Properties of CFTs dual to Charged BTZ black-hole

TL;DR

The paper studies strongly coupled 1+1D CFTs at finite density via their gravity dual, a charged BTZ black hole, computing fermion, scalar, and gauge-field correlators at zero and finite temperature. It identifies an IR AdS$_2$ region controlling low-energy behavior, predicting power-law or log-periodic responses in correlators and a non-Fermi-liquid, potentially Fermi–Luttinger-like, boundary theory due to the background charge. Numerical results for fermions reveal a finite Fermi momentum and linear dispersion near the Fermi surface, with log-periodic and particle–hole asymmetric features consistent with the IR AdS$_2$ analysis; gauge-field results show vanishing Re$\,\sigma$ at small $\omega$ with a divergent DC conductivity due to translational invariance. The scalar sector displays log-periodicity and possible instabilities signaling condensation in the AdS$_2$ throat. Overall, the work demonstrates rich nonperturbative boundary dynamics at finite density, connecting holographic IR physics to 1+1D condensed-matter phenomenology like Luttinger and Fermi–Luttinger behavior.

Abstract

We study properties of strongly coupled CFT's with non-zero background electric charge in 1+1 dimensions by studying the dual gravity theory - which is a charged BTZ black hole. Correlators of operators dual to scalars, gauge fields and fermions are studied at both T=0 and $T\neq 0$. In the $T=0$ case we are also able to compare with analytical results based on $ AdS_2$ and find reasonable agreement. In particular the correlation between log periodicity and the presence of finite spectral density of gapless modes is seen. The real part of the conductivity (given by the current-current correlator) also vanishes as $ω\rightarrow 0$ as expected. The fermion Green's function shows quasiparticle peaks with approximately linear dispersion but the detailed structure is neither Fermi liquid nor Luttinger liquid and bears some similarity to a "Fermi-Luttinger" liquid. This is expected since there is a background charge and the theory is not Lorentz or scale invariant. A boundary action that produces the observed non-Luttinger-liquid like behavior ($k$-independent non-analyticity at $ω=0$) in the Greens function is discussed.

Figures (16)

• Figure 1: Left:Spectral function of spinless Luttinger Liquid $A(k,\omega)$ vs $\omega$ for $\tilde{k}>0$ and $\gamma=1/2$. It vanishes the range $\omega=(-v_F \tilde{k},v_F \tilde{k})$ Right:Spectral function given by eqn.(\ref{['F_SpectralLLmodified']}) for $k=5$, $\alpha=\beta=1/2$, $\Delta_1=.01,\Delta_2=.2$, $v1=-.6,v2=.9$
• Figure 2: $T=0$: Imaginary part of Fermion Green's function for $q=0$.Left:$k=1$. Right:$\omega=20$
• Figure 3: $T=0$: Imaginary part of Fermion Green's function for $q=1/2$.Left:$k=4$ (Compare with what is expected, Figure 1). Right:$k=-4$
• Figure 4: $T=0$: Density plot of imaginary part of Fermion Green's function for $q=1/2$ and $q=-1/2$. Lighter regions have higher value of the function
• Figure 5: $T=0$: Dispersion relation near Fermi point for $q=1/2$