Helical Luttinger liquids and three-dimensional black holes

We use the AdS/CFT correspondence to discuss an equivalence between a helical, strongly coupled Luttinger liquid and a fermion propagating in the background of a topologically charged black hole in three dimensions. The Fermi level is set by the topological charges, thus surmounting difficulties in low dimensions of the standard approach using Coulomb charged black holes. The construction is fully embeddable in string theory, and the microscopic Lagrangian is explicitly known. The retarded Green function at low temperature and energy arises from the geometry very near the black hole horizon, a structure that is universal for all cold, charged liquids with a dual $\mathbb{R}\times \mathrm{U}(1{)}^s$ invariant description in gravity. This explains a subtle relationship between Luttinger physics and the infrared behavior of higher dimensional non-Fermi liquids in the AdS/CFT correspondence.

Figure 1

Spectral density for $m=1.2$, $T_{\pm }={10}^{-4}$, and ${\alpha }_{\pm }=0$. The horizontal axis is $n$ and the vertical $\omega$. The spectral density vanishes rapidly outside the cones defined by $\omega \pm n=0$. Varying ${\alpha }_{\pm }$ shifts the spectral bands in the $\omega -n$ plane.