Fermi surface behavior in the ABJM M2-brane theory

We calculate fermionic Green’s functions for states of the three-dimensional Aharony-Bergman-Jafferis-Maldacena M2-brane theory at large N using the gauge-gravity correspondence. We embed extremal black brane solutions in four-dimensional maximally supersymmetric gauged supergravity, obtain the linearized Dirac equations for each spin-$1/2$ mode that cannot mix with a gravitino, and solve these equations with infalling boundary conditions to calculate retarded Green’s functions. For generic values of the chemical potentials, we find Fermi surfaces with universally non-Fermi liquid behavior, matching the situation for four-dimensional $\mathcal{N}=4$ super-Yang-Mills. Fermi surface singularities appear and disappear discontinuously at the point where all chemical potentials are equal, reminiscent of a quantum critical point. One limit of parameter space has zero entropy at zero temperature, and fermionic fluctuations are perfectly stable inside an energy region around the Fermi surface. An ambiguity in the quantization of the fermions is resolved by supersymmetry.

Figure 1

A cartoon of the parameter space of black holes we consider.

Figure 2

Class 1 fermions for the $(3+1)\mathrm{QBH}$. Fermi surface singularities are shown as blue dots, while zeroes are marked by empty circles. The green hatched region is the oscillatory region characteristic of an infrared instability towards pair production in the bulk. The solid blue contours bound the region of Fermi surfaces with non-Fermi liquid–like excitations.

Figure 3

Class 2 fermions for the $(3+1)\mathrm{QBH}$. These modes are unique in that they exhibit multiple Fermi surfaces for small ${\mu }_R$.

Figure 4

Class 3 fermions for the $(3+1)\mathrm{QBH}$. The poles end at the oscillatory region just before ${\mu }_R=1$.

Figure 5

Class 4 fermions for the $(3+1)\mathrm{QBH}$. For the net-neutral modes, there is a novel transition at the 4QBH state from Fermi surface singularities to zeroes.

Figure 6

Class 5 fermions for the $(3+1)\mathrm{QBH}$. Unlike their net-charged brethren, there exists no oscillatory region for the net-neutral modes but a single “oscillatory point” at the pole/zero transition.

Figure 7

Poles and zeros of the retarded Green’s function for the class I fermion from $0<{\tilde{\mu }}_R<1$ (left) and for the class II fermion from $1<{\tilde{\mu }}_R<0$ (right). Viewed together the two plots depict the entire range $0<{\tilde{\mu }}_R<\infty$ for class I or $\infty >{\tilde{\mu }}_R>0$ for class II.

Figure 8

Poles and zeros of the retarded Green’s function for the class III fermion from $0<{\tilde{\mu }}_R<1$ (left) and for the class IV fermion from $1<{\tilde{\mu }}_R<0$ (right), or the entire range $0<{\tilde{\mu }}_R<\infty$ for class III or $\infty >{\tilde{\mu }}_R>0$ for class IV.

Figure 9

Poles and zeros of the retarded Green’s function for the class V fermion from $0<{\tilde{\mu }}_R<1$ (left) and for the class VI fermion from $1<{\tilde{\mu }}_R<0$ (right), or the entire range $0<{\tilde{\mu }}_R<\infty$ for class V or $\infty >{\tilde{\mu }}_R>0$ for class VI.

Figure 10

Class 1 fermions for the 3QBH. There is both a line of poles throughout the stable region $-\mathrm{\Delta }<\omega <\mathrm{\Delta }$ and a pair of poles nucleating very close to $\omega =\mathrm{\Delta }$ before ending on the oscillatory region (green).

Figure 11

Class 2 fermions for the 3QBH. Here there is a line of poles only.

Figure 12

Class 3 fermions for the 3QBH. No line of poles through $\omega =0$ exists, but a pair of poles nucleate near $\omega =\mathrm{\Delta }$ and end on the oscillatory region.

Figure 13

Class 4 fermions for the 3QBH, with a line of poles only.

Figure 14

Class 5 fermions for the 3QBH, with a pair of poles nucleating near $\omega =-\mathrm{\Delta }$.

Figure 15

Class II fermions for the $(2+2)\mathrm{QBH}$, with $k$ normalized relative to ${\mu }_2$ instead of ${\tilde{\mu }}_2$. The Fermi surfaces all lie at $k_F\approx {\mu }_2/\sqrt{8}$.