Diffusion and instabilities in large-N holographic Fermi liquids: the vector fluctuations of the electron star

TL;DR

We investigate the transverse (vector) hydrodynamic response of the $AdS_4$ electron star as a large-$N$ holographic Fermi liquid by computing two-point functions and the transverse conductivity across temperatures from $T=0$ to $T_c$. The analysis combines full backreacted backgrounds (electron star and electron cloud) with linearized vector fluctuations, revealing hydrodynamic diffusion at both low $T$ and near the transition to the RN phase, but lacking hydrodynamics at intermediate $T$ where we also uncover an IR instability manifested as a pole with $\mathrm{Im}\,\omega>0$. A semianalytical low-energy expansion in a tortoise coordinate yields analytic Green's functions $G_{\pm}$ and a dispersion $\omega = i D_1 |k| - i D_2 k^2 + \cdots$, with a critical momentum $k_c = 3|D_1|$ signaling an upper-half-plane pole for small $k$. The results indicate that the finite-$T$ electron cloud is not the true ground state of finite-density holographic matter, motivating exploration of alternative holographic descriptions of non-Landau Fermi liquids and quantum critical phases.

Abstract

We study the hydrodynamic response of the AdS electron star in the vector sector, and compute the correlation functions and the transverse conductivity of the dual field theory. The system exhibits hydrodynamic behavior at low temperatures and near the critical temperature where the electron star undergoes the phase transition to the RN black hole. However, at intermediate temperatures the hydrodynamics does not exist. Remarkably, the system has an instability, i.e. a pole on the positive imaginary frequency axis at finite temperature. This instability is found both from analytical arguments and from numerics. Its physical meaning is so far unclear but it might mean that the ideal fluid limit for the star is a false vacuum.

Figures (13)

• Figure 1: The background functions $f$, $g$ and $h$ (blue, red, green) for $z=4$, for $T=0$ (left) and $T/T_c=0.65$ (right).
• Figure 2: The dimensionless star density $\hat{\sigma}$ for $z=4$, for $T=0$ (left) and $T/T_c=0.65$ (right).
• Figure 3: The real part (left) and the negative imaginary part (right) of $G_{ytyt}$ for $z=4$, for $T/T_c=0$ and a range of momenta $k/\hat{\mu}\in[0.01,0.14]$ (from blue to red; top) and for $T/T_c=0.23$ and a range of momenta $k/\hat{\mu}\in[0.001,0.091]$ (from blue to red; bottom). For these temperatures we see the typical diffusion peak in $\mathrm{Im}\,G_{ytyt}$.
• Figure 4: The real part (left) and the negative imaginary part (right) of $G_{ytyt}$ for $z=4$, for $T/T_c=0.35$ and a range of momenta $k/\hat{\mu}\in[0.03,0.6]$ (from blue to red; top) and for $T/T_c=0.65$ and a range of momenta $k/\hat{\mu}\in[0.02,0.78]$ (from blue to red; bottom). In the top panels the typical form of a diffusive correlator is visibly deformed. In the bottom panels the diffusion pole vanishes completely.
• Figure 5: The real part (left) and the negative imaginary part (right) of $G_{ytyt}$ for $z=4$ and $T/T_c=0.95$, for a range of momenta $k/\hat{\mu}\in[0.05,0.95]$ (from blue to red; top) and for the RN case at $T/T_c=0.95$, for a range of momenta $k/\hat{\mu}\in[0.05,1.4]$ (from blue to red; bottom). At high temperature the hydrodynamic diffusion is restored.