Emergent quantum criticality, Fermi surfaces, and AdS2

TL;DR

The paper demonstrates that charged AdS black holes at finite density possess an emergent IR CFT$_1$ arising from the AdS$_2$ near-horizon region, which governs the low-energy scaling of boundary two-point functions. By matching the IR AdS$_2$ dynamics to the UV geometry, the authors derive a master formula for retarded Green's functions whose low-frequency behavior is controlled by IR conformal data, notably the dimension $\delta_k=\tfrac{1}{2}+\nu_k$ for each momentum and, for fermions, yields a spectrum of non-Fermi-liquid fixed points with holographic Fermi surfaces. The work uncovers three types of emergent quantum critical behavior: (i) power-law scaling of spectral functions, (ii) log-periodic behavior from complex IR dimensions, and (iii) Fermi-surface physics with quasi-particle poles whose stability and dispersion depend on IR data, including a marginal Fermi liquid regime. It also reveals a rich landscape of non-Fermi-liquid states, including bosonic instabilities and the effects of double-trace deformations, and highlights a connection between IR CFT data and phenomenological descriptions such as Marginal Fermi Liquid behavior observed in cuprates. These results provide a robust, nonperturbative holographic framework for understanding metallic quantum criticality at finite density.

Abstract

Gravity solutions dual to d-dimensional field theories at finite charge density have a near-horizon region which is AdS_2 x R^{d-1}. The scale invariance of the AdS_2 region implies that at low energies the dual field theory exhibits emergent quantum critical behavior controlled by a (0+1)-dimensional CFT. This interpretation sheds light on recently-discovered holographic descriptions of Fermi surfaces, allowing an analytic understanding of their low-energy excitations. For example, the scaling behavior near the Fermi surfaces is determined by conformal dimensions in the emergent IR CFT. In particular, when the operator is marginal in the IR CFT, the corresponding spectral function is precisely of the "Marginal Fermi Liquid" form, postulated to describe the optimally doped cuprates.

Figures (14)

• Figure 1: The motion of poles of the Green functions \ref{['osciP1']} of spinors (left) and scalars (right) in the complex frequency plane. For illustration purposes we have chosen parameters and rescaled $|\omega|$( $\rightarrow |\omega|^\#$ with small $\#$) to give a better global picture. The poles are exponentially spaced along a straight line (dotted line) with angle ${\theta}_c$ given by \ref{['pepO']}. There are infinitely many poles, only a few of which are shown. Left plot: The black dashed line crossing the origin corresponds to the value of ${\theta}_c$ at $k=k_o$ (see \ref{['oenr']}): the boundary of the oscillatory region. As $k \to k_o$, most of the poles approach the branch point $\omega=0$ except for a finite number of them which become quasi-particle poles for the Fermi surfaces at larger values of $k$. The color dashed lines in the right half indicate the motion of poles on another sheet of the complex frequency plane at smaller values of $k$ (see the end of sec. \ref{['sec:3C']} for the choice of branch cut in the $\omega$-plane). Right plot: the two dashed lines correspond to $k=0$ (upper one) and $k=k_o$ (lower one). Again most of the poles approach the branch point $\omega=0$ as $k \to k_o$. These plots are only to be trusted near $\omega=0$.
• Figure 2: A geometric illustration that poles of the spinor Green function never appear in the upper-half $\omega$-plane of the physical sheet, for two choices of $\nu_{k_F} < {{\frac{1}{2}}}$. Depicted here is the $\omega^{2\nu_{k_F}}$ covering space on which the Green function \ref{['spinorG']}, with the $\omega/v_F$ term neglected, is single-valued. The shaded region is the image of the upper-half $\omega$-plane of the physical sheet. The pole lies on the line $2 \nu_{k_F} {\theta}_c = - {{\gamma}}_{k_F}$ for $k_\perp > 0$ and on $2 \nu_{k_F} {\theta}_c = \pi - {{\gamma}}_{k_F}$ for $k_\perp <0$, which are indicated by the purple solid line in the figure. The triangle formed by dashed arrows and solid lines in the upper left quadrant gives the geometric illustration for the equation $\pi - {{\gamma}}_k ={\rm arg} ( e^{2 \pi i \nu_k} - e^{-2 \pi q e_d})$ (following from the first equation of \ref{['phaseE']}), which makes it manifest that for $k_\perp < 0$ the pole lies outside the shaded region. In contrast for a scalar one needs to reverse the direction of the horizontal dashed ! line and the pole lies inside the shaded region. Similarly, the triangle in the lower right quadrant gives the illustration for $- {{\gamma}}_k ={\rm arg} (- e^{2 \pi i \nu_k} + e^{-2 \pi q e_d})$ which is relevant for $k_\perp > 0$. We also indicated the angles ${\alpha}_\pm$ which will be introduced and discussed in detail around \ref{['aloenp1']}.
• Figure 3: Examples of the motion of the pole for a spinor as $k_\perp$ is varied (arrows indicating the directions of increasing $k_\perp$). Left plot:$\nu_{k_F}<{\frac{1}{2}}$, for which the pole moves in a straight line. The plot shows an example where the pole moves to another sheet of the Riemann plane for $k_\perp > 0$ ( i.e.${\alpha}_+ > {\pi \over 2}$); ${\alpha}_\pm$ indicated there are introduced in \ref{['aloenp1']}. Right plot:$\nu_{k_F} > {\frac{1}{2}}$ for which the dispersion (real part of the pole) is linear.
• Figure 4: Spinor Green's function $G_2$ (as defined in Appendix \ref{['app:fer']}) at $k= 0.918$ as a function of $\omega$, computed numerically. We have chosen parameters $r_*=R=g_F=q=1$, $m=0$ and $d=3$ for which the Fermi momentum is $k_F = 0.91853$. The real and imaginary parts are shown in blue and orange dotted curves, respectively. Also shown is \ref{['spinorG']} (solid lines) with $h_1, h_2$ computed numerically using the method of Appendix \ref{['app:vF']} and $\nu_k$ given by \ref{['defNu']}.
• Figure 5: The values of $k_F$ as a function of $q$ for the Green function $G_{2}$ are shown by solid lines for $m = -0.4, \, 0, \, 0.4$. In this plot and ones below we use units where $R=1$, $r_* =1$, $g_F = 1$ and $d=3$. The oscillatory region, where $\nu_k = {1 \over \sqrt{6}} \sqrt{k^2 + m^2 -{q^2 \over 2}}$ is imaginary, is shaded. From \ref{['ne3']} in Appendix \ref{['app:a3']}, $G_1 (k) = G_2 (-k)$, so $k_F$ for $G_1$ can be read from these plots by reflection through the vertical $k=0$ axis. The $m=-0.4$ plot corresponds to alternative quantization for $m =0.4$ following from equation \ref{['altenQ']}. For convenience we have included in each plot the values of $k_F$ for the alternative quantization using the dotted lines. Thus the first ($m=-0.4$) and the third plot ($m=-0.4$) in fact contain identical information; they are related by taking $k \to -k$ and exchanging dotted and solid lines. Also as discussed after \ref{['eep']}, for $m=0$ the alternative quantization is equivalent to the original one. This is reflected in the middle plot in the fact that the dotted lines and solid lines are completely symmetric. All plots are symmetric with respect to $q , k \to -q , -k$ as a result of equation \ref{['jke']}.