $\mathcal{N}=4$ super-Yang-Mills theory with two equal nonvanishing chemical potentials, which is characterized by a singular horizon and zero ground state entropy. We show that fermionic fluctuations are completely stable within a gap in energy around a Fermi surface singularity, beyond which non-Fermi liquid behavior returns. This gap disappears abruptly once the final charge is turned on and is associated to a discontinuity in the corresponding chemical potential. We also show that the singular near-horizon geometry lifts to a smooth ${\mathrm{AdS}}_3\times {\mathbb{R}}^3$ and interpret the gap as a region where the quasiparticle momentum is spacelike in six dimensions due to the momentum component in the Kaluza-Klein direction, corresponding to the final charge.

Figure 1

Cartoon comparison between a regular extremal black-brane background (left) and the extremal two-charge black hole (right). In the former, fluctuations characteristic of a non-Fermi liquid surround the Fermi surface; in the latter, they are separated from the Fermi surface by a gap of size $2\mathrm{\Delta }$ wherein the fluctuations are stable.

Figure 2

Two approaches to the extremal two-charge black hole.

Figure 3

Plots of ${\mathrm{\Phi }}_1(r)$ for various $Q_1$ as $Q_1\to 0$ with $T=0$ held fixed. For all nonzero $Q_1$ one has ${\mathrm{\Phi }}_1(r_H)=0$; note $r_H$ moves left as $Q_1\to 0$. For $Q_1=0$, ${\mathrm{\Phi }}_1$ degenerates to a nonzero constant.

Figure 4

The ratio of chemical potentials ${\mu }_1/{\mu }_2$ compared with the charge density ratios ${\rho }_1/{\rho }_2$. As ${\rho }_1$ approaches zero from above, ${\mu }_1$ approaches ${\mu }_2$ from below, but at ${\rho }_1=0$ strictly we have ${\mu }_1=0$ (heavy dot at origin).

Figure 5

Poles in the retarded Green’s function from $\omega =0$ to $\omega =\pm \mathrm{\Delta }$ (red dashed lines) for fermion $A$.

Figure 6

Fermi surfaces in ($2+1$)-charge black holes as a function of ${\mu }_R$ from [59]; on the left is case 3, on the right case 1. The ${\mu }_R\to 1$ limits (purple dashed lines) match onto the left and right edges of the gapped region in Fig. 5.

Figure 7

Poles in the retarded Green’s function from $\omega =0$ to $\omega =\pm \mathrm{\Delta }$ (red dashed lines) for fermion $B$.

Figure 8

Fermi surfaces in ($2+1$)-charge black holes as a function of ${\mu }_R$ from [59] for cases 4 and 5. The ${\mu }_R\to 1$ limit (purple dashed lines) matches onto the right edges of the gapped region in Figs. 7 and 9, respectively.

Figure 9

Poles in the retarded Green’s function from $\omega =0$ to $\omega =\pm \mathrm{\Delta }$ (red dashed lines) for the neutral fermion.